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1 . . . ._ V V V V .. .
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”flu-s: 9

 

This is to certify that the
thesis entitled
A Development of the Character Tables for
Certain Classes of

Unitary and Linear Groups

presented by

William Albert S impson

has been accepted towards fulfillment
of the requirements for

Ph 0 D 0 degree in Mathematics

 

 

Q, mea 34W

Major pmfessor

Date (LU/5‘”); 3),) fill

0-7639

/ 1/1.

ABSTRACT
A DEVELOPMENT OF THE CHARACTER TABLES FOR

CERTAIN CLASSES OF
UNITARY AND llNEAR GROUPS

. By

William A. Simpson

The classical groups U(n,q2) and GL(n,q) occur as in-
finite families of groups indexed by a dimension n and a prime
power q. It is convenient to develop what might be called
'abstract' character tables whose entities are written as func-
tions of n and q and which describe the characters for the
entire family of such groups. It is too difficult to work.with
both n and q arbitrary, so n is fixed and the character
table which holds for all q is found. In 1955 a method for con-
structing the character table of GL(n,q) for a given n and
arbitrary q was developed by Green [2]. Since then the only
'abstract' character tables constructed have been those for
U(2,q2), U(3,q2) by Ennola [l] and Sp(4,q), q odd, by
Srinivasan [3].

In this paper ten abstract character tables are developed,
representing the group families SL(n,q), PSL(n,q), SU(n,q2),
PSU(n,q2) n 8 2,3 and PSL(4,q) d = 1. Of these, seven have
never been published. The tables for these groups are of
particular value because they often appear as important subgroups

in other larger groups such as many of the Sporadic simple groups,cu'

William A. Simpson

are themselves simple, or are used to construct other simple groups.

This paper has several purposes:
1. The character tables are made available in the sense that the
range of all parameters are given explicitly so that the user can
easily generate the desired table for any Specific q without
searching through the paper for the definition of the various
entries in the table. This degree of explicitness is not present
for the three tables now in print and for this reason they have
been included in this paper.
2. A standardized notation is used for all the tables which
should facilitate comparisons and other inter-connecting uses.
3. The procedure discussed in section III, together with the main
theorem developed in section II should enable one to more easily
work out a Specific character table for any of the groups SL, PSL,
SU, PSU not covered by this paper.
4. A very interesting and potentially important conjecture made
by Ennola [1} relating the generalized character tables of GL(n,q)
and U(n,q2) is extended, in section IV, to the special and pro-
jective Special groups. It is demonstrated that a change of q a -q
in the character table for SL(n,q) or PSL(n,q) will yield the
table for SU(n,q2), PSU(n,q2) reSpectively for the case n = 2,3.
The converse, of course, is also valid for these cases.

The main theorems of the paper, proved in Chapter II, are
the following:

IE, If HR is a central product of groups H, K where K
is abelian, then every irreducible character of HR remains

irreducible when restricted to H. From this theorem, we can

William A. Simpson

obtain the following Theorem for SL(n,q) which is a striking
improvement on Clifford's Theorem.

Th_ Any irreducible character x of GL(n,q) when re-
stricted to SL(n,q) is either irreducible or Splits into t
conjugate, irreducible characters of SL(n,q) of multiplicity l
where t‘d and d = (n,q-l). X has 133%1 associates in
GL(n,q) relative to SL(n,q) if X is reducible and q-l
associates if irreducible. (An identical theorem holds for
SU(n,q2) with (q-l) replaced by (q+l).) This is the main
working theorem of the paper as it tells us just how characters
of GL will split when restricted to SL. Clifford's theorem
only says that the number of components, t, will be a factor of
(q-l) which is of no help. But the above theorem says that
t = 2 or 3 for n = 2 or 3 respectively.

In section III the technique used to formulate the char-
acter tables is discussed. It is shown that each conjugacy class
of SL can be indexed by a Jordan canonical form.

The characters of GL(n,q) are restricted down to SL

and the inner product (x,x) = Téil- 2 x(g) x(g) is calculated.
gESL

If (x,x) = 1 the restricted character is irreducible. Otherwise,
it is reducible and splits up as indicated by the theorem just
discussed.

Once the character table for SL is determined we can
easily generate the table for PSL, since every character of SL
which is constant on Z(SL) is an irreducible character of
PSL 8 SIJZ(SL), and every irreducible character of PSL can be

so obtained.

William A. Simpson

In the sections V through XV, the character tables are
developed. Many of the routine calculations are done once in
detail and merely mentioned or entirely omitted in later sections.

In section XVI the results obtained for PSL(4,q) d = 2
are given and some Space is devoted to discussing the problems
which prevented further progress, problems which were primarily

attributable to the complexity of the table for GL(4,q).

REFERENCES

l. V. Ennola, Characters 2£_Finite Unitary Groups, Ann. Acad.
Scien. Fenn., Ser. A, 323 (1963), 120-155

2. J.A. Green, Characters g£_the Finite General Linear Groups,
Trans. Am. Math. Soc., 80 (1955), 402-477

3. B. Srinivasan, Characters pf the Symplectic Group Sp(4,q)
9 odd, Trans. Am. Math. Soc., 131 (1968), 489-525

' A DEVEmmENT OF THE CHARACTER TABLES FOR
CERTAIN CLASSES OF
UNITARY AND LINEAR GROUPS

BY ‘_ a.

{.\\..-‘ '-
William At Simpson

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Mathematics

1971

ACKNOWIEDGEMENTS

I feel greatly indebted to my major Professor J. Suther-
land Frame for his patience, encouragement, and generous assis-
tance during the period of research. His insight and suggestions
at critical points were inestimable in value. I thank the members
of my committee, Professors W.E. Deskins, D.W. Hall, R. Hamelink,
and N.L. Hills, for their time Spent in reviewing this paper.

I also wish to thank Professor Arunas Rudvalis for his
interest in this problem and valuable suggestions on several
occasions.

Acknowledgement must be made of the kind assistance given
by Professors Robert Steinberg and H. Brinkman through correSpon-

dence.

ii

TABLE OF CONTENTS

Page

LIST OF TABLES v

LIST OF CHARACTER TABLES V1

1. INTRODUCTION 1
1.1 History 1

1.2 Recent Interest 2

1.3 The Problem 3

1.4 General Theory 6

II. A MODIFICATION OF CLIFFORD'S THEOREM 13
2.1 Clifford's Theorem 2 13

2.2 A Theorem Applicable to SL(n,q), SU(n,q ) 15

III. TECHNIQUE 21

3.1 Determination of Conjugacy Classes for
the Linear Groups 21
3.2 Determination of Conjugacy Classes for

the Unitary Groups 24

3.3 Development of Character Tables 25

Iv. ENNOLA'S CONJECTURE 28
V. CHARACTER TABLE FOR SL(2,q), d = l 31
5.1 Character Table for GL(2,q) and U(2,q2) 31

5.2 Conjugacy Class Structure 33

5.3 Calculation of Characters 34

VI. CHARACTER TABLE FOR SL(2,q), d = 2 36
6.1 Conjugacy Class Structure 36

6.2 Calculation of Characters 38

VII. CHARACTER TABlE FOR PSL(2,q), d = 2 46
7.1 Conjugacy Class Structure 46

7.2 Calculation of Characters 49

iii

VIII.

IX.

XI.

XII.

XIII.

XIV.

CHARACTER TABLE FOR SU(3,q2), d = 1

8.1 Character Table for GL(3,q), U(3,q2)
8.2 Conjugacy Class Structure

8.3 Calculation of Characters

CHARACTER TABLE FOR SU(3,q2), d = 3

9.1 Conjugacy Class Structure
9.2 Calculation of Characters

CHARACTER TABLE FOR PSU(3,q2), d = 3
CHARACTER TABLE FOR SL(3,q), PSL(3,q), d = 1,3
11.1 Character Table for SL(3,q), d = 1

11.2 Character Table for SL(3,q), d = 3

11.3 Character Table for PSL(3,q), d = 3
CHARACTER TABLE FOR PSL(4,q), d = 1

12.1 Conjugacy Class Structure
12.2 Calculation of Characters

CONJUGACY CLASS STRUCTURE FOR U(4,q2), PSU(4,q2),
d I 1

13.1 U(4,q2)
13.2 PSU(4,q2), d = 1

CHARACTER DEGREES AND FREQUENCIES FOR PSL(4,q),
d B 2

Summary of Results
BIBLIOGRAPHY

GENERAL REFERENCES

iv

Page
51
51
51
57
6O

60
61

67
71
71
74
77
79
79
81
95
95
97
100
107
109

111

Table

10
11
12
13
14
15
16
l7
18
19
20
21

22
23

LIST OF TABLES

Character Table for GL(Z,Q) and U(2:qz)

Character Table for SL(2,q) a PSL(2,q) a PSU(2,q2) 3
SU(2,q2), d = 1, q even

Conjugacy Class Structure for

SL(Z.Q), d = 2

Induce-Restrict Table for GL(2,q) - SL(2,q), d = 2

Preliminary Character Table for

The Character Table for SL(2,q) a SU(2,q2), d

Conjugacy

Character

SL(2,q), d = 2

II
N

Class Structure of PSL(2,q), d = 2

Table for PSL(2,q) a PSU(2,q2), d

Character Table for GL(3,q)

Character

Character

Table

Table

Induce-Restrict

for U(3,q2)

2
for SU(3,q2) a: PSU(3,q ), d

II
M

II
H

Table for U(3,q2) - SU(3,q2), d = 3

Partial Character Table for SU(3,q2), d = 3

Character

Character

Character

Table
Table

Table

Induce-Restrict

Character
Character
Character
Conjugacy

Conjugacy

Character

Table
Table
Table
Class

Class

for SU(3,q2), d = 3

for PSU(3,q2), d = 3

for SL(3,q)sz PSL(3,q), d = 1

Table for GL(3,q) - SL(3,q), d = 3

for SL(3,q), d = 3
for PSL(3,q), d = 3
for PSL(4,q), d = 1

Structure of U(4,q2)

Structure of PSU(4,q2), d a 1

Degrees and Frequencies for

V

PSI-(49(1): d = 2

Page

32

35
37
4O
42
45
48
50
52,53
54,55
58,59

63

65,66
69,70
72,73
74

75,76
77,78
86-93
96,97

98,99
106

Character Table and/or Conjugacy Class Table

GL(2.q)

GL(3,q)

SL(2.9).
SL(2.q).
SL(3.q).
SL(3.q).
SL(4.q).
PSL(2.q).
PSL(2.q).
PSL(3.Q).
PSL(3.q).
PSL(4.q).
PSL(4.q).
U(2.qz)

u<3.q2)

U(4.qz)

SU(2.92).
SU(2.qz).
800.3).
SU(3.q ).
SU(4,q2),
Psuamz).
PSU(2.q2>.
PSU(3,q ).
PSU(3.q ).
PSU(‘hq ).

momma.

N

NNN

manque.

mammo-

ammo-o.

HWHNH

NHU’HNt—I

HWHNH

F‘ U3 P‘ h) P4

LIST OF CHARACTER TABLES

Page

32
52,53
35

45
72,73
75,76
86-93
35

50
72,73
77,78
86-93
107
32
54,55
96,97
35

45
58,59
65,66
98,99
35

50
58,59
69,70
98,99

I. INTRODUCTION

1.1 History

 

Towards the end of the 19th century G. Frobenius develOped
the foundations for the theory of group representation in a series
of papers dating from 1895 to 1911. In 1911 Burnside published
his book.Theory pf Groups g£_Finite Order in which he independently
rediscovered some results of Frobenius. More importantly he applied
this embryonic theory and obtained some surprising results in group
theory regarding the solvability of certain groups which gained
more serious attention for this new concept. After this initial
Success, however, the theory of group representation lay relatively
dormant for nearly 25 years. It appeared that the applications of
this theory as develOped up to that point had been exhausted.

This disinterest was not universal. Physicists quickly
grasped the importance of representation theory and soon began
using it to advantage. Since representation theory reduces the
abstract properties of groups to numbers, they found that this
enabled them to apply group theory to any system possessing
symmetry that was too complex to handle by classical analysis.

Such fields as thermodynamics, crystallography, wave equations,
quantum mechanics, molecular and nuclear structure are but a few
of the many areas where representation has played an important

role. Thus the theory of representations developed steadily in

the area of applications long after the pure mathematics of its
source ceased to offer much promise. Indeed, during the period

of 1905 to 1955 the growth of new developments in group theory
proceeded at a very slow pace. Felix Klein felt this was due to
the abstract nature of group theory which isolated it from physical
phenomenon which motivated SO much mathematics of that time and as
a reSult only a certain type of mathematician was attracted to the
field. He felt this homogeneity of the researchers was largely
responsible for its slow rate of development, unmarked by
imaginative leaps forward. If this was the reason for the decline
in activity in group theory then the successes won by representation
theory in the physical sciences may well have lessened the stigma
of abstractness and helped in restoring vitality to the field of

group theory.

1.2 Recent Interest

 

In the period 1925-55 a major advance was made by R. Brauer
in the study of modular representations. In 1955 when activity
in group theory suddenly increased, Brauer's work provided some
powerful tools with which to study group structure. Thompson and
Feit in their long paper on groups of odd order used modular theory
with telling effect and thus rekindled interest in representation
theory. M. Suzuki has used considerable character theory in his
work on the classification of simple groups by the structure of
involution normalizers.

At the present time significant advances in group theory

are being made in the area of classifying all simple groups and

here representation theory continues to play a large role since
nearly all theorems on groups using characters establish the exis-
tance of normal subgroups. The first step in this classification
problem was to find all simple groups of order less than some large
number. Initially all the simple groups which were found could

be placed in one of several classes of groups, all groups of a
class sharing some common properties. However, this convenient
pattern was broken when, starting in 1964, people began finding
some large simple groups which didn't fit into any of the known
classes. These groups became known as the 'Sporadic' simple groups
and they attracted considerable attention. At this date the char-
acter tables for nearly all these Sporadic groups are known, to-

gether with some of their larger Subgroups.

1.3 The Problem

There are several reasons for wanting to know the char-
acter table for a group. The most obvious reason is to be able
to study better the structure of the group. Sometimes one has
the character table of a group and desires to know some of its
larger subgroups (see p. 66 [2]). Methods are available to find
the character table of such subgroups (see p. 150 [16]). One
may then be able to recognize this subgroup's character table
as being the same as some known table. Thus a good stockpile
of character tables would aid in identifying subgroups. The
classical groups turn up often as subgroups and so their tables

are particularly valuable.

Wales'paper [22] furnishes an excellent example of the
current manner in which characters of groups, in this case
PSL(2,17), can be used to obtain surprising results.

Another use for character tables lies in using them to
test out various conjectures which might arise. For this purpose
one would like to have tables for a wide variety of groups.

Sometimes groups are Constructed by extensions of several
groups. If the tables of these subgroups are known, it is often
possible to obtain the character table of the constructed group.
Very frequently the groups used in this construction are of the
type discussed in this paper.

This paper deals with the character tables for PSL(n,q),
PSU(n,q2), SL(n,q) and SU(n,qZ) for n = 2,3 and some results
for n = 4. The case of n = 2 was originally done by Frobenius
[12], Schur [18] and later independently by Jordan [15]. This case
is redone here for several reasons. The methods used by these
earlier researchers are considerably different and more complex
than those used in this paper. Also the resulting tables are
written in such a manner that in order to use them, one would have
to go back through the paper to find the definitions and range of
values for the many parameters. In addition, the difference in
notation from one table to another would make comparisons very
difficult. The table for PSL(3,q) was done by Brinkmann [3]
but the results were never published. Thus it was considered
advisable to work out all the tables for n = 2,3 whether they

had been done before or not.

The final result is a set of 10 character tables, only 3
of which have been previously published, each of which describes
an infinite class of finite groups and all unified by common nota-
tion with the range of parameters explicitly given to facilitate
using the tables. The procedure used to obtain these tables is
quite likely the most straight forward one possible and yet little
can be done on the cases for n = 4, not because the method breaks
down but because the character tables become so large and complex
that the details are exhausting.

The character tables develOped here are of a somewhat
unusual nature and a word or two should be said about them. Some-
times it is possible to find a character table which can serve as
the table for a whole class of groups by changing some parameters.
Apart from the economy of Such an 'abstract' character table it is
valuable in that it provides a strong linkage among the groups.
There have been few such tables developed and so it seems that a
few more additional tables would be of value.

One of the most valuable classes of groups for which char-
acter tables are available is the class of symmetric groups S
(also An). Although no abstract character table has been developed
for the entire class an equivalent formulation has been obtained,
namely, a comparatively simple construction method for building the
character table for any of the Specified groups.

In 1896 Frobenius determined the generalized character
table for the groups GL(2,q). In 1949 Steinberg [19] worked out
the tables of GL(n,q) for n = 2,3,4 in a particularly straight

forward fashion. These results and some later work done by

Steinberg led to the develOpment by Green [14] in 1955 of a method
to calculate the characters of GL(n,q) for any n. In 1963
Ennola [6] found the characters of U(n,q2) for n = 2,3 and
recently Srinivasan [21] obtained the table for Sp(4,q) q odd.

This sums up all the results on 'abstract' tables as of this date.

1.4 General Theory

 

In the following short section an attempt is made to run
quickly over the key definitions and theorems of character theory
which are employed in this paper or necessary for its understanding.

DEFINITION. A representation of a group G is a homomorphism

 

p: G a GL(n,() where GL(n,[J is the group of all n X n,
nonsingular matrices over the complex numbers; the group
operation is ordinary matrix multiplication.

Every group has at least one representation, the most trivial one

being p1: G d l. The totality of all representations of a given

group can be narrowed down by defining the following equivalence

relation:

DEFINITION. If p and O are n-dimensional representations of
group G and there exists a nonsingular matrix A s.t. C(g) =
A p(g)A-1 for all g E G then we say 9 and o are
equivalent.

The next step is to show that certain representations of G are

the 'building blocks' from which all the non-equivalent representa-

tions can be constructed.

DEFINITION. A representation 9 is reducible if it is equivalent

* *
to a representation with matrices of the form (-6-+-;70

01(8) Lug)
0 I 02(8)

 

1.e. p: g a

It follows that 01(g) and 02(g) are also representations
of G. If a representation is not reducible then it is said to be

irreducible.

 

THEOREM 1.1 If the representation 9 (over(£) is equivalent to

a reducible representation a

. 01(8) 1 NS)
1.e. o: g .. 0 \02(8)

 

then 9 is equivalent to

the direct sum of the representation 01 and 02.
From this theorem it follows that:
THEOREM 1.2 If p is a representation for G then 9 is equi-
valent to the direct sum of irreducible representations
of G, for p over 0;.
This key theorem says that once we know all the irreducible rep-
resentations of a group G, we have all the possible representa-
tions since they can be constructed by adding together irreducible
representations.
The next surprising theorem tells us that the number of irreducible
representations of a finite group is strictly limited.
THEOREM 1.3 The number of non-equivalent irreducible representa-
tions of G equals the number of conjugacy classes of C.
It would be very cumbersome, and as it turns out, unnecessary to
write down all the irreducible representations of a given group.
To avoid doing this we look for some way to designate each of

these representations. This leads to the next definition.

DEFINITION. If p is a representation of G, then the character
x of p is the mapping from G to a: defined by:
x(8) = trace 9(8)-
If g1,g2 E G are elements of the same conjugacy class and p
is a representation of G, then p(g1) is similar to p(g2) and
since similar matrices have the same trace we can state:
THEOREM 1.4 Any character x of G is constant on conjugacy
classes of G.
Also:
THEOREM 1.5 Characters of equivalent representations are identical.
THEOREM 1.6 If the representation p of G is reducible and
equal to the Sum p1 + 92 +...+ pk of irreducible represen-
ations, then X = x + X +...+-x . In particular,
P 91 92 Pk
every character of G is the sum of irreducible characters
of G.
The above theorems mean that we can identify a representation 9

by specifying its character X which is an r-row column vector

X(81)

X(82) each entry being the trace of a representative matrix

xigr)

9(81) from each conjugacy class of G. We call x(l) the degree
of the character. Theorem 6 says we need only Specify the
irreducible characters of the group since all the others are simply
integral linear combinations of these. Thus we can write down a

square matrix for a group G called the character table for G.

 

The columns of this table are the n irreducible characters of G.

Customarily the identity character 1 is written as the first

1
column (or row) and the first conjugacy Class, i.e. the first row

(or column) is the identity of the group. The ijth entry of the

table is the value XJ(81X gi being an element of the ith con-

3 being the jth irreducible character of G.

jugacy class and x

It should be noted in passing that G 25H implies G
and H have the same character table but the converse is go_t true.
The quaternion group and the octic group are the smallest counter
examples.

An amazing number of relationships exist between the rows
and columns of a character table. In fact, these relationships
are so restrictive that in some cases it is possible to construct
the character table for a given simple group knowing no more than
the order of the group. There is no universal procedure to develop
a character table. After one has armed himself with a thorough
knowledge of the various character relationships and some Special
number theory concepts, he can then proceed only on intuition and
experience. No computer program has yet been devised to calculate
character tables for all groups of order s 200 and yet tables for
groups of this size can be done reasonably quickly by hand.
Frame's paper [5)] displays in detail many of the techniques used
to generate a character table of a group from that of a subgroup.
Although individuals have found character tables of groups with
orders as large as ‘G‘ = 47,377,612,800 the most notable success
measured in terms of magnitude has been the generation of the

table for Conway's group of order 221-39-54-7-11-13-23 by computer.

10

The method used some of the unique properties of the group and
in no way represented a general procedure which would work for
other groups.

Before the more useful properties of the character table are
stated, the following definition is necessary.

DEFINITION. Let x1,x2 be characters of group G. Let

( ) dgf 2 x1(gi)x2(si)
X1’x2 . ]N(gifT

1

 

where gi is an element from the ith conjugacy class and
\N(gi)‘ is the order of the centralizer N(gi) of gi.

In the following let Xi’xj be irreducible characters of G and

9 be a reducible character.
THEOREM 1.7 (Orthogonality relations on columns)
(1). (Xi’xj) = 6ij iff xi,xj are irreducible char-
acters.
(ii). (xi,9) = a1 where 31 is the number of times the
irreducible character xi is contained in 9.
(iii). (9,9) = 2 a? = sum of squares of the multiplicities
of all the irreddcible characters appearing as components
of 9.
The above relations enable one to determine if a character is
irreducible or not and in the case of a reducible character, to
break it up into its irreducible components. Note the following:
(i) implies that if (Y ,Y ) = 1 then Yi is irreducible. If
(9,9) I 2 or 3 then (iii) implies that 9 is composed of 2 or

3 irreducible characters respectively (this case comes up often

in this paper).

11

THEOREM 1.8 (Row Orthogonality)

f’ . .
0 1f g,h are not In the same
conjugacy class

 

M7?

. x‘(g>xi(h> = <

1 l

‘N(g)‘, if g,h are in the same
Lconjugacy class

 

Note: A special case of the above is:
c o 2
E x1(1)x1(l) = 2 (degree of ith character) = ‘G‘ .
i i
The next major topic is the question of how characters of the
group are related to characters of its subgroups.

If x is an irreducible character of G we can easily

 

get a character x‘H of H s G by restricting x to H.
x‘H(g) = x(g) for g being any element in a conjugacy class of
H. X‘H is not necessarily irreducible and often it isn't.

If Y is an irreducible character of H s G we can obtain a

character YG of G by a process called inducing, which is
G H
Y1 x; G th
formulated by: = E where C, is the j con-
\NG\ H G NH 3
j Cf:cj x

jugacy class of G and C: are all conjugacy classes

of H which are contained in the conjugacy class C3.
Again we note that the induced character YG need not be
irreducible in G. The induced and restricted characters of
H and G are bound together by the following theorem:

THEOREM 1.9 (Frobenius' Reciprocity Theorem)

_ G
(lemH — (xd’ )G .

This theorem says that the number of times the restricted

12

character “It contains y as a component, equals the number
of times that the induced character YG contains X as a com-
ponent.

Thus if H s G, and one has available the character table
for C(H), it may be possible to find the table for H(G) by
restricting (inducing) all characters of C(H) to H(G) and
using the orthogonality properties and Frobenius' Theorem to
determine how to Split up all the reducible characters into their
irreducible components.

The preceding has been a summary of character theory which
is applicable to all groups. In the following section some

theorems are discussed which apply only to Special groups.

II. A MODIFICATION OF CLIFFORD'S THEOREM

In this section we introduce some results known collec-
tively as Clifford's Theorem, concerning characters of normal sub-
groups. The particular structure of SL(n,q) is analyzed and a
theorem is obtained which enables us to extend Clifford's Theorem
and achieve even more information concerning the characters of

GL(n,q) restricted to SL(n,q).

2.1 Clifford's Theorem

 

Nearly every textbook which contains a section on Clifford's
Theorem, including most of Clifford's paper [4], states the results
in terms of irreducible G-modules and G-submodules. In this section
we restate these theorems in terms of characters. Such a change
in vieWpoint can be found in Feit [8] and Lomont [17].

The following are various definitions, notations and con-
cepts used in the discussion of Clifford's Theorem. Let G be a
group and suppose p is an irreducible representation of G and
x its character.

DEFINITION. If C E Aut(G) then the representation p0 defined
by: po(g) dgf p(g°)‘v g E G is said to be conjugate to

p.

Likewise the character, x°(g) dgf x(g°) V g E G is

conjugate to the character x of G.

13

14

It can be shown that if x is an irreducible character, then x0

is also. If H'd G and X is an irreducible character of G then

8

xjfl, x are said to be conjugate relative to G. Conjugate char-

in
acters should not be confused with characters x, X' which are

complex conjugate characters defined by x'(g) = x(g) .
DEFINITION. Let H < C. Two irreducible representations of G,

91 and 92 are associates if they have an irreducible

 

component in common when restricted to H.
The following theorems are all modifications of, or an outgrowth
of Clifford's work [4].
THEOREM 2.1 Clifford's Theorem
If H'd G and p is any irreducible representation of G,
then the restriction of p to H denoted by pH is
either irreducible or reducible into irreducible components
of the same degree. If pg is any such irreducible com-
ponent of pH, then all other components are conjugates
of pfi relative to G and in addition, every such con-
jugate of 9H appears as a component of 9“.
This theorem says that if a character XH Splits in H then it
Splits into characters of equal degree and which are identical
on all classes of H which are complete classes of G. The sets
of values of the characters over the other classes of H are
identical but the values are permuted over these classes. Now we
will take up a theorem which says something about the number of
components and their multiplicities.
Suppose XH is reducible. Then associated with this char-
acter is a subgroup G* s.t. H < G* < C. If G/H is cyclic then

the following theorem provides some interesting information.

15

THEOREM 2.2 Clifford's Theorem

Let H be a normal subgroup of G Such that G/H is
cyclic of order k. Then the irreducible components of XH each
have multiplicity 1. The number of components of X is m, and
the number of distinct associates of X relative to H is k/m,
where m = [G:G*] and H S.G* S G.

Since the scalar matrices of GL form a cyclic group,
GL/SL, we know that if a character of CL Splits in SL it will
split into m pieces of multiplicity l, where m divides
[GL:SL] = (q-l). Also, since the number of associates is (q-1)/m,
this means that if XSL is irreducible (i.e. m = 1) then there
are (q-l) characters of GL that restrict down to the same char-
acter of SL. If XSL is reducible into m components
Y1,...,Ym then there are (q-1)/m characters of GL which, when

restricted to SL, also split into the same components.

2.2 A Theorem Applicable to SL(n,q), SU(n,q)

Let X be an irreducible character of GL(n,q). By
clifford's Theorem 2.2, if X‘SL is reducible then it splits into
m conjugate irreducible components of multiplicity 1 where m
divides [GL:SL] = (q-l). In this section we prove that m
divides d I (n,q-1), and this determines m exactly for all the
groups considered in this paper.

DEFINITION. Let H,K,M be groups with M S Z(H) and suppose
there exists an isomorphism 9 of M into Z(K). Then

if we identify M with its image 9(M), there exists a

group G of the form C = HK with M = H 0 K S Z(G)

16

such that H centralizes K. Such a group G is said to
be a centralgproduct of H and K w.r.t M. In diagram
form: T T

Z(H) Z(K)

M a 901)

Lemma 2.3 If H X K is the direct product of a group H with a
group K and Y,9 are irreducible characters of H,K
reSpectively then the character X defined by X(hk) =
Y(h)-9(k) is an irreducible character of H X K. Con-
versely, every irreducible character of H X K is equi-
valent to such a product of characters of H and K.
pf: see Littlewood [16]

Lemma 2.4 If H X K is a direct product of a group H with
an abelian group K and X is an irreducible character
of H X K, then X‘H is an irreducible character of H.
Pf: By lemma 2.3 X(g) = Y(h)-9(k) where g = hk, h E H,
k 6 K and Y,9 are irreducible characters of H,K
respectively. K is abelian so all its characters are of
degree 1. x(h) = X(h-1)= ‘l’(h)~9(1) = ‘i’(h)-1 = ‘i’(h)
for all h E H. Thus X‘H = Y which is an irreducible
character of H.

Lemma 2.5 If HR is a central product, then there exists a
homomorphism f:H X K —+ HK
pf: Gorenstein [13]

Lemma 2.6 Let f be a homomorphism f:G ~ G'. If X is an
irreducible character of G/Ker f then X' defined by

x'(g) = X(f(g)) is an irreducible character of G.

17

Pf: Consider (X',X'):
1
(x'.x') = Z x'(g)
|Gl 86G

‘Ker f‘
= -—TET——' 2 X(8)X(8)
gEG/Ker f

x'<g) = 17% 2 X(f(g))x(f(g))

gEG

1 __
= T__T z: X(g)x(g) = (x.x) = 1
G/Ker f gEG/Ker f

since X is an irreducible character of G/Ker f

.2 (X'X') = 1 implies X' is an irreducible character
of G.

Note: Lemma 2.4 and 2.6 seem to be familiar results but
don't appear anywhere in standard references.

THEOREM 2.7 If X is an irreducible character of a central pro-
duct HK, where K is abelian, then X H is an
irreducible character of H.

Pf: By Lemma 2.5 there is a homomorphism f:H X K a HK
such that HK a: H x K/Ker f .

Let X' be defined by: X'(g) = X(f(g)).

Bnyemma 2.6 X' is an irreducible character of H X K.
Since H is a subgroup of both H X K and HK and
x'(g) = X(f(g)) then x'(h) = X(h) for all h E H.
This implies X"H = X\H.
But X"H is irreducible by Lemma 2.4 so X‘H is an
irreducible character of H.

This theorem says that every irreducible character of a central

product HK with K abelian is irreducible upon restriction to

H and so no Splitting of characters can take place.

We now study the structure of SL(n,q) and use Theorem 2.7 to

18

obtain our major goal.

dk g-l
DEFINITION. Let M(d) = {A e GL(n,q)\det A = 9 k = 1,..., d }

 

where d = (n,q-1) and g is a primitive element of

GF(Q)-
We note that every scalar matrix of GL(n,q) is contained in
M(d) since det pkl = pkn = 9(kn/d)d.
Lemma 2.8 M(d) 4 GL(n,q).
Pf: Let A 6 GL, B E M(d)
det ABA-1 = det A det B det A“1 = det B = pdk
thus ABA“1 E M(d) a M(d) < GL
Lemma 2.9 GL/M(d)ia o(d), a cyclic group of order d.

Pf: Let f:GL(n,q) 03t° 0(a)

where f(A) = (det A)(q-1)/d
Clearly f is a homomorphism and f(GL) 3 c(d). It is
easy to show that the Ker f = M(d) so by the ISt
Isomorphism Theorem GL/Ker fez f(GL) i.e. GlfM(d) a g(d).

Lemma 2.10 M(d) = SL-Z(M(d)) is a central product.

Pf: Z(M(d)) = {scalar matrices A 6 GL}

Izm<d)>1 = Maj-l) = (q-l)

 

s-_1
Z(SL) = 9‘ d )k ‘ ‘
_ k=Luqd> zmm =d
(Sal)k
p J
Consider: Z(M(d)) SL
M a: Z (SL)

(1) M(d) = SL-Z(M(d))
Pf: Let A G SL, B E Z(M(d)), then det (AB) =

k
det Acdet B = l-pd = AB 6 M(d) = SL-Z(M(d)) s M(d).

19

 

 

16L]
_ ° (q-1>
But \SL-2(M(d))\ = LSL‘ngéfimni = (q 1) d = \M(d)‘

.n SL-Z(M(d)) = M(d).
(ii) SL 0 Z(M(d)) = Z(SL)
Pf: Let A E SL 0 Z(M(d))
then det A 8 l and A is a scalar matrix so A E Z(SL).
(iii) Z(M(d)) centralizes SL
Pf: Every matrix of Z(M(d)) is scalar and so commutes
with every element of SL.
Conditions (i), (ii), (iii) = M(d) is a central product
of SL and Z(M(d)).
We now put this all together to obtain the following results.
THEOREM 2.11 Every irreducible character X of GL(n,q) when
restricted to SL(n,q) is either irreducible or Splits
into t conjugate, irreducible characters of SL of
multiplicity 1 where t divides d. X has Xiill
associates in GL w.r.t. SL.

Pf: GL SL x Z(M(d))

d
M(d) B SL-Z(M(d))

SL'_1

d\

SL
Suppose X is an irreducible character of GL(n,q). Now M(d)‘< GL

by Lemma 2.8 and GL/M(d) is cyclic by Lemma 2.9 so by Clifford's

 

 

Theorem 2.2 X‘M(d) is either irreducible in M(d) or it Splits
into t conjugate irreducible characters X1,...,Xt of multiplicity
1 where t divides [GL:M(d)] = d. Since by Lemma 2.10, M(d) is
a central product of SL and the abelian group Z(M(d)), then by

Theorem 2.7 every irreducible character of M(d) is also an

 

20

irreducible character of SL under restriction.

We see from this theorem that any splitting undergone by
the characters of GL takes place in restricting from GL to
M(d). This means that the characters Split into t conjugates
where t‘d. This is marked improvement over Clifford's Theorem
which requires only that t‘(q-1).

We thus have the immediate Corollary:
Corollary 2.12 The characters of GL(2,q), GL(3,q), GL(4,q),

d = 2 must either be irreducible when restricted to

SL or split into 2,3, or 2 irreducible conjugate char-

acters respectively. Every character X of GL(2,q),

GL(3,q), GL(4,q), d = 2 has 3&1 associates w.r.t. SL

where t = 1 if X‘SL is irreducible or t = 2,3, or 2

reSpectively if X‘SL is reducible.

Theorems 2.11 and 2.12 can be restated for the characters of

 

U(n,q2) restricted down to SU(n,qz). We need only define M(d)
by: M(d) = {A e U(n,q2)\det A = pd'uq‘l) k = 1,...,9§l} where

d = (n,q+l) and p is a primitive element of GF(q2).

III. TECHNIQUE

In this section we describe the procedures by which the

character tables are obtained by restriction from GL(n,q) and

U(n.q2)-

3.1 Determination of Conjugacy Classes for the Linear Groups

Dickson [5] did considerable work on determining the con-
jugacy classes of the general linear groups. The following is a
brief description of how the class structure is found.

The elements of GL(n,q) are matrices, with each of
which there is associated a characteristic equation. A familiar
theorem tells us that if matrix A is similar to matrix B then
the characteristic equation of A equals the characteristic equa-
tion of B. Thus we can find all possfllle characteristic equa-
tions and correSponding to them we can associate a conjugacy class
of GL having that equation. However, since the converse of
the above theorem is not true, then some of the characteristic
equations will correSpond to several different conjugacy classes
having that same equation. We can get a full separation by
writing down the possible Jordan canonical forms for each type of
characteristic equation in an appropriate extension field. Each
canonical form corresponds to exactly one conjugacy class and

every class has such a canonical form.

21

22
In [14] Green shows that if A E GL(n,q) and A has
k
1R2 ‘34
1 f2 fN where f1,...,fN are

distinct irreducible polynomials over GF(q) then for every

characteristic.polynomialf

possible partition v1,...,vN of the exponents k1""’kN
there correSponds a distinct canonical form and thus a distinct
conjugacy class.

This means that the number of conjugacy classes of GL(n,q)
can be determined by counting the number of possible partitions.
The final result is a generating function which for a given n
is a polynomial in q with constant rational coefficients expres-
sing the number of classes for GL(n,q).

For example, if we want to find the conjugacy class
structure of GL(3,q) we first write down the possible factoriza-

tions of the characteristic equation of degree 3 and then for

each factorization we write down the possible Jordan forms.

 

 

Types of characteristic polynomials Jordan Canonical Form
irreducible cubic (T q 2)
T Tq
uadratic factor linear factor k L
q ’ p a QB
O
k k
k
3 equal linear factors pk k (9 pk K)(e p k
9 pk 9 19
k
k
2 equal linear factors pk k p p L
9 L1 9
p
3 different linear factors pk p; m)
P
PGGFW)
2
UEGF(Q)

3
T E GF(q )

23

The number of conjugacy classes can now be easily counted by con-
sidering the valid range of the exponents on the primitive field
elements.

Grouping the conjugacy classes in sets according to the
possible Jordan Canonical forms is not only a convenient way to
count the classes, but also serves several other purposes. First
of all, the values which the exponents k,l,m,n assume can be
used to determine the character value for each class, so that
great economy of notation results; each whole set of conjugacy
classes requires only one entry in each character. Thus the char-
acter table of GL(3,q) is composed of only 8 rows, one for each
canonical form. Also, it turns out, as first noted by Steinberg
in [19], that there is a l-l correspondence between the sets of
characters of GL(n,q) having the same degree and the sets of
conjugacy classes of the same canonical form. The order of a
given set of conjugacy classes is equal to the order of the
corresponding set of characters. Thus for the group GL(3,q)
we see that all the characters are of only 8 different degrees and
it is particularly convenient to set up the character table so as
to exhibit this 1-1 correSpondence between classes and characters.
We remark that this l-l correspondence between sets of classes
and characters doesn't hold in SL and SU if d # 1.

Dickson determined the order of the centralizer of an
element for each conjugacy class type of GL(n,q) n = 2,3,4, by
counting the number of matrices commuting with a given canonical

form.

24

Finding the conjugacy class structure of SL(n,q) is
accomplished by determining what canonical forms have determinant
1. Since SL‘Q GL, then SL is composed of complete conjugacy
classes of GL. Thus every conjugacy class of SL has the same
order as it did in GL with the exception of the classes of GL
which Split in SL. These splitting classes were determined by

Dickson for n = 2,3.

3.2 Determination of Conjugacy Classes for the Unitary Groups

 

In [23] Wall obtains results on the conjugacy classes of
some classical groups, the unitary group, U(n,qz), included. His
main results for U(n,qz) are:

THEOREM. (1) X E GL(n,qZ) is similar to an element of

*-1

U(n,qz) iff X ~ X ('~' indicates Simi-

larity and '*' indicates the conjugate transpose)
.. 2 2
(11) Two elements of U(n,q ) are conjugate in U(n,q )
iff they are similar in GL(n,qz)

(iii) The number of conjugacy classes in U(n,qz) is

m A
the coefficient of tn in n -l4i-E—i .
x=11-qt

He also gives a formula by which the order of each conjugacy class
can be calculated.

The above statements say that all the elements of U(n,qz)
which are similar to some A 6 GL(n,qZ) such that A ~ (A*)-1,
form exactly one conjugacy class.

In [7] Ennola refines the formula for calculating the
order of the conjugacy classes. He also Shows that if ARE GL(n,qZ),

q +l _
e —

* -
and A ” (A ) 1 and e is an eigenvalue of A, then 1

25

where k is odd. Thus we can use Jordan Canonical forms with
marks of GF(qZR), fitting the above conditions, to represent the
conjugacy classes of U(n,qz). We use the same type Jordan forms
employed as class representatives for GL(n,q) except now we use
elements such as pk, ck, wk: where p = 0(q-1), T = TiqB-l) and
a, T1 are primitive elements of GF(qz), GF(q6) reSpectively.
As in GL(n,q) we assemble the conjugacy classes of
U(n,qz) into sets, each set corresponding to a Jordan Canonical
form. ‘We likewise place the irreducible characters into sets
according to their degrees. As before, there is a 1-1 corre-
spondence between the sets of classes and sets of characters in

U(n,qz).

3.3 Development of the Character Tables

The character tables for GL(n,q) n = 2,3 are given in
Steinberg [l9] and for U(n,qz) n = 2,3 in Ennola [6]. In the
following we shall describe the procedure for finding the char-
acter table for SL(n,q) and PSL(n,q), with the understanding
that the same procedure works for the unitary case; just replace
GL, SL, PSL by U, SU, PSU throughout the discussion.

The conjugacy classes of GL(n,q) are represented by
all the possible Jordan Canonical Forms. We first find the con-
jugacy class representatives for SL(n,q) by selecting only those
representatives of GL with determinant 1. Then the irreducible
characters Xi of GL are restricted down to SL one at a time
by using only the values Xi(g) where g is from a conjugacy

class of SL. The inner product (Xi‘SL’xi‘SL) is calculated.

26

If (xi‘SL’xi‘SL) = 1 then Xi‘SL is irreducible and we are done
with this character. Otherwise, we know it Splits into d irre-
ducible components Y1,...,Yd which are all equal on those classes
of SL which are complete classes of GL. The values of the Ti
on the other classes can be later determined, after all the char-
acters of GL have been restricted down to SL, by using the
orthogonality relations Theorems 1.7 and 1.8.and Gaussian Sums.
After the character table for SL is found, we can obtain
the table of PSL I SL/Z(SL) by using the following well known
theorems due to Frobenius.
THEOREM 3.1 If Hid G and X is an irreducible character of
G, then X is a character of G/H if and only if it has
equal values for any two elements of G which are eQUi-
valent modulo H.
THEOREM 3.2 If Hld G then every character of G/H is a char-
acter of C.
To determine the conjugacy classes of PSL we simply fuse into a
single class of PSL all the classes of SL which are equivalent
under multiplications by scalar matrices from Z(SL).
Picking out a set of classes of SL which are all equi-
valent to one class in PSL, we then select the characters of
SL which are constant over this set of classes. These are the
characters of PSL.
If one is working with a Specific group character table
then this procedure for finding the character table of a known sub-
group works rather well. The restricted characters can be tested

for reducibility and the use of the character orthogonality relations

27

and Frobenius' reciprocity theorems would enable one to Split
the reducible characters into irreducible components.

It works well because the job of taking inner products
(X,X) is not too arduous a numerical calculation. However, if
one is working with a 'generalized' character table Such as
GL(n,q) or U(n,qz) the entries are not numerical but are poly-
nomials in q and variable roots of unity, so that the task of
calculating inner products becomes extremely tedious. It would
be almost hopeless if many of the restricted characters were re-
ducible with many components. Theorems 2:11 and 2L1; are thus
very important because they show that the Splitting is of a very
Simple nature, and thereby make this process feasible. The pro-
cedure can be best understood by following through one of the
table derivations in detail as done in one of the first sections

of the paper.

IV. ENNOLA 's CONJECTURE

In [20] Steinberg noted that if the conjugacy classes of
GL(n,q) are partitioned into sets according to the type of Jordan
canonical form to which they were similar, and the irreducible
characters were also put into sets according to their degrees,
then there was a l-l correspondence between the conjugacy class
sets and the character sets. Also, the correSponding sets had
the same order. Ennola noticed that if this same partitioning is
done on U(n,qz) then the number of resulting sets is the same
as for GL(n,q).

In a general character table for GL(n,q) and U(n,qz)
we write down only one character for each such set of irreducible
characters and this character has only a single entry for each
set of conjugacy classes. Thus the character tables for U(n,qz)
and GL(n,q) contain the same number of 'generalized' entries.
Ennola conjectured for all n, and proved for n = 2,3, that if
q is everywhere replaced by -q in the character table for
GL(n,q) and the resulting character changed in sign if the degree
function is negative, then the character table U(n,qz) results.
He used Brauer's characterization theorem [1] to prove it for
these two cases but the complexities of GL(n,q) prevent one
from carrying forward his method. If this conjecture is true,

then Green's method for constructing characters for GL(n,q)

28

29

would serve for U(n,qz) also. One would only have to use -q

in place of q in all the class functions constructed and change
the sign if the degree becomes negative. Since GL(n,C) is
closely related to Un (GL is a topological product of U and
the Space of positive definite Hermitian matrices) we might expect
some connection between their character tables, but not such a
simple one.

Ennola's conjecture appears so trivial that one would
expect an equally simple proof. However, like many other theorems
involving character relations, such a simple proof is not forth-
coming. It is quite possible that the most direct proof would be
through the use of Lie theory. One interesting observation which
indicates some difficulty is that a permutation character of GL
is 22£_transformed into a permutation character of U. Thus some
reducible characters split up differently than their images under
the q a -q transformation.

In this paper we extend Ennola's conjecture to SL and
PSL and show that it holds for all cases under consideration.

A result which is similar to the above conjecture, but

which is proveable is the following:

Un n GL
THEOREM 4.1 Ci (q) = (-1) C)\ (-q)
U
where an(q) = the polynomial in q which expresses the
order of the Ath conjugacy class of
2
U(n,q )
GL . . .
CA (q) = the polynomial In q which expresses the

order of the xth conjugacy class of

GL(n.q)

30

and the xth conjugacy class of GL has the same type

canonical representative as the xth class of U(n,qz).
This says that q a -q will transform the polynomials expressing
the conjugacy class orders of GL into those for U(n,qz). Com-
paring Lemma 2.4 in [14] and Definition 2 in [6] gives the above
theorem. It appears that Ennola used this to obtain his tables

for U(2,q2) and U(3,q2) but he never explicitly mentions it.

We shall use this theorem in Chapter 15.

V. THE CHARACTER TABLE FOR SL(2,q) d = 1

In this section we develop the character for SL(2,q)
d = 1, q = 2K. This is the most trivial case because for d = l
GL(2,q) = SL(2,q) X Z(GL) and every character of GL is irre-
ducible upon restriction to SL. This one table serves for all

the linear and unitary Special and projective special groups

2
since SL(2,q):2 PSL(2,q) a PSU(2,q )2: SU(2,q2) for d = l.

5.1 Character Tables for GL(2,q) and U(2,q3)

All the character tables developed in the next three
chapters are obtained from the table for SL(2,q) which is found
in Steinberg [19]. Although the table for U(2,q), as found in
[6], is not used, we furnish it for completeness. It is con-
venient to combine these two character tables as in Table 1 on
the following page. This also demonstrates Ennola's conjecture

regarding these two character tables.

31

32

Table 1. Character Tables for GL(2,q) and U(zeqz)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

conjugacy canonical parameters number of centralizer
class representative classes order
(K) p" a
C. ( p") k-1,---. (q-S) q- «r q(q-5) (q+5)
m) (9" ‘)
2. 3) 1“ P k-1,"',(q-a’) q-J q(q-b’)
(K. (P 91) k,’-l,"-, (q- 5) .1. z.
3 k < 1 2(q-6)(q-1-6) (ca-6)
(K) H
c. <‘ a“) ;;;.(---3<q23.g ... 1.
mo + - - -
Cmec‘flgfl 2 q + )(q 6) (q+5)(q 5)
parameters t-l, , ((1-5) t,u=l, - -- ,(q-S) t-1,-'- ,qt-l (Adi-‘5“)
t<u tiO (mod q+J)
number of 1
characters 9" 9' ‘9‘ I 5(9'5) (9'1’5) Egg-1+5) (9‘5)
char. (t) (f) Ha“) (I)
clas 3 ’X' X ‘ X 9‘ vx ‘t-D
{to Eth O J E ((+U)K -55?“
ch") Etflcht) 5£f(K*’) 5(8tk+u3+ EMK+ t1) 0
C:K\ Eff“ ~5€5tK O _5’(4sz+’,7:th)

 

 

for GL(2,CI)=

+1

5

p a primitive element of GF(q)

o - primitive element of GF(qZ)
q+1
'= e

q-l

e ‘13“

q2-1 =

1.11

for U(2,q2)3

5=-1

-l
p = OH

+1
sq

=1,'nq-

o = primitive element of GF(qz)

21 q

-1
‘n =

=1, e

 

33

5.2 Conjugacy Class Structure

For d = l (q = 2K), SL(2,q)sa PSL(2,q), so we know that
no conjugacy classes of GL(2,q) will Split in SL(2,q). The
class structure is determined by observing the canonical repre-
sentatives of GL in Table 1. and selecting only those whose
determinant is 1. These classes are then counted. Since
SL‘Q GL every conjugacy class of SL is of the same order as
th

it was in GL. If \le 8 order of the centralizer for A

conjugacy class and ‘Ck‘ = order of xth conjugacy class, then

 

SL . SL _LcLl/(q-1) =_1__ CL =_;_ GL
‘Nx I +074 19,1 q_1 C). q_1 ‘Nx |. Thus we

divide the order of all the centralizers in GL by (q-l) to

get the centralizer orders for SL.

k¢>am QM

Only the classes C; require any calculation.

 

Cék’L) O 'Ik,L = l,...,q-l is equivalent to
0 pt<k#L
(Ht) ' (Cl-1)
(c‘kd) .. Cask)
(10
C3

(pk 0 ) k=1,...,%(q-2)

ok(q+1) = k E O(mod q-l) we get:

cfik) ok 0 k (q-I).2<q-1>,....<q+1)(q-I) or
[ kq) k a! 0(mod q+1)
C(k) . C(kq)

{k - (q'1)32(Q'1):° 0°:Q(Q'1)
C&)=&M)

34

Now let k - k'(q-l) where k' = l,...,q
k'(Cl'l)
0

SO 0 '
also ok'(q-1)(q+1) . 1
, ok'q<q-1>ok'<q-1) g 1
-. ok'q(q-1) _ O-k'(q-1)
so akq = Ok'q(q-1) a o-k'(q-1)
(k) ak(q-l) ) 'k = l,...,q
thus an equivalent form for C4 18: ( O'k(Q'1) C(k) = C(-k)

or k = l,...,%q. The resulting list of conjugacy classes appears

in Table 2.

5.3 Calculation of Characters

Each of the characters of GL are now restricted down to

SL.

(1) x‘t)|SL = I, = (e2(q'1)t.e2(q'1)t.et(k'k),ek(q'1’t> = (1.1.1.1)‘V t

(11) x“’ . (q,2<q-1>t O et(k-k),_etk(Q-1)

q ‘SL 8 Yq )= (qs0313'1)V t 0

There is no need to test for irreducibility Since SL is a
quotient group of GL for d - 1 and so by Frobenius' Theorem
3.1 we know that all the characters of GL restricted to SL

are irreducible.

(111) xéil‘SL a Y(t)

q+I )e(t+u)(q-l),e(t+u)(q-l),etk-uk +_€-tk+uk

= ((q+1 .0)

'3 (q+1:133k(t-U) + 6-k(t-u),0)

(t) kt -kt

let t = (t-u) then Yq+1 - (q+19133 + 6 ,0)

so t = 1,...,%(q-2)

-1 1 -1 1 k -1
(iv) Xéfl‘SL . wit: - ((q_1)nt<q )(q+ > -nt(q )(q+ ’.0.-(n (q )c

-k
e

+ nktq<q'1’)> = ((q-1>,-1.0.-<ekt +» t>> where

35

q+1

e - 1. Thus t runs over the values l,..

.,’5q.

The resulting character table appears below.

I
Table 2. Character Table for SL(2,q) a: PSL(2,q) a PSU(2,q2) e:

SU(2,q2), d = l, q even.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

conjugacy canonical parameters number of centralizer
class representative classes order
| c
C. (e O) 1 q(q+1)(q-l)
I e
C! (a I) 1 q
(m a
c, (g 3...) k-1.---.s(q-2) é(q-2) (1-1
C20 (can-u $4?» k-1,---,§q h q+1
parameters t.1,...’;(q_2)F t-l,"',§q
(t)-(-t)
number of
characters 1 1 %(q-2) 43
c ar W ‘ (17 (0
class W. ‘ W9.— W14
C: 1 q (1+1 q-l
Ca 1 0 l t -1
- K
C;” 1 1 E!!! j c 0
no (It ~13
4 1 '1 ° -6 - 9
q-l
p - primitive element of GF (q) s a 1
a - primitive element of GP (<12) eq+l - 1

 

v1. CHARACTER TABLE FOR SL(2,q) d = 2

In this section the character table for SL(2,q).a SU(2,q2)
d = 2 is develOped. In some reSpects this is a difficult table
to handle since there is a Sign fluctuation throughout the table

which depends on whether q = : 1(mod 4).

6.1 Conjugacy Class Structure

 

(1) Consider classes of the form (pk pk) . Since 21(Q-1)
then k = (q-l) and k = %(q-1) will be the only forms
with determinant 1. In SL the 1-1 correspondence be-
tween sets of classes of the same canonical form and sets

of characters with the same degree does not exist and so

we will separate CEq-l) and cik(q-l)) into classes
labeled Simply C', C" .
1 k
(ii) Consider the classes of the form (9 k) . Again the
1 p

only ones in SL are k = q-l, £(q-l). We know from

Schur [18] that this type of class will Split into

k k k
(p k) s (p k) . We Shall label the form (9 k)
1 p p 9 k 1 9

k = $(Q-1):(Q-1) as Cé(k) and the form (9 k)
9 P
k = %(q-1).(q-l) as 03(k).

k
(iii) In order for (p 91,) to have determinant l the following

conditions must hold on the indices:

36

L,k = l,...,q-l

37

 

 

k+L = q-l
k.t # q-l. %(q-1)
C(k) = Ca.)
(k) k
which is equivalent to C3 (p p-k) k = l .,%(q-3).
k
(iv) The indices on (é qu are = (q-l),2(q-1),...,q(q-1)
k 9‘ %(q+1)(q-1)
C(k) = C(kq) .
k(q-1)
This is equivalent to [ O-k(q-l) k = l,...,%(q-l).
The above calculations give us the classes as listed in
Table 3.

Table 3. Conjugacy Class Structure for SL(2,q) d = 2
conjugacy canonical parameters number of centralizer
class represrntative classes order

’(K\ 1
C. It's-1 1 0(q --D
“(to P"
. (p :) kném-l) 1 (1011-!)
mo 9" 0
z I 9" k'%(q—1),(q-1) 2 2C4
"m p" 0
C2 9 k=%(q-1),(q-1) 2 2e
k,2.1,.u’q-1
It!) " . ..
‘ P M q 1 %(q-3) q-1
3 k ,H q-1,%(q-1)
éiKl: cl.)
(K) 6x6 k-(q-l) 2(q-1)..
C q(q-1) %(q-1) q+1
4 1
k¢ g(q+1)(q-1)
Cum, claw

 

 

 

 

 

 

9,0 = primitive elements of GF(q), GF(qZ)

respectively

 

38

6.2 Character Table

The characters of GL are again restricted down to SL
as in Section 5.3, except that since some of the characters may
now be reducible, the inner product (Y,Y) must be calculated

on all the characters.

2t(q-l) 2t(1%l) 2kt k=q-1 2kt k=q-1
e ’6 ’6 teem-1):e k=s<q-1>’

t(k-k) tk(q-l)
6 ,6

(i) xiWSL = <

)

(1.1.1.1.1.1.1.1) = ‘1’1

(11) xi” = (q.q.0.0.0.o,1.-1> x xit) in CL.

('1) _ ._
Thus Xq ‘SL " (q,q,0,0,0,0,1,'1) - Yq

2 2
eqmq) 2—-‘1—q(q 4) + 2 <q_1)+ 2 q“ 1

so Yq is irreducible.

_ 3:;
(m) xéilUMSL = xiii”) = ((Q+1)e(t+u)(q 1).(q+1)e(t+u)( 2 ).
e(”“)“ 1<=s<q-1>.<q-1)new")k teem-1), (q-l);

etk-uk +'€uk-tk,0)

((q+1) . «no out“. (-1>““.1,<-1>““.1.

ek(t-u) + €-k(t-u),0)

replace (t-u) by t; since t-u = t+u (mod 2) we can also

replace (t+u) by t and get:

’31 = (q+1.(q+1)(-1>‘,<-1>t.1.(-1)t,1 .6“ + e'tkfi)

t = l,...,q-2
t¥%(Q-1) t=1,...,3;—3
YCC) = Y('t)

39

Now we check for reducibility:

 

 

2 (q-3)/2
2(q+l) 4 2kt -2kt
(Y ,Y ) = --§--+ -' + z (e +'e + 2)/(q-1)
+1 +1 2
q q q<q -n q k=1
- 9-_3
= 4 + 1 (q 3é/2( 2tk +_ -2tk) + 2‘ 2 )
= 4 + 9'3 + 1 (CE-U(fl) _ __2_
q-l q-l q-l k=1 2 q-l
1 (q'l) 2tk
- 1 +' _1) Z 3
(q k=1

= = ' = 1:1

(Yq+l’Yq+l) 1 + l 2 If t c( 2 )
9-1
= l + 0 = l for t # C( 2 )
3:1
< 2 >
Thus we see that Yq+l is reducible and Splits into two con-
9:}.
. . ( 2 ) '
jugate, 1rreduc1ble characters for SL. Yq+1 - Y +1 +-Y +1.
2 2

Since (t+u) can equal 3%; for 3%1' different values of t

and u we see that Xq+1 has 3?— associates in CL relative

to SL which we already know by Theorem 2.12.
This means that Vii: is irreducible for the remaining %(q-1)(q-3)
values of t # 5(Q-1).

(iv) In much the same manner we get:

k -k
Xéfi‘SL = viii = (q-1’(q-1)(‘1)ta'12'('1)ta‘1:'(‘1)ts09'9 t'e t
where t = l,...,qZ-Z QQ+1 = l
t ¥ mult. (q+1)
(t) = (tQ)

We check for reducibility:

40

2 (q-1)/2
2 -1 4 1 2kt -2kt
(Yq_1.vq_1) - 417L+ 59+ 2 (e + e + 2>/(q+1>
cm -1) k=1
9:1
_ 4 2(2) _1__(‘1+1)/2(92kt+ -2kt ___2_
q+1 (q+1) q+1 R; 9 ) q+1
+1
_ 1 ‘1 2kt
— 1 + q+1 g 9
.. g = ail
(vq_1,wq_l) 1 + 1 2 if t c( 2 )
= 1 + o = 1 if t # (SEE) .
Thus Y(t) = Y +-Y' for all t = multiple of 9:13 which
q-1 9:.1 £1.21 2
-1 2 2 2 (t)
occurs 35- times. The remaining %(q-1) characters xq_1

restrict down in sets of q-l to the %(q-1) irreducible char-
acters, Yéfi, of SL. The restrict table for CL to SL is

given below.

Table 4. The Induce-restrict Table for GL(2,q) - SL(2,q) d = 2

 

GL(2:9}

%(q—1)(q-2) eds-1)

 

I
end q—l elf-use) ans-1) yea-1) Hq-I)

 

 

 

 

114°” '11-.

 

 

 

 

 

l l

 

41

51;}. 9:1
(2) <2) _
At this p01nt we know that xq+1 and xq-l split
. I d I
into 2 characters of SL, Y +1, Y +1 an Y _1, Y _1
2 2 2 2

reSpectively. By Theorem 2.12 Yi(g) = Y;(g) = Xéfil for i = q+1

and q-l on all conjugacy classes of SL which are complete
-1 +4
(L) (L5)

classes of GL. Thus and Xq_1 Split in half on all

2
xq+1
C£(k), C3(k). We now fill in the known

classes of SL except
portions of the character table and try to determine the missing
values. The Sign alternation on some of the characters depends
on whether 1%; is odd or even. It is convenient to let

iii
6 = q(mod 4) and replace (-1) 2 by ;:6. Initially we will

set up a separate character table for 6 =;: l and later combine

them. At this point the character table is as follows:

42

 

 

 

 

 

 

 

 

 

 

Table 5. preliminary Character Table for SL(2,q) d = 2
6: 1
lasscmr k” we webs) Wei?) WM“) Wail?" iwi‘i" 1- Vi, ‘1‘”
/ 1 q q+1 q-l %(q+1) %(q+1) 2(q-1) 2(9-1)
C; 1 q (q+1)(-1)t (q-1)(-llr 93rd) %(q+1) -%(q-1) Jam-1)
I'Hr') 1 o l -1 k,+kz k\-k2 2,412 x.-9e
C243”) 1 o (-l)t -(-l)t t3-k4 k3+k4 2,414 51,4,
(3’ ‘1'” 1 o l -l k,-k,_ k,+kz 51,-12 R, +92
(fl—(12") 1 0 (-1)t -(-1)t k3+k4 kB-kq $1344 25-24
2““ l 1 EM+ £5“ 0 (--1)K (--l)K o o
:3") 1 -1 o - K: M o o —(-1)K -(-1)'<
C4
6 = *1
‘ cl:::r WI We, WE)! W1: Winn) Li’s-(w; Win-n Win-l)
C, 2km) 3.;(q+1) ‘5;(q-l) =}(q-l)
C; -1r(q+1) -i(q+1) Jim-1) 2(0-1)
c" ‘7'") k. +k1 k."kz Rule 9.412..
I? 1;." k3'k 4 k3+k4 Q3411 1344
C7- (same as above)
C'z' ‘7’") k, -k2 k. +k2 SI. 4?, 1, +51,
Z (1;) k3+k4 k3-k4 513%, 23-314
(3" (-1)" (-1)K o 0
C5,” 0 0 +1)" +1)“

 

 

 

43

The missing entries in Table 5 can be filled in using the
orthogonality prOperties of the character table; however, the
details are rather laborious. A more elegant approach is the use

of Gaussian sums.

The elements in classes C2'(k), 03(k) are of order q.
. . |
We know that the miSSIng entries for Y +1, Y +1 are all composed
2 2

t
of sums of $(q+l) q h roots of unity and the entries for

Ygzlf Yézl are composed of sums of 5(Q-1) qth roots of unity.
2 2
th ,
We let 6 be a q root of unity .
2 61 (the exponents are quadratic residues
i=k2(mod q) of q)

i
e (the exponents are non-residues of q).

Also let x

and let y 2
i¢k2(mod q)

Thus x and y are sums of %(q-l) roots of unity. In partic-
ular they are Gaussian sums and the following theorem applies.

THEOREM 6.1 With x and y defined as above, then

Xy = %(1 - 6Q)

x = %(-l +q/Eq) y = %(-1 -,/Eq) where 6 = q(mod 4) = i l .

For the entries of the characters Y , Y' we use x,y as

51;].- Ll
2 2
. I = _ _ _ - l =
defined above if Y _1 +'Y _1 l and x, y if Y _1 +-Y _1 +1.
2 2 2 2

If we let x' = x+l and y' = y+l then x',y' are sums
of %(q+l) roots of unity and we have the following corollary.

Corollary 6.2 With x' and y' defined as above, then

44

x' +-y' = l
X'y' = 3&(1 - 5Q)

X' = 5(1 +/VI) y' = 52(1 mffi) -

For the entries of the characters Y +1, Y'+1 we use x',y' if
2 2
' = .. .. ' ' = ..
Y +1 +Y +1 1 and x, y if Y +1 +Y +1 1.
2 2 2 2

Using this method the missing table entries can be easily
filled in for the cases 6 = l and 5 = -l and then these two
cases can be merged to complete the block as it appears in Table 6.

Now that we have determined the table for SL, there is
no need to use notation such as C5, C3 etc. to link SL with

GL. The resulting character table is more simply written as

follows.

45

Table 6. The character Table for SL(2,q) a: SU(2,q2) d = 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

conjugacy canonical parameters number of centralizer
class representative classes order
C1 (23 f) 1 QUIZ -l)
C, 11) (l) 1 q(q"-1)
(K) (-].)K K
C3 ( 1 (4)) k-1.2 2 2q
(K) (~1)k K
C4 < P H) ) “'1’?- 2 2‘1
K) x
C; ( P p") k'1."'.%(q-3) MCI-3) q-l
C(GK) (‘Kflrip-mr”) kill," .,%(q-1) %(q_1) Q+1
parameters t-l, - ' ' ,Mq-B) t-l, ' ' ',%(q-1)
number of
characters 1 1 2 2 flq-B) §(q-l)
ha . {I
classc r “’1 W1, Yin“) Wilt-S) W‘éfi W21).
C1 1 q %(q+ 6 ) %(q- S ) q+1 q-l
C2. 1 «1 sure) -%(q-6) (-1)t (q+1) (4)"(«1-1)
K
C‘,“ 1 o harm) give-$21371) Hf" <-1>““'
Cf,“ 1 0 was?) 9:21)“ (4153) (-1)‘ K (-1)“""
l -
Ci,“ 1 1 (nine) (:1? (1-5) L)?“- T" o
‘2' 2. | tl<
) + K+ 't ..
C2“ 1 -1 (51521-6) g1) (1+6) 0 11,12“. U1
2. ‘Z
6 = q(mod 4) (Ag-1 - l £J§+1 = 1 p,a = primitive elements of

CF (q) , CF (qz) respectively

 

VII. CHARACTER TABLE FOR PSL(2,q), d = 2

In this section the character table for PSL(2,q)sa
PSU(2,q) d = 2 is constructed from the table of SL(2,q) given

in Tab 1e 6.

7.1 Conjugacy Class Structure

As in SL, the fact that q =3: lOmod 4) creates a prob-
lem not only in finding the characters themselves but also in

determining the conjugacy classes. The scalar matrices of SL

-1 O

l 0
are (0 1) and ( 0 _1)

so the classes of GL will combine in
pairs to give the classes of PSL except for the self-equivalent
classes (these are classes of SL which map into themselves under
matrix multiplication by one of the above two scalar matrices).
(k) (k)
We note that the number of classes of type C5 (C6 ) in Table
6 is 'even' when 6 = -l (6 = +1) and 'odd' when 6 = +1 (6 = -l),
where 6 - q(mod 4). Any time the number of classes of a certain

type is 'odd' this means that one such class is self-equivalent.
k

Thus there is a self-equivalent class of type p O_ > when
0 p
Ok(Q'1) 0
6 = +1 and one of type k 1 when 6 = -1. We
0 a- (q- )

also note that the order of a self-equivalent class is a half
of its order in SL Since the class collapses. We now calculate

the number of classes of type Cék):

46

47

if 6 = -1 no. classes

5(3539 = 33:22.22;

4 = 9-4-5
4
if 5 = +1 no. classes = a9;- - 1) = Elia—Ll
Similarly for Cgk);
' = = 11 = (51-22 + 1
if 6 +1 no. classes %( 2 ) 4 = -2+6
4
if 5 = -1 no. classes = 5(iél - 1) = S3;£%—l—l
The value of the index k for which the class cék) or cék)
is self equivalent is:
C(k) (when 6 = +1): k = %(3-—3 + 1) = g..._1
5 2 4 k = 3:9
Cék) (when 6 = -l): k = $(3%1-+ 1) = 321

We can now determine the form of the conjugacy class of PSL
which consists of the self-equivalent class of SL.

Let C' denote this class.
6 9-1

P 4 -1 w 4
+1 then - 92- ~ ( 05-1) Q) 6 GF(q} Q) = 1

if 6 p

is self-equivalent

(q-1)(q+1)

 

 

4
if 6 = -1 the“ O _ (Q'1)(g+h ~ ¢ '1
o 4 K ¢
2 4
o E GF(q ) o = 1 .
Thus Cé is of the form (CD -1> $4 = 1 and Q) E (GF(q; if 5 = 1
¢ GF(q ) if b = -1

The conjugacy class structure of PSL is given on the following

page. The classes of SL which combine in PSL are indicated.

48

 

 

 

 

 

 

 

Table 7. Conjugacy Class Structure of PSL(2,q) 2
conjugacy conjugacy number of centralizer
class in class in classes order
SL PSL

I o
C' (O ') C] (I '0)
(Lira) ' (3 l 1 3Q(Q1-1)
C2 0".
(K) f-l)K (3 (u’ (I (D)
C3 I (4)" Z ' ' 1 q
K
4 p (a) 3 p l 1 ‘1
(K) 10“ o /(K) 9" 0
-K C ( .K
5 O P 4 0 P
k-l, 3%:(q-3) k-1,'~,i-(q-h-5) 31(q-b-5) ism-1)
(K) («(14) 0 Cl“) (Incl—c) 0
C6 (0 ~?(‘Z‘l) 5 (O ':‘K(?"))
k'ls s%(9']) k'1:"3%(Q'?+d;) %(Q'2+d;) 2(q+1)
/ ¢ 0
(1; ( C) 954)
9’)" '1 1 (q- 5)
‘CEGF(q) inf-+1
¢1Gr'(q2) if Jul
p = primitive element of GF(q)

Q
ll

primitive element of GF(qZ)

 

49

7.2 Calculation of Characters

 

Every character of SL which is constant on the classes
C1 and C2 is a character of PSL and even more, every char-
acter of PSL can be obtained in this way. Thus the characters

X1: qu X +6’ ii: for t even, X(E ) for t even, are all

2
characters of PSL.

The only difficulty is in finding the values for these
characters on the class C6 since the canonical form of C6 and
hence the character values, depend on 6.

We will concern ourselves with the character values

Xi(8) where g 6 C6 so K = 3:9 as shown in 7.1.

4
(i) (g) - +1 (if 5 = +1)
xq } x (g) =
-1 (if a = -1) q
9:_1 1:1
.. (t) _ kt -kt _ (t) _1_+g (4” ‘(efl'
(11))tq+1(g)1+(,)1 (o — +1) x q+1(g g) — ( 2 )(ISI +w1 )
= 0 (6 = -1> = (Hm-1)t
1__ +1
(t) _ (t) (z, )t "(i— "‘)l
(1.11) Xq_1(8) =0 (6 = +1.) xq_1(s) = (T )(H +li) )
= «(32“ +6.5?) (5 = -1) = (o-1)(-1)
211 11
k 1 1 1 4
(iv) xme) = (-1) 45—5— <a = +1>= <-1) “(J-L): (-1) 1%
2 1 El+_1_1 1 1 L2? = H)
- <-1>k+1 —,9 (a = -1>= H) (j) = <-1>

where [%] = greatest integer S % .

We can now write down the character table for PSL(2,q).

50

2
Table 8. Character Table for PSL(2,q)sz PSU(2,q ), d = 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

conjugacy canonical parameters number of centralizer
class representative classes order
1 2
C. ( 1) 1 ‘%q(q -1)
l
(:2 (l l) l q
1
(:3 (F’ 1) 1 q
C4“, (pk or k=1.-~,+(q-11-5) inn-11-5) ice-1)
dug-n-
C500 ( 6””) 1<=11"'.%(q-2+ c5) Tim-2+5) %(q+1)
Cc (at as" 1 (q-S)
parameters t-l’ - ° - I talf . . .
' ' 3%(9‘13‘3) " 21(Q'2+S)
number of , l J‘( 5)
characters 1 l l L 4(q-1:’-5 ) 4 ‘27::
Mam!“ \H W1 w‘tl‘i-HT) Y-sz’) W221 wet-1
C. 1 q luv-6) %(q+5) q+1 q-l
C: 1 0 so J53) Jew ~53) 1 -1
c1 1 0 1(5 4271) in +1371) 1 -1
K -Kt
C4." 1 1 %(-1)K(1+6) 2(43‘ (1+5) (113.11), 0
+ +1 ~Kf
C?“ 1 1 1m; ‘(1-5) an? (1-6) 0 «If-(«A
3r 2!
C; 1 5 (-1)[ 4] (-l)[ 4] (4; (1+6) (-1)t (5 -1)

 

 

 

 

 

 

 

 

 

2 .
p,a = primitive elements of GF(q), GF(q ) reSpectively

wim'l) = 1 93““) = 1 o4 1 ¢ 6 GF(q) if 5 +1 5 = q(mod 4)
o e GF(qZ) if 5

II
I
....s

VIII. CHARACTER TABIE FOR SU(3,q2) d = 1

2
8.1 Character Table for GL(3,q), U(3,q )

In the next six sections all the character tables will
2
be derived from GL(3,q) and U(3,q ) as given in Steinberg
[l9] and Ennola [6]. For convenience these are listed on the

following pages.

8.2 Conjugacy Class Structure

The determination of the range of the class parameters be-
comes more complicated when we move up to 3 X 3 matrices so it
is best to indicate how these calculations are made.

The fact that d = 1 implies 3 * (q+1) and the condition
that the determinant must be unity for all matrices in SD impose
strict limitations on the values which the class parameters can
take on.

The only classes of type in SD are

9

(k) (k)
C C2

(k)
1’ Ca

those for k = q + l.

The classes have determinant 1 only if

(k1L) (k1t)
Ca ’ C5

2k +-L a 0(mod q+1) i.e. L = -2k(mod q+1). Once k is selected,
L is uniquely determined. We can discard the L parameter and
the value k = q+1 (Since k a -2k = L for this value). Thus for

Cik), Cék) in SU the valid range is k = l,...,q.

51

52

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56

(hum)
6

and L. With k,L selected there is a unique value of m such

+1
For classes C there are (q2 ) ways to select k

that k +'L +'m 0(mod q+1). However there are q choices of k
and L which will force m to equal k or L so we must discard

these values. Thus the number of classes equals

q+1

_(._2__)_:_:=l(-1)
3 6qq '

In order for classes of type Cék’L) to have determinant
unity, pko-L(q-1) = pkg-L = 1 which implies L = k(mod q+1). Thus

L is determined once k is chosen. Now L i 0(mod q-l) or else

db = ot(q?1) = pt and we are back into the classes of a type already

considered.
Thus Cék) in SU is indexed by k = l,...,q2-1
k i 0(mod q-l)

(k) {-qk)

C = C

The number of such classes is %[(q2-1) - (q+1)] = %(q-2)(q+l).

In order for classes of type Cék) to have determinant l
we have the following:
3 4 2 6
k -1 + -1
Tm )(q q+1)___1=,rq

2 2 3
Tk(q + q+1) (q - q+1) = Tq +1

1 which implies k E 0(mod q+1).

Thus k = (q+1),2(q+l),...,(q3+l) and so the number of classes
k 1‘ 0(mod q2 - q+1)

2 4
C00 = C(kq ) = C(kq )

l 2 1
equals §[(q - q+1) - 1] = 3 q(q-l).
The complete class structure has thus been determined and appears

in Table 11.

57

8.3 Calculation of Characters

The characters of U(3,q2) are now restricted down to
SU(3,q2) by letting k = (q+1) on classes Cik), Cék), Cék);
L = -2k for Czk’c), Cék’t); and L = k for Cék’L). We keep in

. +1 .
mind that eq = 1. In several cases we can eliminate one of the

character variables t,u,v by a suitable Substitution. In X(C’U)‘SU

(t,u) u . (t,u)
- d
and x ‘S we replace (t u) by t an in XS ‘S we

substitute t' - (q-1)t for u and obtain a simplification of the

characters.

(t.U)

The calculation of XSp ISU

is done below as an example.

XS(f),u) ‘SU = (sp e(t-lu) (q+1) ’€(t+u) (q+1) , e:(t-l-u) (q+1) ’Seuk-Zkt

euk-Ztk tk uk -quk

90:6 (n + n )30)

= (Sp’1,1’86(u-2t)k’€(u-2t)k,o,nk[(q-l)t+u] + T‘k[(q-1)t--qu],0)

(replace u by t' - (q-1)t)

t'k t'k kt' -kqt'

= (Sp31,1,S€ ,6 ’0’“ + n ,0) = Y(t )

SP

None of the characters can be reducible so inner products are not
2
calculated. The character table for SU(3,q } d = 1 appears in

Table 11.

58

 

 

 

 

 

 

 

 

p=o ,T=T

(q-l)

3

(q -1)
1

Table 11. Character Table for SU(3,q2 )_ PSU(3,q2 ),d - 1
'conjugacy canonical parameters number of centralizer
class representative classes order

l
1
C. ( ' A 1 q3rs p
| 3
CZ I I I 1 q 3
l
‘ 1
C3 ( i I) 1 q
(K) p‘ K 2
C4 9 (52k k-l, ,q q qrs
(K) p“ K
[5 I P ém k-1.°°',q q q s
(mflgn) PK 9 k,2,m-l,°'°,(q+l) 2
C P 1.. Rd‘ m z'q(q-1) s
6 P k+9+m=0 (mod q+1)
m P“ K ) k'16”’,(qz-1) M >( )
~K kf mod q+1 -2 +1 rs
C7 6 6 ‘5 Cm) C(-1K) (modif- 0 2. q q
7* . k=(q+1), 2(q+1),"
C“) 7"“ ,3(q +1) qu(q-1) p
8 m2“ kfimult. q“ -q+ +1
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IX. CHARACTER TABLE FOR SU(3,q2), d = 3

9.1 Conjugacy#Class Structure

(i) For classes Cik),...,C§k) the only values of k for which
the determinant is unity are %(q+1), %(q+l), and (q+1).
k
Dickson [5] shows that classes of type 2 pk in
k
1 P

GL(3,q) will split into three conjugacy classes in SL(3,q). Since
the classes of U(3,q2) are indexed by certain classes of GL(3,q2)

gk) of U(3,q2) will

this means that the corresponding classes, C ,
Split in SU(3,q2) into three classes with 1,3,52 as the off
diagonal entries, where B is any non-cube root of unity in G(q2).
(ii) For the classes Cék’b),C§k’L) we require that 2k +-L a 0
(mod q+1), that is, L = ~2k(mod q+1). Also k a9 multiple of
%(q+l), since this value would result in a class of type

C1,C2 or C3.

1) ways

Cék’L’m) we select k,L in

(iii) For the classes (q:
and m is uniquely determined since k +-L +-m E 0(mod q+1).
However, there are (q+1) - 3 values of k and L for
which the determined m value is the same as k or L.

We must discard these values. We thus get the class count:

fi-[cq’z’b - (cl-2)] = %<q+1><q-2) + 1 .
(iv) Since (k-L) E 0(mod q+1) for classes of type Cék’L) we

can let L = k and let k = l,...,(qz-l). However k
60

61

cannot be equal to a multiple of (q-l) or else we would
have a canonical form of one of the classes already counted.
Thus we discard the (q+1) possible multiples of (q-l) in

l,...,(qz-l) and get the class count:

fled-1) - <q+1>l = %(q-2>(q+1>.

(k)

(v) As shown in Section VIII, k, for the classes of type C8 ,

must be a multiple of (q+1) and not a multiple of
(q2 - q+1) of which there are 3 in l,...,q3+l. The total
class count is %{(q2 - q+1) - 3] = %(q-2)(q+l).

A complete listing of these results can be found in Table 14.

9.2 Calculation of Characters

 

The process of restricting the characters of U(3,q2) down
to SU(3,q2) is little different from that of the case d = 1. We

let k = l(q+1), %(q+l),(q+1) for classes cik), (320‘), (:00.

3 3 ,
L = -2k for Cék’L), Cék’b); L = k for C§k’L) and keep in mind
that eq+1 = 1. Making these substitutions we can obtain the

restricted characters. A further substitution of t' = (t-u) and

I = - (tau) (tau)
t (t+2u) made in both Xp ‘SU and XQp ‘SU

w(t,u) = w<t+2u) if w3 = 1. A

will simplify
these characters. We note that

(

similar simplification of XSS’U)‘SU is made by letting
t. = u + (q'1)to
All the characters thus obtained are then tested for reduc-

ibility and it is found that only the sets of characters of type

er and X 2 contain any reducible characters. The inner products
5 r

for these characters are calculated below.

62

 

 

or .v )-§-—P—+3(2q'iz+ 9 H” +§—>21+1-2-22

2
rp rp q3r82p q38 3q

2
= (18 +'2 £1 +-22)/s

where 21 ' 2 \ Z e(t+u-2v)k‘2’ 22 g 2 l 2 etkfiuLivm‘Z
k=1 (t,u,v) k,L,mfll [t,u,v]
160(3/3) k<L<m

k+L+m=0
In this expression the numerator is an integral multiple of sz=(q+1) .
As suggested by Steinberg, this integer is the coefficient of the q2
terms of the numerator. Only 22 will have such a term.
1’

etk-hlL-l'vm -tk-uL-vm

22-6 2 [ 2 ( z: e )3
[t ,U,V] k,L,m=1 [t,u,V]
kw
k+L+m50(s)
a l. 8 [ Z (1+E(t-v)k+(u-t)L+(v-u)m +_e(t-u)k.+(u-v)L+~(v-t)m

6 [t,u,v] k,L,mFI

+_e(u-V)(L-:)+t(t-v)(k-m)+e(t-u)(k-L)] +-linear function of q

S
=% I: [82 + :33 (t-lu- 2v)k(s z e(2u-t-v)L)+ 26 (2t -u -v)k(8 z

[t, u,v] k=18 L-l k=l L=l
S S

+% 2 [a g (u “was 2: (c -u)k+8 z (t-v)k

[flu-W] L'l k=1 k-l

c(t+u-2v)L)

1+ linear function of q

The second term vanishes since t i u #‘V, so:

2 [ :3 (t+u- 2v)kz e(2u-t-v)L
(t, u,v)kFl L=le

e(t+u-2v)k2 3(2u-t-v)L=82, which =-v-u=u-t=s/3

sun:

2
22 = 8 +' ] + linear function of q

3 if 2
Thus (er,er) -

'. er splits into 3 components for u=t+s/3, v=t+28/3, t=l,...,s/3.

The maximm value of x<§>lSU a Y“?
s r s r

k # multiple of 3. For such a value of t we get:

4 2 2
B 35 r 35 9 9(2-32 _
s r s r q rs p q 8 3q

1 otherwise

will occur when t = g2

Thus Y(§) Splits into three irreducible components when t =.§
r r

 

63

and a different set of three components when t - éfi .

We can now construct the restrict table.

Table 12. Induce-Restrict Table for U(3,q2) - SU(3,q2) d = 3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

veil
E;Q(q+l)(q-1) ‘;q(q+1)(q-1)
é(q-2) (q+1? sum) sow) %(q-2) (q+1)z (q+1)
vs.- - ls. is. in mm )(a. x... m. «...-m.
1+! 1+, .... ,..
‘sz s“ WSW, LPS‘I’Is Why, WSW, Wit/3 [133/5 \er ‘H—p Wm, Wye/3 “PH;
=1; (q+1)(q-2) 3 3 é(q-2) (q+1) 3
SU(3,q2)
U(3,q2)
q+1 q+1 q+1 q(q+1) q(q+1) ski-2) (q+1):L
”Kn/XI ax‘L-I’X‘L Xian-Wt, ‘X‘KP... ’er ’XSP... ’Xsp
1" 1“ 1“ \t“ w w y.
W. ”“1. we LE" ‘11? was” \KP
1 1 1 q q §(q+1)(q-2)
SU(3.q2)

The reducible characters er and Y 2 will Split into thirds
s r

on all classes of SU which are complete classes of U.

leaves only the values on the classes 05(k), 63(k), cg(k)

determined.

The missing block is:

This

to be

 

 

64

Table 13. Partial Character Table for SU(3,q2) d = 3

L ‘ '- 1.
L“ We Wr- 9/3 git-F73 \PSIH) LPS 173 W5 ‘73 q/St‘ya W5 "/3 W8 ‘7 3
2K

 

 

 

 

 

 

 

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(K) x x x LK 2K 1k
- - 2 -1 - -qu -qu - a) —shJ -qu
C2 (2‘; 1) (2‘1 1) £1: ”5“” ’5 =3 % 3' =3
C/M a b c a’ b ’ c ’ ” b” c”
3
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CW" c a b c/ a/ b c / a / b

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//

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CW") 1 or 1 or 1 or
6 -2 -2 -2

(:lK)

7
o M
C8 1 l l l l 1

 

 

 

 

The easiest way to fill in this block is to look at the table for

III

PSL(4,q), where only the characters Y' 3, ng/3, er/B appear.

rp/

It can be shown that the values for these characters on C5, 3,
G; must all be rational values. Thus this 3 X 3 block of values
can be quickly found using the orthogonality relations. The values
of er/3 on C5’ Cg, Cg suggest that the reducible characters

Y 2 Split up in a similar manner. The remaining 3 x 6 block
3 r
can then be determined using orthogonality relations.

The only remaining detail which needs to be mentioned is

| u III

rp/3’ er/3

the values which er/B’ V have on the various classes

of type Cék’L).

The reducible character Y:;,u,v), t = 1,...,%(q+l), u=t+%(q+l),

- - k,
v=t+§1q+l), equals - 2 e<t v)k+(u V)L on Cé L)

[C,U,V] k # L

k,L = l,...,(q+1)

65

 

 

 

 

 

 

 

 

Now if k. E L (mod 3) then €(t-v)k+(u-v); = l V’t,u,v
permutations so Yr p(C6)= otherwise e(t-v)k+(u-v)L = w‘ for
3 of the 6 permutations of t,u,v and the other 3 permutations give
w2 .3 er(C6) = -3(w' + wz) = -3(-1) = +3. From this we finally
get:
‘10:) -2 if RE L (mod 3)
rp/3(C6) =
1 otherwise
The complete character table can now be written down. It appears
below.
Table 14. Character Table for SU(3,q2) d = 3
conjugacy canonical parameters number of centralizer
clean representative classes order
(R) U“ K
C; W to" k“(3,3 3 q’ra'p
() k”
C: 1 Ukwx) k-1,2,3 3 gas
’00 't x 1
C, 1 w L; k-1.2.3 3 3g
1 DJ‘
C3“) (9 L9} to") k-1’2’3 3 3g;
wuo L{‘ K 2
C3 9 (b-Jz UK k-1,2,3 3 3Q
K
"0 t° . k-1.---.<q+1) _ 2
C4 ( P p" )' we (mod sum) > 9 2 q”
(K) 6" k 1 ( 1)
‘ . .U q+
C5 1 9 2K kiO,mod’ §(q+l) 9’2 ‘13
pK k Q m.]_ ((1,1)
Comm) 9’ (Jun, ’ é(q+1)(q-2)+1 32
5 55" k+a+m=o (mod q+1)
9“ k 1.. ‘
' 9 sq ’1
"" ( s“ ) ky‘Omod ‘13 1.-<q-9)<q+1> rs
7 6' “(K’ C‘ (mod q -l)
(k) 5K ,_ k-l . . qu
Ca ( 6”” «1.4) kiO mod 3(q -q+1) 5(q‘2)(9*1) P
5 Cut): c(K1‘)=C(x14)
2
a = primitive element of GF(qz), p = 09.1, sq -q+l = l 9 E GF(qZ) 93 ¢ 1

 

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H.a.ucouV qH mHan

x. CHARACTER TABLE FOR PSU(3,q2) d = 3

The center of SU(3,q2) consists of the scalar matrices
(1 l 1) , C” m w) , (92 w2 w2) w3 = 1. To determine the conjugacy
classes of PSU we identify classes of SU which are equivalent
to each other under multiplication by the above three scalar
matrices since PSU = SU/Z(SU). Thus, except for a single case,
the classes of SU combine in sets of 3 to give the classes of PSU.
The exception lies in the set of classes of type Cék’b’m).

The number of such classes in SL is %(q+1)(q-2) +-1 which is

not divisible by 3 and so we know that one class is self-equivalent

+1 2 +1
(‘13, (3‘ ).q+1)

6
2 .
which has the canonical representative (w w i) . This class 18

under scalar multiplication. This class is C

broken out and relabeled as C9 in PSU.

To find the characters of PSL we take all characters X
of SU Such that x (wk wk wk) is constant for k = 1,2,3. Thus
Y1, qu, and Yq3 are characters of PSU. When t is a multiple
of 3 then Y(t), Y(t) Y(t), Ylt) are characters of PSU. All

9 qp ’ rp r28

three characters Y and all Y(t,u,v)
Sp/3

when t + u +-v is
Sp , < >

a multiple of 3, are characters of PSL also. The table for PSL
is thus very easily obtained once we have the characters of SL.
2
We note that the tables for PSU(3,q ) d = 1,3 agree with

Frame's results in[J£fl on the number and order of the conjugacy

67

68

classes and the frequency and degrees of the characters.

The number and order of the conjugacy classes for SU(3,q2)
d = 1,3 given by Dickson [5], page 571, agree with our results
for these groups.

Several character tables for specific values of q were

generated and checked against existing tables for errors.

The character entry Yfit,u,v)<cg) = - E wt+2u+3v _
2 P [tau ,V]
t+
- 2 w u can be simplified by observing that if t E u (mod 3)
_ [t 3“ 3"]

then the condition that t + u +~v E 0 (mod q+1) implies

t E u E v (mod 3). In this case t +i2u E 0 (mod 3) so

YEE’U’V)(09) = -6. Otherwise, t + 2n i 0 Cmod 3) and
(t ,u,v) = _ _ 2' ___ (t ,u,v) = _
er (C9) 3w 3w +3. Thus er (C9) 6 or 3.

69

Table 15. a 3
Character Table for PSU(3 q ) d
9

 

 

 

 

 

 

 

 

conj. canonical
class representat parameters
C 1 ive number of centrali
‘ ( 1 1) classes order zer
1
1 1 3 2
C. ( I 1) MP
, 1
C3 (1: 1) 1 {388
C" <6 1 1
z
.., 1..) q
“ (
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Pk
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p k-l,“' ‘L(
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5 (1. PK ~2K) 3 ’3ng
p k=1 L
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kiifm-1,...,q+1
C(Kflp} F3K k+R+jn1 O
‘ 9’ m kfk’RJ-R’ 29d qd‘)
e Sm.” .m (mod‘;(q+1))"’(q+1)
R! I 37180118]! ‘8 ((1-2) 1.82-
R 3(q+1) 3
#%(q+1)
mi (q+1)
M px simultaneously
c l “'K 1" I
7 q:- K1 1‘41, : 3(q2-1)
?Kmod (Q‘l) ‘i(
00 gr C =C<-'<?) 6 q-2)(q+1) 431's
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Ce ( J“! . k-I - -'-
6K!) C(K’) £3211, (K 4 ‘
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a 1 32'
= primiti
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2 3
eecF(q).e +1.6q'Q+1=1
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H.e.ucouV HH IHHuH

x1. CHARACTER TABLES FOR SL(3,q), PSL(3,q) d = 1,3

The structure of the character tables for the unitary and
linear groups are so similar in form that the details of obtaining
the characters of SL and PSL from GL differ very little from
those described in Sections VIII-X for the corresponding unitary
groups. Once the general method is established the differences
between the unitary and linear cases can easily be handled. For
this reason no calculations will be given for any of the tables

in this section.

11.1 Character Table for SL(3,q) d = l

 

The calculations to determine the number of conjugacy
classes of each type and the range of the class parameters are of
course quite different from the calculations for SU(3,q2) but
they are of the same nature. Since no restricted characters can
Split, the development of the character table poses no difficulty.

Table 16 is the character table for SL(3,q2) d = 1.

71

72

 

 

 

 

 

 

 

Table 16. Character Table for SL(3,q) EEPSL(3,Q) d = l
conjugacy canonical parameters number of centralizer
class representative classes order

I
Cl ( ' I ) l qzrzsp
I
C2 ' | l ) 1 (131‘
I
(:3 (I l ‘) 1 Q:
(K) PK PK 2
C4 (51" k'L'” ,(q-2) q-2 qr s
(k) f“ K
C.5 " p'ZK k-1.~-,(q-2) q-2 qr
(K’p'l’H) pK Q k,9,m-1,... ’(q-l) 2.
C6 9 PM Man g.(q-2)(q-3) r
k+9+m-O (mod q-l)
II . T
C(K) (p 0,-K K) k-lg.”3q '1 1
-.L .-
7 5' EtMOJHE-SKg'il §q(q 1) 1‘8
) ‘1'" .
C(K 7K1 k 1 k‘(Q"1),2(Q"1), I I
8 ... 7- m ( ;',(q‘tq)(q-1) ijq(q+1) p
(: : qu'= Cfxt)

 

 

 

p,a,T are primitive elements of GF(q), GF(qZ), GF(qa) respectively

+
P ' 09 1

2
_ Tq +q+l

r B q - 1

s = q+1

p=q2+q+1

73

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74

11.2 Character Table for SL(BLq) d = 3

The induce-restrict table for GL(3,q) to SL(3,q) given

in Table 17 demonstrates the close similarity between the unitary

and the linear cases.

 

 

Table 17. Induce-Restrict Table for GL(3,q) - SL(3,q) d = 3
. GL(3.Q)
q-1 I q-1 ]q -1 l(q-1)(q-2) [(q-1><q-2)Lq<q-1)
Y.” -.'X les 'thryz’ "‘Xfl 7(9 " 1X1?” ~7< prIXI-p ’er

  
  

    

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3

 

 

JI::(q--1I) (q-1)l 3 Ifi<q+2) (CI-1) 3

SL(3.q)

 

It should be noted that the 1-1 correSpondence between sets
of classes of similar canonical form and sets of characters of the

same degree, breaks down in SL and 80. This is because the

 

75

classes that Split do not correspond to the characters which split.

Thus, the extreme orderliness of the character tables for

and U(n,qz)

GL(n,q)

is lost in the special and projective Special Sub-

 

 

 

 

 

 

 

 

groups. The complexity becomes more pronounced with an increase
in n and d since more class and character Splitting occur. The
character table for SL(3,q) d = 3 appears in Table 18.
Table 18. Character Table for SL(3,q) d = 3
conjugacy canonical parameters number of centralizer
class representative classes order
(K) to" w“
C\ w" k-1,2,3 3 qsrzsp
W) “J7 <
C; (1 L...) Lu) k‘1,2,3 3 qu
I(K) U‘ I
C, ( W w) II-1,2,3 3 sq‘
”(In (Lefty;
C3 IJ k-II2.3 3 3q‘
’"(K) L~’:
C3 0 Lb): w“ k-1,2,3 3 3‘]:
CH0 {0 PK ":) k.1,.oo.q-1
4 P kf 0 mod. 5(q-1) q-h qr‘s
PM
CM) 1 PK k'19"'SQ‘1 .
5 9 k! 0 mod 3(Q-l) q-b qr
III... r0" II,I ...-1,... ,I-I .
9"“ k< m 'g(q-1)(q-h)+l :-
‘9 k+2+ma 0(mod q-l)
I(
(K) ‘0 67" 191’... “12-1
C, 5'3“ kI‘O mod q+1 z iq(q-l) rs
C0" __. CW!) (mod q -1)
ch‘) SKJKz a. k- ”1 'z,q +q
8 5m: IIIIo mod s(q +q+1‘) 5(q+?)(q-l) p
c‘“: cm“.— c“? ’moap
, 2
p,o = primitive elements of GF(q), GF(q ) respectively
2
3 3 +1 3 +1
eecF<q). e #1. seem ). 6““ =1,II =1, D'Oq

r‘Q'la saq+13 P=q2+q+l

 

76

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for

unitary case.

SL(3,q)

77

11.3 Character Table for PSLLQLQ) d = 3

The table for

PSL(3,q)

The table follows.

is quickly obtained from the table

in the same manner as described previously for the

 

 

 

1‘ Table 19. Character Table for PSL(3,Q) d = 3
conjugacy canonical parameters number of centralizer
class representative classes order
1
Cu 1 1 1 J3 Q3 1:: 3p
1
1
C2 1 1 1 éqar
/ 1'. 1 2.
C3 1 1 1 q
n (5 )
1 2
C3 0 1 1 q
(a )
I 1 L
'5 6‘ 1 1 q
(K) Pr k
P . K | I
C‘ ( 91 ) k'lf”, §(q-h) 3 (Cl-h) 3 gr s
(m 2.9“ )
C, p. k-1,'“,3 (q-h) é(q-h) éqr
k,?,m-l, sq'l
c”w”) p" 9 k<Q<m
6 (0 pm k+R+m E 0’ (mod q-l) z
kik; 1*1, mim {g(q-1)(q-h) é r
simnltaneously
ki 3(q-1)
1n- §(o-1)
m- (q-l) simultaneously
(K) or .L 2
p -K k-1,...’ 3(q -1) _I_ - 1
C7 ( q ugk ka (mod q+1) 6 q(q 1) 3 r8
6 C“): C(Kz)(rnod%(g’—IU
6“ W1 (lg-'1)
“t J. - 1
CB 6 5“: CM: (“1): C (“‘1‘) 9 (‘3‘?) (q 1) 3 p
C“ (1' o; ;) 1 r1
q A)

 

 

 

 

 

 

 

p,a - primitive elements of GF(q), GF(th reSpectively

2
e e GF(q). e3 i 1. 6 E GF(q3). sq '1 = 1, w

3

r B q-l, s = q+1, p = q2 + q+1

+1
=1’p=aq

78

 

 

 

 

 

 

 

 

 

 

 

 

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x11. CHARACTER TABLE FOR PSL(4,q) d - 1

In his thesis, [20], Steinberg calculated the character
table for GL(4,q). This table has 22 types of classes and char-
acters and many of the entries are quite complicated. Due to its
size the table will not be reproduced here.

Since SL(4,q) a PSL(4,q) for d = 1 there are no class
or character splittings and no need to calculate inner products.
In Section 12.1 the parameters on the canonical representatives
are determined. In Section 12.2 a single example is given to
show how these parameters are used to restrict the characters of
GL down to PSL. In many cases an unnecessary parameter on the
character entries can be eliminated by a suitable substitution;

however, such a substitution is in some cases not readily apparent.

12.1 Conjugacy Class Structure

Considering only the diagonal elements of the canonical
representatives of GL(4,q), we have eleven types with which to
work.

k k
(i) det (f 9 pk k) = 1 implies 4k = 0(mod q-l). But 2,4
p

do not divide (q-l) so k = (q-l) is the only possible

(k) (k)
1

value. Thus for C ,...,CS k = (q-l).

k
(ii) det (p pk pk I.) = 1 implies (3k + L) E 0(mod q-l)
9

thus L = -3k. Since k = l,...,(q-l) - 1 there are

79

(iii)

(1V)

(V)

(vi)

(vii)

(viii)

80

(q-2) classes of each of the class types C(k) C(k) C(k).

k 6 ’ 7 ’ 8
det (9 pk pL L) = 1 implies (2k + 2L) E 0(mod q-l)
D
so L = -k. Since C(k) = C(-k) we see that k = l,...,aéz'
and so there are %(q-2) classes of each type Cék), Gig),
eff).
k

det (p pk 9L m) = 1 implies (2k +~L +-m) E 0(mod q-l).
p .

We thus get m = -(2k + L). There are (q-l)(o-2) ways

to select k,L. But -(2k + L) = k (which gives a class of

type C6) for (q-2) choices of k,L so these values must

be discarded. Thus the number of classes equals

1 l 1

-2-[(q-1>(q-2> - (q-m - 5(q-2) = §<q-2)(q-3>

det (0 9L pm a) = 1 implies (k + L +-m + n) E 0(mod q-l).
9

Since k < L < m < n k,L,m,n = l,...,q-l we get

%Z(q-2)(q-3)(q-4) classes of type C(k,L,m,n).

14
det (p pk 0L qt) = 1 implies (2k + L) = 0(mod q-l).
0
Thus L = -2k and there are %-q(q-1) classes of type
(R) 00
C15 ’ C16 °
k

det (9 pL om HR) = 1 implies (k +-L +~m) a 0(mod q-l).
0’

Thus m = -(k +AL) and k,L = l,...,qZ-l, k # L and

k +-L # multiple (q+1). This gives %~q(q-l)(q-2) classes

of type ny’L).
k

det (G okq ck kq) = 1 implies (2k)

0(mod q-l). But

2.* (q-l) so k E 0 mod(q-l). Since k = l,...,qZ-l and
k i 0 mod(q+l) there are q values which are multiples of

(q-l). Since C(k) = C(RQ) we get %-q classes of type

81

(k) 00
C18 ’ C19 '

k
(ix) det (0 qu CL LQ) = 1 implies (k +.L) E 0(mod q-l) so
0'
1 (k)
L = -k. We get g-q(q-2)(q+l) classes of type C20 .

k

(x) det (9 TL TLq 2) = 1 implies (k + L) =—.-. 0(mod q-l)
TLq
1 (k)
so L = -k. We get :3’q(q-l)(q+1) classes of type 021 .
k
2
(Xi) det (u) wkq qu 3) = 1 implies (q3 + q + q + 1)k a
w kq
w
0(mod q-l) which means k = 0 mod(q-l). There are

%-q2(q+1) classes of type C§:)'

These results are tabulated in Table 20.

12.2 Calculation of Characters

There are 22 types of Jordan canonical forms for PSL(4,q)
and if we look only at the diagonal of these types, as above, there
are only 11 types to consider. In the same manner we can consider
the characters of PSL to fall into 22 sets consisting of char-
acters of the same degree. However some of these sets are related

in the sense that the character parameters t,u,v,w run over the

(t) X(t) ’ X(g)

same range, e.g. sf’

all contain (q-2) characters.
qs f q sf

There is a rather close resemblance among these sets of characters

and they behave in much the same way upon restriction to SL. Thus,
in a sense, there are only eleven different type characters to handle.
The details of restricting GL to SL are not of Sufficient interest

to warrent writing out more than one example calculation.

(t a“ 9")

We consider the character Xsfp

of GL(4,q). Each entry

xsfr(gi) is understood to be the value associated with the corresponding

class Ci’

82

_ 2
i = l,...,22. As before, 8 = q+1, r = q-l, i = q +1,

2
p = q + q+1.

(t,u,v> =
sfp

sfp

€(v+u+2t)k where t,u,v = l,...,(a-l)

()R

(q3 + 3q2 + 2q+1)€ t < u < v

32€( )k x(t,u,v) = X(t,v.u)

(2C1+1)e( ”
( >k

6

 

(t+u+v)k+tL (v+2t)k-+uL
e + p e

2

(U,V)

3P

(2q+1)€( )k+tt+s 2 €( )R'HJL

(u,v)

( )k+tL ( )k+uL
e + Sum/)6

 

S2 2 e(v+t)k+(u+t)L + s E

(u,v) (k,L)
(v+t)k+(u+t)L 2tk+(v+u)L
e + e

e(v+u)k+2tL

s[ E + e(v+u)k+2tL

(U,V)

( )k+( )L ( )k+q+{.
Z(u,v)e +tz(kaé)€

]

 

( [(v+t)k+uL+tm] + e[(v+t)k+tL-l-um])

S Z(uav)€

+ s e(v+u)k+tL+tm + Z e2t+k+vL+um
(U-V)

2:(u,v)( )+ J 1+ '2: 8‘: ]

 

2 evk+uL+tm+tn
[kaLaman]

 

(v+u)k+tL
3 €

( )k+tL
e

 

vk+uL+tm uk+vL+tm
a +6

 

i0

83

We now use the relations between the various class parameters

(k)

to restrict Xsfp to SL. For classes Cik)"°°’CS k = q-l; for
Cék’L)...Cék’L) L = -3k; for Cék’L)...C{:’L) L = -k; for

0105””), cg’L’m) m = -(2k + L); for cfz’é’m’“) n = -(k + L + m);
for Cig’L), Ci:’£) L = ‘Zk; and for ka,L,m) m = -(k +-L). Since

xsfp(g) = O for C18,...,C22 we need not be concerned with the

parameters for these classes. Using the above relations and keeping

 

 

in mind that eq-l = l we obtain the following restricted character.
(t U-V) _
Xsfp ‘SL - Sfp
(q3 + 3q2 + 2q + l)
2
s
(2q+l)
l
(u+v-2t)k (v-3u+2t)k
s +
P e P 2(u’v)€
(2C1+1)e[ 14-ng 1
Si 1+2; 1
2 (v-u)k (v+u-2t)k -(v+u-2t)k
s 2%”- + s[e + e 3
+u-2t k -
503.} 3+5" ))+e()

2 e[ 1 + e[ ] +‘e-E ]

 

(v-t)k+(u-t)L + €(v+t-2u)k+(t-u)L) + S €(v-lu-2t)k

S 2 (e

(U-V)

2(t -u)k+(v-u)L
+ 2mm“

2:( )+eE 3+ze[ 3
(v-t)k+(u-t)L +6(v-t)k+(u-t)m +_e(v-t)L+(u-t)m

 

2(uav)

+
Z(k-L-m)€
(v+u-2t)k
S e

(u-vn-(v-t) (1mg 2 €(v-un-(v-t) (lem)
(k,L 3m)

84

( )k
e

(v-t)k+(u-t)L (u-t)k+(v-t)L
e + e

 

 

0

We now observe that the substitution t' = (u-t), u' = (v-t)
will eliminate one of the character parameters and simplify the char-

acter o

' '

(t ’u )‘SL 3 Vii-3U) = Sfp
3 2

(q + 3q +-2q+l)

52 where u,t = l,...,(q-Z)
(2q+l)

l

 

Sp e(u+t)k + p 2(u.t)‘(u-3t)k

(2q+1)e(u+t)k + e(u-3t)k

e(u+t)k

8 2

+823( )k

32 z c(u-t)k + sum-ml- + e-(u+t)k

2 e(ml-Dk + 8 C(u-Pcn- + €-(u+t)k

(u-t)k + e(--+c)1- + e-(u-O-t)k

 

1

S

Z(uat)€

 

uk-HZL + ea-zmwu, €(u+t)k

) + s

(u-t)L-2tk

+ ~’3<u,t)"’

uk+tL +. (u-Zt)k-tL
€

3 Beau)“

) +6(u+t)k

(u-t)L-2tk
+ 201.06

Z(t’u)(e

 

85

ukfitL

23(u.t)€
(“UM-“(Hm (U't)L'-U(k‘|m)
+ 209th + 2(k,L,m)€
s e(u+-t)k

+Ieulc+>tm

+’€uL+tm

 

(u+t)k
6

 

ukfitL
Z(u,t)€

 

00000

The remaining characters of PSL(4,q) d = 1 are developed in a

similar manner. The complete character table appears in Table 20.

86

 

 

 

 

 

 

 

 

Table 20. Character Table for PSL(4,q) d = l
conj. canonical parameters
class representative ggmzlzsses 2322:8117”
Cl ( 11 j
C (1 1 1 1 q‘r’s‘fp
z 1 )
C (l 1 1) 1 q‘rzsp
3
1 i 1 1 q'rs
C (- - )
c (1 )
‘ g l 1 1 q,
(x) P“
C < P" . _ ) k--3k
‘ P P "‘ k'1-'° ' .q-2 q-2 q‘r'sp
on F" a
C7 (1 P P" 6):)
u q,rt
C2" ('3’ f9" )
.13 I
p q r
C30 (9' P: p" ) C“). CH0
o“ g a.
p k l, $(q-2) k--k §(q_2) q‘r’s"
C(K) (git PI ) Cf”: COR)
p" .. . ..
m P k l, - ,q-2 k'-k q-2 qtrls
(00 (q' p“ can: Cc-n)
" 7 P") ”If" -§(q-2) 10'“ liq-2) (1‘!-
(K.” (9.9" P, ) l ,W ' ,q-1
. dun-D k ..
'- 9 Ca}! Sm‘glC-QKBJ MCI-2H?” qrss
(ml) 9"
c ( I P" ' >
:3 9 -(a~o!)
9 qr1
“-9-” ”) (pip! Pm ) :1<’§7:an'11 ' ' 39‘].
men J- - -
" f’a k+2+m+nlo (mod q-l) 24“; ”(Q ”(q-h) r,
C‘m (9 9'0"“ , ) 1:16? A -1
w W W cm) 3023;“ kHz-l) ("‘81
P"
on I
<1 w >
G'- n qrs
ad.) 9' ’ k 9 a
C". ( 9 tum) ) kgalo"':q '1
4M) ' - -
6 ‘ k+9i0 (mod Q+1) IQ“ 1‘)(q 2) r‘s

 

 

 

 

 

 

Pad-T-w - primitive elements of GF(q). GF(qz), GF(qB), GF(qa)
. u,(c1+1) (q2+1)

2
‘rq «1+1

r - (1'1:

reap.

a - q+1, f-02+1, p=q2+q+1

87

Table 20 (cont'd.)

A

 

 

 

 

 

 

 

 

 

 

 

conj. canonical Parameters number of centralizer
class representative classes order

(U0 6“ «g k-l,-",qz-l

Cue ( d 6" Sn) kvo (mod q+1) in qzrazf

k-O (mod q-l)

Co” («'63‘ .- qzs

M c“ .- k-1,---,q"-1
C20 ( 6' 1‘-“ «g “‘0 (mod qgl) gq(q-2)(q+1) ra"

‘ kH- (mod q-l)
P“ 'K k-l .o- 3-1

W) 'T - 9 sq

5:. ( T "Zr-«6) kfo (mod g‘+q+1) §q(q-1)(q+1) :1)
can: C‘“‘ = c‘K“)

m ‘4" n . k-<q-1),2<<.;-1).~--q‘-1

C21 < co “U." a") kIO (mod q‘+l) , %q7'(q+l) sf
(a) C‘“' C(w‘ 2 Cuff-3: C(K‘)
2 3 2

Q‘1 -1 +1 - +1
6 l’nq _1,eq =1,Tq 1=l,¢(<1)(<1+1).__1

q + +1 +1 -1
T q = q _ 3: “q = 6

 

88

Table 20 (cont'd.)

v..._-_L

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

t_1,...’(q_2)
2m:r off 1 1 1 1 1 q-2 q_2 q_2
Y th- V.» -».- w- t: «a as.
C. 1 1.? 2'9 1‘6 1‘ g; 15:; 235;
C; 1 1’5 ‘3 it 0 P ‘is‘ 1;-
C; 1 1 0 ‘v' 0 S q; 0
C. 1 $ 0 O 0 S 1 O
,4 1 o o o o 1 o 0
C2" 1 P 1? 15 1‘ 9655"" 596515?" 195% 1’ €25“
c‘,” 1 s 1, o 5612'“ $3355.“ 1.”:
Cc: 1 1 0 O E..- * an E" 0
cf“ 1 (21-1) rut-z) c ‘3‘ 5(5‘1'2‘") 5'(5"‘.g‘“) figs-+5.)
c2," 1 s 1 1 0 €35?" 5(6‘5 e‘“) 35‘“
(3" 1 1 o 1 0 2‘52"" fig“ 0
C(29) 1 (in) (22*!) 6 g * 5.2153.) Egg-34mm” fishy-mu)
Ci?” 1 Z 1 1 O gal-f..- fégréumm) En-
1 3 3 2 1 521:5?» i‘éi'fi»? 22:59
(2‘: l 1 '1 —* “3 52‘“ o _5£--
CT 1 o -1 1 o e“ o _E..
d?” 1 1 1 o -1 (".2" 0 ~(Ea-HZ")
C3-" 1 *1 '2‘ F a o o 0
c3," 1 -1 0 1 O 0 0 0
CS) 1 -1 -1 z 1 O 0
cf," , I 0 o -1 1 a“ -5“ a"
CT: 1 '1 1 0 -1 o o o

 

 

89

Table 20 (cont'd.)

 

 

 

 

 

 

 

 

1 Pt) t-l’. ,(q-2) 1 t) 1:)

W11 ' W Wm" gr 4"": Y

%(Q-2) q-2 §(q-2)

(f) (t,

W13? “’11:? W1“!

F P <13? 2‘49
(2‘161f‘) 2P its

P 1 it

5 ‘1. o

1 0 0

- 1 - -
P(£""+£ L" ) 1P “2 95‘" (”($sz 5 1" )
5(cuf*éth) €£:“t+S€th %( )
Kt 4.x! -
E1 +£ 8 1K? 0
z - -

5341“" 1- 17"“) s'+1(€""+ 2"") 53+ 1‘ (8“? e l”)
5+(etwn e-tkt) 5+1E-1Kf 5

1+(Etkt+£‘lkf) 1 1

fix”), €‘t(K+1)) 5(£t¢~+lI,C-tm+fi) 5(Et1mlfiE-1mu)

+(E1Kfve-tkt) +£€uttr€zxt ‘%(szt+€-2kt)
( )+( ) ( )+ £1“ t Erma” E-tmrfl)
€t¢K+R)* tfkom) f(KOH) awkm.etut+m)+£flkon) €t(K+J|+E¢(k+mT*Et(Kn-)

* c'ttl'hl" E't(K§IM) ‘- e-t(n.n}

- - («on
+£¢"‘"‘, at )' E'thflu)

‘t I) -’t(KH-n -t )
€11" ‘e *E (bun

 

 

 

 

 

g «at - _

eta + Q. 1£1Kt_ a"? -z(e‘"+ E 11“)
£1K*+ €139 - -l.x(' 0

g“'“”«- E-uxu) Etna!) E-uml) _ Etna“- E. um!)
€ -F ¥

1 1 1

~tK ~ .

EtK*€ ‘5“:- E *K Etw+€ {K

0 o o

O o o

 

 

90

Table 20 (cont'd.)

t,u-l,--c,(q-2)

 

 

 

 

t<u
§(q-2)(q-3) ‘r(q-2)(q-3)
(nu) (tau)
fih “Raw
5% «,st
13317-12141 1(26’021“)
1
S 15
21” 1.
1 O
(n \K (t-SulK a) - 3K
592 “ +925." spew 'i-pfmnfi“ "
(flow ”4“ l“ u- uut t-s 1
(21“)6 +3 ($1.11 (2311);; . “37.05:: «A K
(innK C (1' 3“ “‘ (tn-3K
E *Elun) £-

 

51. €(u-t)K+£-(u-FW)
+5 (C(h+f)K‘€-cu+flk)

s‘cflI-tflt * e-(u-HK)
+ S G'.-nut”: * Eturtn:

()+()

51‘ Ema“: ,, E-(u-TT?)
+138 ( £0.01)" P €‘(HfflkJ
S(
+ %E(M+f)K

(14-: )K -(u-t)K
E + E

 

52w."(Ermunfl,e(e-tu)k—u)
-tuk* -
4' $2“‘““‘ + Ttu.t)e- (e “u
Em." ( ~2ukott-u19
4» €"“"“‘' 1' [mane
U.

 

$17.7: - -
5:11am) (€ l' a“ 1“ 12:13::- (t u)
4. se‘"“"‘ 1- 1 Zane ) E

flan) (1410K- H4 (143)"
Z(Mptl(E +€ ) + E

 

1

 

 

 

 

LEI-Lb) Eli-unfutwnuw * Z1«.1.m,£m-fl’ 'M(K+m) ___’
sc}“**’“ -321u411x
EN”)K -Euuflx
i(£::+u1 _E(K.netxvu1
o O

o O

O O

a O

0 0

 

 

91

Table 20 (cont'd.)

 

 

 

 

 

 

 

 

 

 

 

 

 

t,u,v,w-1,° --,(q-1) t-1,---,q‘-1
t<u<v<w th (mod q+1)
t+u+v~HII 0 (mod q-l) W“): WW“ (mod (14)
&.(q-2)(q-3)(q-h) Mq-I) §q(q-1)
(t,u,v,w) 1'“ (t)
Kg“? WPPP Wt???
51" HP 114p
31341511431” 1111’ ‘ '%
(2:105 ..; gr
3‘+l -1 -2
1 -1 0
(touovowfl:
591mm,“ , Has" we“
1 31¢
(21‘ "i (Saar. w) " 21K ' E“
c m
2 (egunhv) -EtK -E‘K
(fou-v-w’K - K K -
5"ZIttzmmw] V(C"+E 1 ) QHE‘ *‘E a)
l )K -
sztgu‘ww’ YE‘:£”‘ _?Etk
( )K -tk
szflflqw) -6": E O
(t’u-‘zwpt o (v—wu
57! 1Lu. mu) *5" *8“
< )Ko-t )1 « tK
[widow] ' E: - E
rm «Jun» +wn
(K,',M,hj O 0
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0 1c‘T-(‘7'1 11“”) +£"‘—z("z “112 1f"‘)
0 _£'¢K~ ) EtK
0 Jim”: «mint ’7. mm_ Tim'JHt
0 -+‘(e’+ a“) He“ 6“)
0 -(e‘ 1e“) (19"+ 0")
tr 11" -flr «K :1: tr -1 «or
0 ”(~11 nz A: ‘ 7 +7 ‘11 “+7 1
O 0 O
O 0 O

 

 

 

92

Table 20 (cont'd.)

 

 

 

 

 

 

 

 

 

 

 

t'ln' ' ‘ ’q1_1 t-19 ' ' ' :91 '1
u-1."-.q-2 +410 (mod q+1)
ti 0 (mod q+1) Wm: WH$I
w‘f,“l' v(‘t,0)
ig(q-1)(q-2) ‘51 h
(fin m ‘0
"H-sh W199 “Pf"?
1»st 1-‘9 111‘?
111-15434 “‘ ~11?
-5 9.2-1“ 2‘
'5 ‘i‘ O
-1 1 0
FP(€(WO: f:H-300K) O O
_(€(LNHK*£(t-- 3u1K) o O
_(E(u+£)k+£(f~3uh( O o
“flaunt”: E-(u-tw) *1 y-7-
Fem-“'2 584th 4. -1.
_(£(u-t)k + E-(u-tik) 1 1
renauuD)‘ t- EH-Zu‘m- u) 0 O
_E(£k+ul)_ Eli-Udk-ua' O O
O O O
_5€ul( (7‘11K+ 1?" K1) -\-(9fl‘* etki) _+(etk+ tug)
_Eux(41-1tw+ ”fifkt) (9"; end.) (9mk 92kg)
- ”Ht _
-(€““1e“‘)(°7 W + 47 Wm) 0 0
hi mama“) £+i[£(e“+e“*)

—1—s (em-””1 e““'*"}]

pm H )J

I ‘ .
”S (6N1 m , arm" )n

MU 1+1 )1

 

 

O
O
O
O
O

 

9“!" ‘ étlrg
O

t f '- 3
iz-(e‘eb‘gtt‘et‘)

e-IK.‘ 9.th
O
1

t
“:1 (9’19“+9"4 9?t )

 

93

Table 20 (cont'd.)

 

t,u-1, ' . . ,q¢-1
tin-I 0 (mod q+1)
1.1mm: Lp‘tfm): qflfitujz W‘tmu)

t-l’ ' ' "q3-1
tJO mod qz+q+1

w (t: w‘t‘ )3 wfitl’

t'(9'l)a2(9'l) a ' ' 'aq4‘1
tI‘O (mod q1+l )
Wm: Won). Wtq‘), we!”

 

 

 

 

 

 

 

 

 

 

 

 

t q(q-Z) (q+1) i q(q-l) ((111) i q'(q+1)
mu) (1T 11)
413:1» Wy‘s‘! LPF’SP
L"? 1351+. F359
44 -15 +‘s
2111“ +5 1
.4. 1 k
‘1 1 1
o 958" 0
o _4_£tlt 0
0 e" O
F1(E(t.um+ {ht-um) O O
44 ) o O
( ) O 0
O o 0
o o 0
O 0 0
t“ . mt - u x
+2.”, w n: * 1 J o 0
In
2.5, ( I o 0
o O 0
Heaewxen e") o vsw‘w")
( X ) 0 49%“)
(It, tn an: -uu¢
ZS}, ”‘7 ”(‘1 *"I 1) o o
O ftx+ftq+ffltt O
o O _¢t_ ¢t‘_ ¢t‘: ¢tg3

 

 

 

94

Checking the character table for PSL(4,q) posed a problem.
It checked with PSL(4,2), the only existing table for the case d = 1.
However, when q = 2 only 11 of the 22 character types are present
so this verifies only half the table. It is difficult to complete
the check by generating the next larger character table and seeing
if the various orthogonality properties hold, because PSL(4,4) has
82 irreducible characters. The problem is surmounted in the follow-
ing way. Suppose Ennola's conjecture holds. Using the conjugacy
class table for PSU(4,q2) d = 1 given on page 98 and changing q
to -q in the table for PSL(4,q) we should obtain the table for
PSU(4,q2) d = 1. Having done this, we let q = 2 and compare the
resulting table with PSU(4,22) given in Frame [11]. The two tables
checked. It very conveniently happens that PSU(4,22) contains 8
additional character types not checked by PSL(4,2). Thus we have
verified 19 of the possible 22 character types. In the process we
have demonstrated that Ennola's conjecture still holds. Considering
the complicated nature of the characters for PSL(4,q), it is nearly
inconceivable that we could get the correct table for PSU(4,22) if
there was either an error in the table for PSL(4,q) or if Ennola's

conjecture is not valid for arbitrary q.

2 2
x111. CONJUGACY CLASS STRUCTURE FOR U(4,q ), PSU(4,q ) d = 1

2
13.1 U(4,g l

 

As before, we use the results of Ennola's paper [7] to obtain
a set of Jordan Canonical forms in 61(4,q2) which bear a one-to-
one correspondence with the conjugacy classes of U(4,q2). We need
only use simple combinatorial methods to count the number of possible
canonical forms of each type. The order of the centralizers are
obtained by the application of Theorem 4.1 which says that we need
only change q to -q in the expression for the order of the
centralizer of each class in GL to obtain the centralizer order
of the same type class in U(n,qz).

To ensure that the resulting conjugacy class table for
U(4,q2) is correct, it was checked for U(4,22) having order 77,760
with 60 conjugacy classes and for U(4,32) of order 52,254,720 with
188 classes, respectively.

The details of the above work will be omitted since they are

similar to ones already described.

95

96

2
Table 21. Conjugacy Class Structure of U(4,q )

 

canonical

 

 

 

 

 

 

 

 

 

 

 

conjugacy parameters umber of centralizer
close representative classes order

(I) "

C. ( P“ 0" 9") k'lr" .qfl q+1 q‘rza4fp
(x)

C" (\ ) H u (1‘ 1'83 P
(K) 1

C3 ( \1) H " 95 1":
(K) 1 ,

c. (\) u ..
00

C5 (11\1) H H 933

Cm” (9 WP" 2) :3’32'1- "-.q+1 q(q+1) qzra‘p
‘ 9
(n3) 1 "

C7 (\) Q383
at,” u

C; (i\\I\\\\\) “ Qfs‘
(K!) 9‘ K k,9-1,-o-,q+1

C. ( p 9' 9') w com c”"" squad) q'r‘s*
m,” 1 (km (2.x)

Cw ( \ C i C q(q+1) qzrs?’
(u,t) 1 («,n (1.x)

C“ (\1) C : §Q(q+1) q‘s‘l

0‘ x k 1 m-l --- q+1

Clamp») ( p 9’ m k; (n O ’ h(q+1)(q-1) qrsq
"- Q C(K,1.m) : C(k'm'”
('flLM) L n

Ca ( \) " qua

K

6"“ (p 9’ .. :2’1’1’311;‘” "“1 (“I *

\4 9 69‘ .4 3
n 9" K k 1 q+1

w. . D O . . 9 2

C ( 9 «‘ ~11) -1 q*-1 . -‘(~+1) ( -2> ‘ 3
n - ) 2 a q qr 3
’ 6 9¢0'(mod'q-1).C"’= c‘ ‘
(«,I) 1

C.. (\ ) H II qmz
(m...) exp: Eg'l'm'qd 2

M
C” v 6“" m-1,'” ’91-]. (m) (flu-0‘ §q(q-2)(q+l) 1'33

m‘O (mod q-l) C =

 

 

 

97

Table 21 (cont'd.)

 

 

conjugacy canonical parameters number of centralizer
class representative classes order
m ‘K “<1. k-l, ,q -1 («1 2 2 z
m ( a qkq'“) ”‘0 (”0‘1 ‘14) Cm;- 'C ‘ §(Q*1)(q-2) q r s f
(.0 «K W-K‘ u ' u 2
C... (1 1 6‘64“.) q rs

 

 

‘ - k, R-l q -1
Cw") 6 ° “1 k3” 'k.93 0 (mod q-I) fi<q+1)(q-2)(<f-q-h) r231
3" CMflCum) d-rg,1)_dx-g1)

 

1’10 (modq z -q+1)

pK k-l q+1 2

(«.11 7’ 1. 3.1: (135 q(q-l) (q+1) 829
C11 ‘4

m . Mei) (11“)

 

 

to“ - k-l q“ -1
(K) K‘ ’ ’ 2 2._ f
c,, ( “ were) «2 ... .. «a v ~

 

 

 

 

 

a,Tl,w - primitive elements of GF(q2 ), GF(q 6), GF(q4 ) respectively

-1 -l 2 2
930(q ),T= T(q3 ),r=q-1,s=q+l,p=q 'q+19f=q+1

13.2 PSU(4,q2) d = 1

By requiring that the determinants of the canonical repre-
sentatives of U(4,q2) be unity, we can obtain the classes of
PSU(4,q2) a:SU(4,q2) d = 1. The details of 'counting' the number
of conjugacy classes are similar to those written out in detail
for PSL(4,q) and are not of sufficient interest to give here.

The resulting table was checked against PSU(4,22)
having order 25,920 with 20 classes and PSU(4,42) of order
1,018,368,000 with 92 classes.

As mentioned in the preceding section, this table for the
conjugacy classes of PSU(4,q2) together with the character table
of PSL(4,q) with q replaced by -q convincingly appears to give
the character table for PSU(4,q2) d = 1. At least it checks for

q = 2.

 

98

 

 

 

 

 

 

 

 

Table 22. Conjugacy Class Structure for PSU(4,q2) d = 1
conjugacy canonical parameters number of centralizer
class representative classes order
1
‘1
C. ( 1 1) 1 q‘rzsafp
1
C2 (1 1 1 1) 1 game.
1
1 1 5
C3 <1 11 1) 1 q rs
1 1
C4 ( 1 1 1) 1 q‘s
C 1 1
5 1 1 1 1 93
c‘” (P p" “ )
‘ p P-sx k-1,--.,q q cars}?
FR
(M 1 “ .
C, < P pKP-jK) ' q qssz
K
cm) 61) P“ x
e 1 P Pan (1 C123
p" 1r 1
m p" -.. ,2. ,.-'-,q 1 2 z 3
Cq ( P ‘5“) dc}, Ensign) sq q r 8
PK
m x
_ - )
Cw (1 p P'p‘w) Cm" C” q azm2
p“
U” 1 'l (K) {-00
Cu < P ‘2.” par) C t C h (123
P“ K k’2-1,. ° q+1
1K,” ’
C12 ( p (38 -2R-1) k”? 1§Q(G-1) qrs3
pk
(K1!) 1 K 9 1'
C13 < P P 63"“) i'flq-l) <18?"
V“ 1“]! m < n -'- q(q-l)(q-2) s
'4 P P" k+1+m+n=0 (mod q+1) 2‘
(0" k-l 2-1
(x) V 3 sq
C2,, ( P 6" avg) H0 (mod q-l) %(q+1)(q-2) qrzs?’
p“
(x) V
C“ (1 p ‘1'“ -1“! H 3(Q*1)(Q'2) qrs
W” pk? 2: tég'l’qu-l 1 ( >< ) 1
. m v-q q+1 q-2 rs
C” 6 «men kiddo (mod q-l) A
C1092), C(‘flO

 

 

 

 

 

 

 

99

Table 22 (cont'd.)

 

 

 

 

 

 

conjugacy canonical parameters number of centralizer
class representative classes order
(K) «K 41‘ k 1 ... 1'
' a sq ‘1
C'5 ( “K "“1) Rio (mod q-l) }(q-2) sirzsf
«I k-O (mod q+1)
Cr) (1 (“‘W‘ -K‘) II §( -2) 2r
1 o q q
2.
(R) 0'“ - b1." " ,q -1
a, ( °' fl -.. ) ks‘O (mod q-l) 601-2) (qz-q-h) r28
‘3' 3H. Cw, CH‘L Com: CM"
9‘ q: k'l "' q3+l
( j 0 D
C; ( 7' qut‘ ,> k*0 (mod q‘-9+1) §q(q-1)(q+1) op
.Kt
q.
no wk are k'(q+1)s2(q*1)s ‘ " sq‘ ‘1 2
C21 W th‘l 3 k'fiO (mod q“+l) iq (q-l) rt
“1

 

 

 

 

 

O’Tl’w a primitive elements of GF(qz), GF(q6), GF(qa) respectively

P

, 0(q-1)

,T='r

(QB-1)
1

3 r = Q‘l,

2
s = q+1. p = q

- q+1, f = q2+l

 

XIV. CHARACTER DEGREES AND FREQUENCIES FOR PSL(4,q) d = 2

The existing character tables for PSL(4,q) and PSU(4,q2)
d = 2 are those having q = 2 or q = 3. These tables are of a

reasonable size and most of the entries are integers. The non-

a ifi/b

integer entries, if irrational, are of the form -_:T—_' and if
complex, are simple linear combinations of cube roots of unity.

Thus one is easily misled into thinking that a generalized char-
acter table for these groups would involve entries on the same order
of complexity as was the case for n = 2,3, that is, most of the
entries would be constants or polynomials in q. These hopes are
soon forgotten when we consider the table for PSL(4,q) d = l
which we have just developed. We see for example that the entry

Y35(015) = 2 for PSL(4:2) COrreSponds to the entry retk _

(“-Ztk + n-Ztkq

) in the generalized table and realize that the
simple nature of the existing tables is due entirely to the fact
that the various combinations of roots of unity, which are dependent
on q, are relatively elementary when q = 2 or 3. We also note
that the number of classes and characters can be expressed as poly-
nomials in q. At the level n = 4 these polynomials are in gen-
eral of a higher degree than those for n = 2 or 3. As a result,
the number of classes and characters increases at a much faster rate
with an increase in q. Thus we find that PSL(4,2) has 8 char-

acters, PSL(4,3) has 29, but PSL(4,4) has 82 characters and

100

101

PSL(4,7) has 407.

The smallest projective special groups PSL(4,q) for which
no character tables have been developed are PSL(4,7) and PSL(4,5)
for d - 2,4 reSpectively. These are very large groups, having
over 100 conjugacy classes, and so their character tables would be
rather difficult,to work with even if they were available. If it
were necessary to have the table or a partial table for one of these
groups it would be much easier to develop it from. GL(4,q) with q
equal to the appropriate value rather than trying to get the abstract
table for arbitrary q because the calculation of inner products
involving numerical values is far easier than performing the same
calculation with polynomials in q and sums of (q-l)St roots of
unity.

Aside from these practical considerations there are some
theoretical difficulties which would impede progress. For the cases
d 8 2,4 both classes and characters undergo some splitting. Pre-
liminary checks reveal‘this splitting is much more extensive than
on the n = 2,3 levels. There is no correlation between the
classes which Split and the splitting characters. We have already

seen in the case of SL(3,q) d = 3 that classes of type C Split

3

while the characters which Split correspond to the classes C6 and

C Thus, determining the splitting of the classes and characters

8.
are two separate problems. For the cases n = 2,3 we had Dickson's
results to give us the class splits, but he did not work out the

case for n = 4. In the case for d = 2, if a class of GL Splits

in SL we do not know if it splits into 2 or 4 classes.

102

Taking inner products to determine which characters are
reducible is very difficult with the abstract characters of
GL(4,q) because they are so much more complicated than the previous
cases considered. Also, for the case d = 4, characters can split
into 2 or 4 components and so the inner products must be calculated
with more care than previously. We finally note that after we
determine which characters split, we cannot immediately fill in
their values on the classes which are not complete classes of GL.
Since there are many splitting classes and characters this means
we are left with numerous large holes in the table to be filled in
by the tedious process of using the orthogonality properties of the
table.

In this section an attempt is made to at least determine the
degrees and frequencies of the characters for PSL(4,q) d = 2.
We now use a different route and proceed from GL to PCL.a:GIJZ(GL)
to PSL.

GL.
q-l q-l

PGL a- GL/Z (GL)

\7‘

PSL e: SL/Z(SL)

The work is based entirely on Clifford's theorem which tells us how
many associates the irreducible and reducible restricted characters
have. Since PSLIQ PGL and [PGL : PSL] = 2 we can apply Clifford's
theorem. If a character of PGL is irreducible upon restriction to

PSL then it has 2 associates. If the character is reducible then

103

it has only 1 associate and splits in half.

We now assume tentatively that if 2 divides the number of
characters of PGL of the same degree, then they all restrict down
in sets of 2 to irreducible characters of PSL. If 2 does not
divide the number of characters of the same degree we will pull off
one character and assume this character Splits in PSL. We do this
for q - 3,7,ll,... and hOpe that a clear pattern emerges. In
Steinberg's thesis [20] the frequency table for the characters of
PGL(4,q) is given.

The only characters which require any Special consideration

('2) (t)

are X 2 and X . We consider then separately.
3 fp r fp

The number of characters of degree szfp alternates from
even to odd. This means that either the number of Splitting char-
acters alternates from O to l with every q, or the number increases
by one for every q value. This situation did not occur in the
tables for n = 2,3 so we have no precedent by which to go; however,

the second alternative seems most likely.

104

number of characters

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

q of degree 8 fp in PGL number of characters number of
%Z(q-3)(q2-6q+11) in PSL reducible chars.
3 o o 01
7 3 2 l
1~_\i 1
? n-LB
ll 22 20 10 4
1
l ___1 2
l
l
———1
19 172 168 84
———l
l ————____~_.______________1
l
1.=:;
1
It 1
1
1 1

 

From this we conclude that of the %Z(q-3)(q2 - 6q + 11) characters

of degree Szfp in PGI(4,q) d = 2, 3i2- of them are reducible,

when restricted to PSL and the remaining %Z(q3 - 9q2 + 23q - 15)

will restrict down in sets of 2 to irreducible characters of PSL.
The same situation occurs for the characters of degree

2
r fp except that the number of characters alternates from odd to

even out of phase with the number of x
s fp

105

number of characters

q of degree r fp in PGL number of characters number of
l 2 in PSL red cible chars.
§<q+1)<q -2q-1) “

 

3 l-===:::::::::::::::i 11

7 34 32 16

11 147 144 72

 

From this it appears that of the %(q+l)(q2 — 2q - 1) characters
of degree rzfp in PGL, SE1. of them are reducible when restricted
to PSL.

The frequency table for PSL(4,q2) d = 2 is correct for
PSL(4,3), however we could only be sure it is correct if the table
for PSL(4,7) was available. The calculation 1; 011(1))2 = ‘c‘
proves correct for q = 4,7. however this still d::s not indicate
whether the number of Splitting characters is correct because if
Y, a character of FGL, does not Split in PSL then the k asso-
ciates of Y restrict down to g-Y characters in PSL. If

Y‘PSL is reducible then the k associates of Y restrict down to

2k characters of half the degree of Y. However

%(Y(l))2 a 2k(1éll)2 so the sum of squares of char-
irreducible case reducible case

ter degrees is unchanged by a Splitting.

 

106

The total number of characters for

2 +'7q + 23)

PSL(3,q) d = 3 is

l 2
3-(q + q + 10) and we get for

l 3
Z(Zq + 20 PSL(4,q)

which appears to be of the same form.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 23. Character Degrees and Frequencies for PSL(4,q) d = 2
10111.0) 0-2 PSL(h,3) 251.01,?) PSL(h.<1) 0:2—
umber 01' character no. degree no. of degree umber of character
characters degree char. char. characters degree
q-3 W
2 2 1 1 1 1 1 1 1
2 qp 1 39 l 399 l qp
2 2 cf: 1 90 1 21.50 1 «fr
2 2 q’p 1 351 1 19551 1 q'p
2 2 q‘ 1 729 1 11761:? 1 q‘
0 1; st 0 2 1100 1(0-3) sf
0 h qs‘r 0 2 221100 2(0-3) 08‘!
0 h q‘ar 0 2 137200 Mq-B) q’sf
0 a up 0 1. 22000 Mel-3f up
0 a 00:0 0 1 159600 1(0-3)’ gm
0 3 {2 a‘rp 0 1 1821100 4L,(q3-9q‘+23q-151 s‘rp
1 0 2 91200 1(0-3) gen»
1
2 5h rsfp 1 10110 27 136000 Ski-3) mp
112 s‘ r' r h 61.0 56 115200 * q(q‘ -1) 05%
1 5 h :0 2 65 2 2850 2(0-3) no
1 2 11.25 2 hp
1 5 h 0% 2 585 2 139650 1(0-3) ‘rp
l 2 69825 2 gq‘fp
2 10 qu l 390 5 19950 (q-Z) qu
2 18 rfp 1 260 9 17100 §(q-l)2 rfp
2 18 qrfp 1 780 9 119700 31(q-1)2 qrfp
1 3h 32 r‘fp 2 260 16 4.02600 f§(q3-d5-5Q‘3) r'fp
1 {2 51300 Had) ir‘rp
1 2 ,
3 2 7 6 r’p l 52 3 2052 §(q-1) r‘p
1 1 2 26 2 1026 2 h'p
3 2 7 6 q‘r‘p 1 1168 3 10051.0 1(0-1) if?
1 1 2 23h 2 502711 2 r p
8 96 r sp 1; I116 116 981196 é(q-1)(q+l)z rJep
29 he? 1(2q‘+2q’+7q+23)

 

107

Sumary o f Resu It s

 

The primary purpose of this paper was to determine the
abstract character tables for the groups SL(n,q), PSL(n,q), SU(n,qz),
PSU(n,q2) n = 2,3, PSL(4,q) d = l and display the tables in such
a manner as to facilitate their use. The procedure developed could
also be used to find additional tables for Specific n and 0 values.

The secondary goal was to investigate the possibility of
Ennola's conjecture holding for the Special and projective special
subgroups. The fact that the conjecture did hold for all the groups
under discussion was very surprising. For example consider the groups
PSL(3,S) and PSU(3,5). Their character tables are of completely
different form because d = 1 for PSL(3,5) whereas d = 3 for
PSU(3,5). Thus the transformation q a -q clearly does not apply
to literal values since in the above case 5 a -S does not change
the table for PSL(3,5) into the table for PSU(3,5). The con-
jectured transformation could better be written as q a -q' where
q is not necessarily q'. Even if the conjecture is true, however,
the relationship between the tables of the unitary and linear groups
is not completely determined because the conjugacy class structures,
are not related in any known way; i.e. the polynomials giving the
number of classes and irreducible characters of the linear groups
cannot be transformed into the correSponding polynomials for the
unitary groups. Also, the conjecture does not appear to be of any
value if the tables in question are for specific values of q be-
cause the transformation only operates on the abstract symbol q

and not numerical values.

108

The modified version of Clifford's theorem, although by no
means essential to the problem of developing the character tables,
was nevertheless a considerable aid. The task of determining the
number of components in a reducible character becomes more laborious
without the theorem because the inner products, (x,x), must be
calculated considerably more exactly in some cases.

The theorem'was also used to arrive at a conjectured table
of character degress and their frequencies for PSL(4,q) d = 2
which checked for the case PSL(4,3). It would be interesting to
determine if the table is also correct for PSL(4,7). This is

such a large group that a check here would be very convincing.

 

B IBLIOGRAHIY

 

10.

11.

12.
13.

14.

15.

BIBLIOGRAPHY

Brauer, R. Characterization Lf Characters Lf Groups, Ann. of
Math., 57 (1953), 357 -377._

 

Brauer, R. and 38h, C. Theory g§_Finite Groups, w.A. Benjamin
Inc., 1969.

 

Brinkmann, H.W. Group Characteristics pf LF(3,q) and other
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Clifford, A.H. Representations Induced in Invariant Subgroups,
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Dickson, L.E. Linear Groups, Dover, 1958.

 

Ennola, V. Characters Lf Finite Unita_y_Group_, Ann. Acad.
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. Conjugacy Classes Lf the Finite Unitary Groups,
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Feit, W. Characters of Finite Groups, W.A. Benjamine, 1967.

 

Frame, J. S. The Classes and g_presentations Lf the Groups Lf
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. Some Irreducible Monomial Representations 9£_Hyper-
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. The Simple Group Lf Order 25920, Duke Math. J.,
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Frobenius, G. Uber Gruppen Charaktere, Berliner Sitz, 1896.
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Green, J. A. Characters Lf the Finite General Linear Groups,
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Jordan, H. E. Characteristics Lf Various Linear Groups, Am. J.
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109

 

16.

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19.

20.

21.

22.

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110

Littlewood, D.E. Theory pf_Group Characters, Oxford, 1940.

Lomont, J.S. Applications g£_Finite Groups, Academic Press, 1959.

 

 

Schur, I. Untersuchungen Uber Die Darstellunnger Endlichen
Gruppen Durch Gebrochene Lineare Substitutionen, J. Reine
Angew. Math., vol. 132 (1907), 85-137.

Steinberg, R. The Representations 9£_ GL(3,q), GL(4,q), PGLfi3,q),
and PGLfi4,g), Can. J. Math., 3 (1951), 225-35.

Representations pf the Linear Fractional_Gropps,
Thesis, U. of Toronto.

Srinivasan, B. Characters g£_Symplectic Group Sp(4,q), 3' odd,
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Wales, D. Simple Groups pf Order 17-38-2b, J. of Alg., 17,
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Wall, G.E. The Conjugagy Classes pf Classical Groups,
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GENERAL REFERENCES

Boerner, H. Representations 9i Groups, North-Holland Publishing
Co., 1963.

Davis, C. Bibliographical Survey p£_Simplg_Groups p£_Finite
Order, 1900-1965, New York Univ., 1969.

 

Hall, M. The Theory pjncroups, Macmillan Co., 1959.

Rotman, J. The Theory p£_Groups, Allyn and Bacon, Inc., 1965.

111

    

 
 

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3 1293 0317