• No results found

The Open University s repository of research publications and other research outputs

N/A
N/A
Protected

Academic year: 2021

Share "The Open University s repository of research publications and other research outputs"

Copied!
24
0
0

Loading.... (view fulltext now)

Full text

(1)

Open Research Online

The Open University’s repository of research publications

and other research outputs

Real and strongly real classes in SL¡i¿¡sub¿n¡/sub¿(q)¡/i¿

Journal Article

How to cite:

Gill, Nick and Singh, Anupam (2011).

Real and strongly real classes in SLn(q).

Journal of Group

Theory, 14(3), pp. 437–459.

For guidance on citations see FAQs.

c

2011 de Gruyter

Version: Version of Record

Link(s) to article on publisher’s website:

http://dx.doi.org/doi:10.1515/JGT.2010.054

Copyright and Moral Rights for the articles on this site are retained by the individual authors and/or other

copy-right owners. For more information on Open Research Online’s data policy on reuse of materials please consult

the policies page.

(2)

J. Group Theorya(2010), 1–23 DOI 10.1515/JGT.2010.054

Journal of Group Theory (de Gruyter 2010

Real and strongly real classes in SL

n

(

q

)

Nick Gill and Anupam Singh

(Communicated by W. M. Kantor)

Abstract. We classify the real and strongly real conjugacy classes in GLnðqÞand SLnðqÞ. In

each case we give a formula for the number of real, and the number of strongly real, conjugacy classes.

This paper is the first of two that together classify the real and strongly real classes in GLnðqÞ, SLnðqÞ, PGLnðqÞ, PSLnðqÞ, and all quasi-simple covers of PSLnðqÞ.

1 Introduction

Let G be a group. An element g of G is called real if there exists hAG such that

hgh1¼g1. Ifhcan be chosen to be an involution (i.e.h2 ¼1) then we say thatg isstrongly real. In all cases we say thathis areversing elementforg. Ifgis real (resp. strongly real) then all conjugates of g are real (resp. strongly real), hence we talk aboutreal classesandstrongly real classesinG.

Tiep and Zalesski [9] have listed all quasi-simple groups of Lie type for which every element is real. In this paper we generalize one part of the work of Tiep and Zalesski by identifying exactly which conjugacy classes are (strongly) real in the quasi-simple groups which cover PSLnðqÞ; furthermore we count these classes.

This paper is structured as follows: in Section 2 we outline results concerning a spe-cial class of polynomials. In Section 3 we use information from Section 2 to classify the real and strongly real classes in GLnðqÞ. While this work is not new, the theory

that we develop in the process of studying GLnðqÞ forms the foundation for the

rest of the paper. In Sections 4 to 6 we classify the real and strongly real classes in SLnðqÞ.

In a companion paper to this one [1] we classify the real and strongly real elements in PGLnðqÞ, PSLnðqÞand those quasi-simple groups which cover PSLnðqÞ. To

under-stand reality in PGLnðqÞand PSLnðqÞwe need to study thez-real elementsin GLnðqÞ

and SLnðqÞ; the z-real elements are defined in Section 2 of this paper, and we will

study them in parallel with real elements throughout Sections 3 to 5.

Sections 4 to 6 all include a theorem at the end, which summarizes the main results of the section. An interesting consequence of this paper and [1] is the following state-ment, the proof of which is scattered throughout the two papers.

(3)

Theorem 1.1.Let G be isomorphic toGLnðqÞ, PGLnðqÞ,or a cover ofPSLnðqÞ. Then

all real elements are strongly real if and only if G is in the following list: (1) GLnðqÞ;

(2) PGLnðqÞ;

(3) SLnðqÞ=Y with n22ðmod 4Þor q even;here Y is any central subgroup inSLnðqÞ;

(4) SLnðqÞ=Y with n12ðmod 4Þand q11ðmod 4Þ,here Y is any even order central

subgroup inSLnðqÞ;

(5) PSL2ðqÞ; (6) 3:PSL2ð9Þ.

As we have mentioned, the work on GLnðqÞis not new: Gow has already

enumer-ated the real classes for GLnðqÞand given a generating function for this count [2]. The

work on SLnðqÞis partially new; a version of Proposition 4.4, which deals with the

case whenn22ðmod 4Þorq23ðmod 4Þ, was first proved by Wonenburger [10].

Re-sults concerning the remaining case, whenn12ðmod 4Þandq13ðmod 4Þ, are new.

2 Self-reciprocal andz-self-reciprocal polynomials

As we shall see, real elements in GLnðqÞwill turn out to correspond to sequences of

self-reciprocal polynomials. In this section we define what a self-reciprocal polyno-mial is and we gather together some basic facts about such polynopolyno-mials.

We also introduce the notion of az-real element inH, a subgroup of GLnðqÞ, as

follows: fixz, a non-square inFq. We say thatgisz-real inH iftgt1¼g1ðzIÞfor

sometAH; we say thatgisstronglyz-realiftcan be taken to be an involution. Once

again we say that tis areversing element forg. The z-real elements of GLnðqÞwill

turn out to be of vital importance when we come to examine the real elements of PGLnðqÞand PSLnðqÞ in [1]. Note that we will sometimes abuse notation and, for

an elementgAGLnðqÞ, writezgwhen we meanðzIÞg.

It turns out that z-real elements will correspond to sequences ofz-self-reciprocal polynomials. Thus in this section we also examine these polynomials. Throughout what followszis a fixed non-square ofFq.

2.1 Definitions. Consider a polynomial fðtÞAFq½t of degree d with roots

½a1;. . .;adinFq, the algebraic closure ofFq. We say that fðtÞisself-reciprocalif

½a1;. . .;ad ¼ ½a11;. . .;ad1:

We say that fðtÞisz-self-reciprocalif

½a1;. . .;ad ¼ ½za11;. . .;za

1

d :

For both definitions, by½;. . .;we mean an unordered list of roots, taken with mul-tiplicity. Note that, sincezis a non-square inFq, when we talk aboutz-self-reciprocal

(4)

2.2 Self-reciprocal polynomials. We are interested in Td, the set of self-reciprocal

degreed polynomials inFq½twith constant term equal to 1.

It is easy enough to prove thatTd ¼FdUGd where

Fd ¼ ffðtÞ ¼tdþa1td1þa2td2þ þa2t2þa1tþ1g;

Gd¼ fgðtÞ ¼ tdþa1td1a2td2þ þa2t2a1tþ1g;

and theaivary overFq. Note that ifqis even thenFdandGdcoincide. We definenq;d

to be the number of self-reciprocal polynomials f inFq½tof degreed which satisfy

fð0Þ ¼1.

Lemma 2.1.The number nq;d is given in the following table:

q is odd q is even d is odd 2qðd1Þ=2 qðd1Þ=2

d is even ðqþ1Þqd=21 qd=2

Before we leave self-reciprocal polynomials we make one more definition. Let pðtÞ ¼tnþa

1tn1þ þan be a monic polynomial over k. We define

~

p

pðtÞ ¼a1

n tnpðt1Þ; this is the monic polynomial whose roots are precisely the inverse

of the roots of pðtÞ. Thus a monic polynomial pðtÞ is self reciprocal if and only if

pðtÞ ¼ ~ppðtÞ. Note too that pðtÞis irreducible ink½tif and only if ~ppðtÞis irreducible ink½t.

2.3 z-self-reciprocal polynomials.Letq be odd and let f be az-self-reciprocal poly-nomial with roots in½a1;. . .;adAFq. Suppose that

ai¼zaj1; aj¼zak1:

Clearlyai¼ak. Thus the roots of f can be partitioned into subclasses of size at most

2. Observe that, within these subclasses,aiaj¼z. Now if the subclass is of size 1 then

a2

i ¼zand so does not lie inFq. We conclude thatd must be even.

Now, ford even, define the setSd to be the union of the following two sets:

fðtÞ ¼ 1 zd=2t dþa 1 1 zd=21t d1þa 2 1 zd=22t d2þ þa 2t2þa1tþ1 ; gðtÞ ¼ 1 zd=2t dþa 1 1 zd=21t d1a 2 1 zd=2t d2þ þa 2t2a1tþ1 ;

(5)

Lemma 2.2.The set Sd is precisely the set ofz-self-reciprocal polynomials of degree d

with constant term equal to 1. Moreover,jSdj ¼nq;dsd wheresd equals1 if d is even

and0otherwise.

Proof. Let hðtÞ be a polynomial of degree d and write the list of roots forhðtÞas

½a1;. . .;ad. Clearly tdhðz=tÞ is a polynomial of degree d and its list of roots is

½za1

1 ;. . .;zad1. Thus hðtÞ will be z-self-reciprocal if and only if hðtÞ is equal to a

scalar multiple oftdhðz=tÞ.

Examining fðtÞ and gðtÞ given in the form above in Sd we observe that

fðtÞ ¼ ðtd=zd=2Þfðz=tÞ, while gðtÞ ¼ ðtd=zd=2gðz=tÞ; hence all elements of Sd are

indeedz-self-reciprocal.

We must now show that Sd contains all z-self-reciprocal polynomials. Let hðtÞbe

az-self-reciprocal polynomial and consider the roots½a1;. . .;adofhðtÞsplit into

sub-classes of size at most 2 as described above.

Let us consider the subclasses of size 1; these have form fagwhere a2¼z. There are an even number of these. What is more, since a is a root of fðtÞ, so is aq. As

aq2

¼aAFq2 this yields a pairing on the set of subclasses of size 1. Thus we can

join subclasses of form fag and faqg together, so that all subclasses have size 2.

Note, moreover, that ifa2¼z, then

aq¼ ða2Þðq1Þ=2a¼zðq1Þ=2a¼ a;

thus in this case our subclass has formfa;ag.

In general then our subclasses satisfy eitheraiaj¼zoraiaj¼ z, and in the latter

caseai¼ aj. Let us consider these two cases. Firstly ifaiaj¼zthen we can multiply

the corresponding linear factors, tai and taj, to obtain an Fq-scalar multiple

of

1

zt

2þatþ1

where aAFq. Alternatively, ifai¼ aj1 andaiaj¼ z, then multiplying the

corre-sponding linear factors yields anFq-scalar multiple of

1

zt

2þ1:

If we multiply such pairs together we generate polynomials of the following forms:

fðtÞ ¼ 1 zd=2t dþa 1 1 zd=21t d1þa 2 1 zd=22t d2þ þa 2t2þa1tþ1; gðtÞ ¼ 1 zd=2t dþa 1 1 zd=21t d1a 2 1 zd=22t d2þ þa 2t2a1tþ1;

(6)

for someaiAFq. NowhðtÞis of this form and lies inFq½t; in other words, forhðtÞ,

the coe‰cientsai lie inFqand we have the required form.

The formula for the size ofSd is an easy consequence of its definition. r

We make one more definition: for pðtÞa monic polynomial of degreed ink½t, de-fineppðtÞto be the monic polynomial which is a scalar multiple of tdpðz=tÞ: Clearly

pðtÞwill bez-self-reciprocal if and only if pðtÞ ¼ppðtÞ.

3 Background information and GLn(q)

We start by collecting some basic facts which we will need in the sequel. Recall that, forgan element of a groupG, we denote the centralizer ofgbyCGðgÞ. We define the

reversing groupofg,

RGðgÞ ¼ fhAGjhgh1¼gorhgh1¼g1g:

WhenGcGLnðFÞ, for some fieldF, we define a related group: fixzto be a non-square inkand let

RG;zðgÞ ¼ fhAGjhgh1¼gorhgh1¼zg1g:

It is easy to check thatRGðgÞandRG;zðgÞare indeed groups and that they contain

CGðgÞ, the centralizer of ginG, as a subgroup of index at most 2. In fact, ifg201,

the index ofCGðgÞinRGðgÞ(resp.Rz;GðgÞ) is 2 if and only ifgis real (resp.z-real) in

G; in this caseRGðgÞnCGðgÞ(resp.RG;zðgÞnCGðgÞ) is the set of all reversing elements

forg.

WhenGis a subgroup of GLnðFÞwe can use theJordan decompositionof elements of GLnðFÞ. We need only a few basic facts about it (more details can be found in [8, p. 20]). Any element gAGLnðFÞ can be written uniquely asg¼gsgu where gs is a

semi-simple element,gu is a unipotent element andgsgu ¼gugs. We have the

follow-ing generalization of [7, Lemma 2.2.1]:

Lemma 3.1.Let g¼gsgube the Jordan decomposition of g inGLnðFÞ. Let G be a

sub-group ofGLnðFÞwhich contains g. Then g is real(resp.z-real)in G if and only if gsis

real(resp. z-real)in G and g1

u is conjugate to xgux1 in CGðgsÞwhere xgsx1¼gs1

(resp. xgsx1¼zgs1Þ.

Proof.We prove this statement for the situation whengisz-real; for the case whereg

is real we simply remove all instances ofz. Suppose thatgisz-real. Then

hgh1¼hgsguh1¼ ðhgsh1Þðhguh1Þ ¼zg1 ð1Þ

for some hAG. Now the Jordan decomposition of zg1 gives ðzg1Þ

s¼zgs1 and

ðzg1Þ

(7)

commute, we have already given a Jordan decomposition of zg1 in (1). Since this decomposition is unique we must have

hgsh1¼zgs1 and hguh1¼gu1:

This implies thatgsisz-real, and thatgu1is conjugate tohguh1inCGðgsÞ(in fact, in

this case, the two are equal).

Now for the converse: suppose thathACGðgsÞsatisfieshgu1h1¼xgux1. Then

ðh1xÞgðh1xÞ1 ¼h1xgx1h¼h1xgsx1xgux1h¼zgs1g

1

u ¼zg

1: r In the next subsection we will discuss the real conjugacy classes in GLnðFÞ, and we will need an understanding of Jordan canonical forms. We mention one last fact closely related to Jordan canonical forms, but of a slightly di¤erent nature. Let

gAGLnðqÞwhereqis a prime power. Writeg¼gsgu, the Jordan decomposition ofg.

To apply Lemma 3.1 we need to understand the structure ofCGðgsÞforG¼GLnðqÞ.

We describe its structure in a particular case.

Lemma 3.2.Take gAG¼GLnðqÞand suppose that the characteristic polynomial of g

and the minimal polynomial of g coincide and are equal to fðtÞrwhere fðtÞis an irre-ducible polynomial of degree d and n¼dr. Then

GLrðqdÞGCGðgÞcRGðgÞcGLrðqdÞ:hsi

wheresis a field automorphism of order d.

Proof. Let Vq be the vector space of dimension n over Fq on which GLnðqÞ acts

naturally, and let Vqd be anr-dimensional vector space overFqd. There is a natural Fq-vector space isomorphismj:Vq!Vqd which induces an embedding ofFqd into

G(as the centre of GLðVqdÞcGLðVqÞ) such thatglies inFqd.

Now suppose that hAGLðVqÞcentralizesg, i.e.hgh1 ¼g. We demonstrate that

h lies in GLðVqdÞ. It is clear, first of all, that hðv1þv2Þ ¼hðv1Þ þhðv2Þ for all

v1;v2AVqd; this follows from the linearity of the action ofhonVq. We need to dem-onstrate thathpreserves scalar multiplication, for scalars inFqd.

Observe that hgi has a well-defined action on V1 where V1 is any 1-dimensional subspace ofVqd. Again we can think ofV1 as a vector space overFqorFqd; the ele-mentgjV1 acts as anFq-vector space endomorphism with minimal polynomial fðtÞ,

and as an element ofFqd by multiplication.

Suppose thatgjV1fixes a proper subspaceWofV1. Letvbe a non-zero vector inW

and consider the elementsv;gv;g2v;. . .:Since these all lie inW and dimW ¼c<d, there exist scalars aiAFq such that

(8)

Sincev00 this implies thata0þa1gþ þacgc¼0 which contradicts the fact that

fðtÞis the minimal polynomial ofgjV1. Thus, for anyvAV1, there is anFq-basis for

V1of formfv;gv;g2v;. . .;gd1vg.

Now takevAW andaAFqd. Note thatacommutes withg since they both lie in Fqd. Then

av¼ ðb0vþb1gvþb2g2vþ þbd1gd1vÞ; for someb0;. . .;bn1AFq. This implies that, fori¼1;. . .;d1,

agiv¼giav¼giðb0vþb1gvþb2g2vþ þbd1gd1vÞ

¼ ðb0Iþb1gþb2g2þ þbd1gd1Þgiv:

Thusa¼b0Iþb1gþb2g2þ bn1gd1:Now observe that, forhACGðgÞ,vAV,

hðavÞ ¼hðb0Iþb1gþb2g2þ þbd1gd1Þv

¼ ðb0Iþb1gþb2g2þ þbd1gd1ÞhðvÞ ¼ahðvÞ: We conclude thatCGðgÞcGLðVqdÞ. It follows immediately that

CGðgÞ ¼GLðVqdÞGGLrðqdÞ:

Now we wish to study the normalizer ofCGðgÞ. Writed ¼d1. . .dl whered1;. . .;dl

are primes. Then GLðVqdÞ:hdi is a maximal subgroup of GLðVqd=d1Þhd1i where d (resp. d1) is a field automorphism of GLðVqdÞ (resp. GLðVqd=d1Þ) of order d (resp.

d=d1). (Details can be found in [4, §4.3]; see, in particular, [4, p. 116].) Similarly GLðVqd=d1Þhd1iis a maximal subgroup of GLðVqd=ðd1d2ÞÞhd2iwhered2is a field auto-morphism of GLðVqdÞof orderd=ðd1d2Þ, and so on. We conclude thatCGðgÞis nor-mal in GLrðqdÞ:hsiwheresis a field automorphism of GLðVqdÞof orderd.

Therefore NGLnðqÞðCGðgÞÞ must contain GLrðq

dÞ:hsi where s is a field

auto-morphism of GLrðqdÞ of order dlogpq. Now any element of GLrðqdÞ which

normalizes CGðgÞ must normalize Fqd ¼ZðCGðgÞÞ and so must induce a field

automorphism on Fqd; these are all accounted for and so we conclude that

NGLnðqÞðCGðgÞÞ ¼GLrðq

dÞ:hsi. Therefore N

GðCGðgÞÞ ¼GLrðqdÞ:hsi, as required.

r

Note that, in the language of Jordan canonical forms,gis semi-simple and conju-gate to a Jordan block matrix. Note too that we could replaceRGðgÞwith RG;zðgÞ in the statement of the lemma and it would remain true. Finally observe that, for

g to be real (resp. z-real) in GLnðqÞ, we must have d even and xðgÞ ¼g1 (resp.

xðgÞ ¼ ðzIÞg1) wherexis a field automorphism of GL

rðqdÞof order 2.

3.1 Real conjugacy classes in GLn(F). Let Fbe a field. The conjugacy classes in GLnðFÞare determined using the theory of Jordan canonical forms. We will assume

(9)

a basic understanding of this theory (more details can be found in [3]). The theory is based upon the idea that, given an element g of GLnðFÞ, we can create an n -dimensional F½t-moduleVby defining a scalar multiplication,

t:v¼gv; forvAV;

and extending linearly. The isomorphism classes of V constructed in this way are in one-to-one correspondence with the conjugacy classes of GLnðFÞ. They are also in one-to-one correspondence with the set of all multi-sets of form

ff1ðtÞa1;. . .;frðtÞarg

where, for i¼1;. . .;r, fiðtÞis a monic irreducible polynomial in F½twhich is not equal to t, ai is a positive integer, and Pir¼1degðfiÞai¼n. These correspondences

allow us to classify conjugacy in GLnðqÞ.

Before we state the main result that we shall need, we introduce some notation. A

partitionnofnis a finite multi-set of positive integersn¼ fn1;. . .;nrgthat sums ton;

we sayjnj ¼n. Writen¼1n12n23n3. . .to mean that

n¼1þ þ1 |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} n1 þ2þ þ2 |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} n2 þ :

The following theorem is classical.

Theorem 3.3.Let g be an element ofGLnðFÞ. TheGLnðFÞ-conjugacy class of g is the GLnðFÞ-conjugacy class of the matrix 0pJnp;p where the sum is over all irreducible

factors p of the minimal(or characteristic)polynomial. Herenp¼ fr1;. . .;rkgis a

par-tition,andPpjnpjdegðpÞ ¼n;the matrix Jn;pis Jr1;pl lJrk;pwhere each Jri;pis a

Jordan block matrix involving a companion matrix corresponding to p.

This theorem allows us to construct the multi-set which corresponds to the conju-gacy class ofg. It is simply6

pðtÞfpðtÞ

r1;. . .;

pðtÞrkgwhere the union is taken over all irreducible factors of the characteristic polynomial ofg.

To classify real conjugacy classes of GLnðFÞ, first we need to look at a Jordan block. Recall that, for a monic irreducible polynomial pðtÞ, ~ppðtÞis the monic irreduc-ible polynomial whose roots are the inverses of the roots of pðtÞ.

Lemma 3.4.Let pðtÞbe an irreducible polynomial of degree d. Then J1

r;p is conjugate in

GLrdðFÞto Jr;pp~,whilezJr;p1is conjugate inGLrdðFÞto Jr;pp.

Proof.The theory of Jordan canonical forms tells us thatJ1

r;p andzJr;p1 must be

con-jugate in GLdrðFÞ to Jordan block matrices. Now the characteristic polynomial of

Jr;p is pðtÞrand so, by considering roots, the characteristic polynomial of Jr;p1 must

be pp~ðtÞr. ThusJ1

r;p is conjugate toJr;pp~as required. Similarly, the characteristic

poly-nomial ofzJ1

r;p must beppðtÞ r

. ThuszJ1

(10)

Proposition 3.5.A matrix gAGLnðFÞis real if and only if g is conjugate inGLnðFÞto 0 p0~pp ðJn;plJn;~ppÞ l 0 p¼pp~ Jm;p :

A matrix g inGLnðFÞisz-real if and only if g is conjugate inGLnðFÞto

0 p0pp ðJn;plJn;ppÞ l 0 p¼pp Jm;p :

Heremandnare partitions which vary with p.

Proof. Let g, g1 be conjugate to J

g¼0pJn;p and Jg1¼0

qJn;q respectively.

Lemma 3.4 implies that there is a matching between the Jordan blocks of Jg and

Jg1 which takesJr;ptoJr;q¼Jr;pp~. Similarly there is a matching between the Jordan

blocks ofJg andzJg1 which takesJr;ptoJr;q¼Jr;pp. This yields the given formulae.

r

Proposition 3.5 asserts that a matrix gAGLnðFÞ is real (resp. z-real) if, for any invariant factoriofgand for pirreducible ink½t, p andpp~(resp.pp) occur as factors ofiwith the same multiplicity.

3.2 Real conjugacy classes in GLn(q).We follow the notation of Macdonald in [6]

where he gives another way to classify conjugacy classes; we will use Macdonald’s method to give a criterion for an element of GLnðqÞto be real. We begin by stating

Macdonald’s result regarding GLnðqÞ.

Theorem 3.6 ([6, (1.8), (1.9)]). Let C be a conjugacy class in GLnðqÞ. Then we can

associate C with a sequence of polynomials u¼ ðu1;u2;. . .Þ satisfying the following

properties: (1) uiðtÞ ¼anit

niþ þa

1tþ1AFq½tfor all i with ani00; (2) Piini¼n.

This gives a one-to-one correspondence between conjugacy classes in GLnðqÞ and

se-quences of polynomials with the given properties.

Note that the sequenceðn1;n2;. . .Þis equivalent to a partition,n¼1n12n2. . ., ofn; the conjugacy classCdescribed in the theorem is said to beassociated withthe parti-tionnand an elementginCis said to beof typen.

We need to describe how the correspondence given in Theorem 3.6 works. Let us start with a conjugacy classCin GLnðqÞ. As we described above, this can be

associ-ated with a multi-set of polynomials,

(11)

where, for each i, fiðtÞis a monic irreducible polynomial inFq½twhich is not equal

tot,aiis a positive integer, andPir¼1 degðfiÞai¼n.

Now define

uiðtÞ ¼k

Y ffjðtÞ:aj¼ig

fjðtÞ:

HerekAFqis chosen so thatuiðtÞhas constant term 1. SouiðtÞis simply the product

of all irreducible polynomials in the multi-set which have associated exponent equal to i. That this construction gives a one-to-one correspondence with the conjugacy classes of GLnðqÞis the content of Theorem 3.6.

Proposition 3.7. An element gAGLnðqÞis real(resp. z-real)if and only if each of the

polynomials ui in the sequence u¼ ðu1;u2;. . .Þ (associated uniquely to the conjugacy

class of g)are self-reciprocal(resp.z-self-reciprocal).

Proof. Suppose that all of the uiðtÞ are self-reciprocal. This means that if pðtÞ is

a monic irreducible polynomial dividing uiðtÞ, then either pðtÞ ¼ pp~ðtÞ or ~ppðtÞ also

divides uiðtÞ (with the same multiplicity). Each monic irreducible divisor of uiðtÞ

corresponds to a Jordan block within the Jordan canonical form for g. Referring to Proposition 3.5 this means thatgis self-reciprocal.

The converse works the same way: ifgis self-reciprocal then we can apply Propo-sition 3.5; thus if pðtÞ divides uiðtÞ then either pðtÞ ¼ ~ppðtÞ or else pp~ðtÞalso divides

uiðtÞ(with the same multiplicity). This means that theuiðtÞare self-reciprocal.

The same argument applies for the z-real case except that we replace ~ppðtÞin our argument withppðtÞ. r

Hence to count the number of real conjugacy classes in GLnðqÞwe need to count

sequences of polynomials

u¼ ðu1ðtÞ;. . .;uiðtÞ;. . .Þ such thatuiðtÞ ¼aitniþ þa1tþ1

are self-reciprocal polynomials over Fq with constant term 1 satisfying Piini¼n.

Thus for a given partitionnwe writegln¼Qni>0nq;ni for the number of real GLnðqÞ -conjugacy classes of typenin GLnðqÞ. Then we have

Theorem 3.8.The total number of real conjugacy classes inGLnðqÞis

X fn:jnj¼ng gln¼ X fn:jnj¼ng Y ni>0 nq;ni:

Furthermore all real classes inGLnðqÞare,in fact,strongly real.

We have not proved the statement about strong reality; however this follows from Corollary 4.5 which we prove later. In fact Wonenburger [10] has proved that

(12)

reality is equivalent to strong reality in GLnðFÞ for all fields F of characteristic not 2.

Gow (who e¤ectively proved Theorem 3.8 in [2]) makes in [2, Lemma 2.2] the in-teresting observation that the number of equivalence classes of non-degenerate bilin-ear forms of rankn overFq is equal to the number of real classes in GLnðqÞ; hence

Theorem 3.8 also provides a formula for the number of these equivalence classes. Note that the functionð1þtÞð2;q1Þ=ð1qt2Þis a generating function forn

q;d, and so the function Yy r¼1 ð1þtrÞð2;q1Þ 1qt2r

is a generating function for the number of real conjugacy classes in GLnðqÞ.

Finally, forq odd we can also write a formula for the total number ofz-real con-jugacy classes in GLnðqÞ: X fn:jnj¼ng Y ni>0 nq;nisni: 4 SLn(q),n22 (mod 4) orq23 (mod 4)

We count the real conjugacy classes of SLnðqÞ using the correspondence given

by Macdonald, who proves that a GLnðqÞ-conjugacy class of type n¼ fn1;. . .;nrg

contained in SLnðqÞ is the union of hn conjugacy classes for SLnðqÞ where

hn¼ ðq1;n1;. . .;nrÞ; see [6, (3.1)].

Proposition 4.1. Let n be a partition of n. Then the total number of GLnðqÞ-real

GLnðqÞ-conjugacy classes of typencontained inSLnðqÞis

sln¼

Q

ni>0

nq;ni; if q is even or niis zero for i odd; 1

2

Q

ni>0

nq;ni; if q is odd and there exists i with ini odd;

fnðqÞ Q iodd;ni>0 qni=21 Q ieven;ni>0 nq;ni; otherwise: 8 > > > > > < > > > > > :

Here fnðqÞ ¼12ððqþ1Þrþ ðq1ÞrÞwhere r is the number of odd values of i for which

ni>0.

Proof. As outlined in Section 3.2 we write n¼1n12n2. . . with jnj ¼n. Let c be

a conjugacy class of GLnðqÞ of type n given by u¼ ðu1ðtÞ;u2ðtÞ;. . .Þ where

uiðtÞ ¼aitniþ þ1. Suppose that cis real and so uiðtÞis self reciprocal for all i;

this means, in particular, thatai¼G1 for alli. Now detð1tgÞis that scalar

(13)

detð1tgÞ ¼Qid1uiðtÞiand so detg¼ ð1ÞnQni>0aii; see [6, (1.7)]. Thus a real

ele-ment of GLnðqÞmust have determinantG1. If p¼2 this means that all real GLnðqÞ

-conjugacy classes lie in SLnðqÞ.

Ifq is odd andniis zero foriodd thennis even, and detg¼Qni>0a

i

i ¼1; hence,

again, all real GLnðqÞ-conjugacy classes lie in SLnðqÞ.

Now suppose thatq is odd and that there existsisuch thatiniis odd. Lemma 2.1

implies that there are 2qðni1Þ=2possibilities foru

iðtÞ; half of these will haveai¼1, the

other half will haveai¼ 1. Then it is easy to see that exactly half of the sequences

associated toncorrespond to elements with determinant 1.

Finally suppose that q is odd and that ni is even wheneveri is odd, with at least

one such ni>0. Clearly n is even, so we require that Qni>0aii¼1. When i is

even,ai

i ¼1 so we must ensure that

Q

iodd;ni>0aii¼1. So let us suppose, for the

mo-ment, that we are dealing with a sequence of length rof self-reciprocal polynomials

ðui1ðtÞ;. . .;uirðtÞÞwhereij is odd for all jand deguijðtÞis positive and even for all j. Lemma 2.1 implies that the number of such sequences isðqþ1ÞrQiodd;ni>0qni=21. Those sequences which correspond to a matrix with determinant 1 will have an even number of leading coe‰cients1. Counting such sequences is equivalent to counting terms in the expansion ofðqþ1Þrin which 1 turns up an even number of times; thus the number of such sequences is fnðqÞQiodd;ni>0q

ni=21; here

fnðqÞ ¼arqrþar2qr2þar4qr4þ

where ðqþ1Þr¼arqrþar1qr1þ þa1qþa0. It is an easy matter to see that

fnðqÞ ¼12ððqþ1Þrþ ðq1ÞrÞ.

Now, if we return to the case where ni may be non-zero for eveni, it is clear that

there areQieven;ni>0nq;ni polynomials corresponding to eveni; these make no di¤er-ence to the determinant of our element, hdi¤er-ence we obtain the given formula. r

Next we need to know how a real (resp.z-real) conjugacy class of GLnðqÞsplits in

SLnðqÞ.

Lemma 4.2. Let gASLnðqÞ and suppose that the conjugacy class of g in GLnðqÞ is

C¼C1U UCr where the Ci areSLnðqÞ-conjugacy classes. Take xiAGLnðqÞsuch

that gxi AC

ifor i¼1;. . .;r. Then g is real(resp.z-real)inSLnðqÞif and only if gxi is

real(resp.z-real)inSLnðqÞfor all i.

Proof. Suppose that tgt1¼g1 for tASL

nðqÞ. Then txgxðtxÞ1 ¼ ðgxÞ1 for

xAGLnðqÞ. Suppose tgt1¼zg1 for tASLnðqÞ. Then txgxðtxÞ1¼zðgxÞ1 for

xAGLnðqÞ. (Recall that, forgAGLnðqÞ, we writezgwhen we meanðzIÞg.)

In both cases tx is in SL

nðqÞsince SLnðqÞis normal in GLnðqÞ. Hence we obtain

our result. r

For our analysis of the real classes in SLnðqÞ, we need an easy arithmetical lemma.

We writejkj2for the largest power of 2 which divides an integerk. That is,jkj2¼2r

(14)

Lemma 4.3.LetFq be a finite field with q odd. Then there existsaAFq withan¼ 1

if and only ifjnj2<jq1j2.

Proof. Recall that Fq is a cyclic group of order q1. Consider the homo-morphism Fq!Fq, x7!xn. Its kernel is fxAFqjxðn;q1Þ¼1g, hence its image

fxAFqjxr¼1g has r¼ ðq1Þ=ðn;q1Þ elements. Now 1 is contained in this image if and only ifris even or, equivalently, if and only ifjnj2 <jq1j2. r

Proposition 4.4.Suppose that n22ðmod 4Þor that q23ðmod 4Þ. Then a real(resp. z-real)conjugacy class ofGLnðqÞwhich is contained inSLnðqÞis again a union of real

(resp.z-real)conjugacy classes inSLnðqÞ.

Proof.We generalize the proof of Wonenburger [10].

First consider the even characteristic situation. Let g be a real element in

G¼GLnðqÞ (remember that, for even characteristic, z-real elements do not exist)

and consider the groupRGðgÞ, as defined in Section 3. Ifgis not real in SLnðqÞthen

RGðgÞVSLnðqÞcCGðgÞ. This implies that RGðgÞSLnðqÞ is a group which contains

SLnðqÞ with even index. But this is impossible since RGðgÞSLnðqÞ is contained in

GLnðqÞwhich contains SLnðqÞas a subgroup with odd index. We conclude thatgis

indeed real in SLnðqÞ.

From here on we assume that the characteristic is odd. Letd1ðtÞ;. . .;dnðtÞbe the

invariant factors of g in Fq½t. Suppose thatg is real (resp. z-real), so each diðtÞis

self-reciprocal (resp. z-self-reciprocal). The Fq½t-module V which we discussed in

Section 3 decomposes as V ¼0n

i¼1Vi, where each Vi is a cyclic submodule of V. Writegi¼gjVi; theng¼0igiand the characteristic polynomial ofgiis the polyno-mialdiðtÞ.

Step 1. We shall construct involutions in GLðViÞ, conjugating gi to gi1 (resp.

zg1

i ). If dimVi¼2m then this involution will have determinant ð1Þm, while if

dimVi¼2mþ1 then we will construct two such involutions, one of determinant 1

and the other of determinant1. This construction is enough to prove the proposi-tion except when q11ðmod 4Þand n12ðmod 4Þ; we deal with this exception in

Step 2.

When g is real we can write the characteristic polynomial of gi as

wgiðtÞ ¼ ðt1Þrðtþ1ÞsfðtÞ where fðG1Þ00 and Vi¼W1lW1lW0, where

W1, W1 and W0 are the kernels of ðgi1Þr, ðgiþ1Þs and fðgiÞrespectively. To

produce the involution hi on Vi as above, it su‰ces to do so on each ofW1, W1 andW0.

When g is z-real we can write wgiðtÞ ¼ ðt2zÞr

fðtÞ where the degðfÞ ¼2mand

fð0Þ ¼zm; then Vi¼WzlW0, where Wz and W0 are the kernels of t2z and

fðtÞ respectively. To produce the involution hi on Vi as above, it su‰ces to do so

on each ofWzandW0.

It is su‰cient to find a reversing involution in the following situations. Letk be a cyclic linear transformation on a vector space W with characteristic polynomial

(15)

(1) wkðtÞis self-reciprocal (resp.z-self-reciprocal) and the degree ofwkðtÞis even, say 2m. In this case we deal with thez-real, and real, situations simultaneously. We will write the proof for the z-real situation; the proof for the real situation is obtained by replacingzwith 1. We assume thatwkð0Þ ¼zm.

(2) wkðtÞ ¼ ðt1Þ2mþ1orðtþ1Þ2mþ1. (3) wkðtÞ ¼ ðt2zÞm

withmodd.

We claim that in Case 1,kis reversed by an involution with determinantð1Þm; in Case 2 there are reversing involutions with determinant1, and with determinant 1; in Case 3 there is an involution with determinant1.

Case1. SincewkðtÞisz-self-reciprocal, our assumptions imply that

wkðtÞ ¼t2mþa1zt2m1þa2z2t2m2þ þamzmtmþ þa2zmt2þa1zmtþzm: SinceW is cyclic, there is a vectoruAW such thatE¼ fu;ku;. . .;k2m1ugis a basis ofW. By substitutingkmu¼ywe getE¼ fkmy;. . .;y;. . .;km1yg. Let

B¼ fy;ðkþzk1Þy;. . .;ðkm1þzm1kmþ1Þy;ðkzk1Þy;. . .;ðkmzmkmÞyg:

We claim thatBis a basis of W. To see this observe first that, fori¼1;. . .;m1,

ðkiþzikiÞy and ðkizikiÞy span the same 2-dimensional subspace as kiy and

kiy. Also y is independent ofkiy andkiy, hence we need only demonstrate that

ðkmzmkmÞy is linearly independent from the rest. Were this not the case,

how-ever, we would have

kmðyÞ ¼zmkmðyÞ þfðkmþ1;kmþ2;. . .;km1ÞðyÞ

and hence

ðk2mfðk;k2;. . .;k2m1Þ zmÞy¼0;

where f is some linear function. But this implies that

wkðtÞ ¼t2mfðt2m1;t2m2;. . .;tÞ zm

which contradicts the form ofwkðtÞgiven above.

We denote the subspace generated by the firstmvectors ofBbyPand that by the lattermvectors byQ. Now observe some facts:

(1) ðkþzk1ÞðkiþzikiÞ ¼ ðkiþ1þziþ1kðiþ1ÞÞ þzðki1þzi1kði1ÞÞ; (2) ðkzk1ÞðkiþzikiÞ ¼ ðkiþ1ziþ1kðiþ1ÞÞ zðki1zi1kði1ÞÞ; (3) ðkzk1ÞðkizikiÞ ¼ ðkiþ1þziþ1kðiþ1ÞÞ zðki1þzi1kði1ÞÞ; (4) kmþzm km¼ a 1ðkm1þzm1kmþ1Þ a2ðkm2þzm2kmþ2Þ .

(16)

Now (1), (2) and (4) imply thatðkþzk1ÞðPÞJPand thatðkzk1ÞðPÞJQ. If we applykzk1to both sides of (4) we find thatðkmþ1zmþ1km1ÞðyÞAQ. This, along with (2), implies thatðkþzk1ÞðQÞJQ. Similarly, applyingkþzk1 to both sides of (4) we obtainðkmþ1þzmþ1km1ÞðyÞAP. This, along with (3), implies that

ðkzk1ÞðQÞJP. Now leth¼1jPl1j

Qand note that

hðkþzk1Þh¼kþzk1; hðkzk1Þh¼ ðkzk1Þ:

This implies thathis an involution which conjugatesktozk1 and has determinant

ð1Þm.

Case2. In this case, we have the characteristic polynomialwkðtÞ ¼ ðteÞ2mþ1where

e¼G1. SinceW is cyclic, there is a vectoruAWsuch thatE¼ fu;ku;. . .;k2mugis a

basis. By substitutingkmu¼ ywe get

E¼ fkmy;. . .;y;. . .;kmyg:

We consider the basis

B¼ fy;ðkþk1Þy;. . .;ðkmþkmÞy;ðkk1Þy;. . .;ðkmkmÞyg:

We denote the subspace generated by the firstmþ1 vectors ofBbyPand the latter

m vectors by Q. Examining the equation ðkeIÞ2mþ1¼0 and applying kGeI

to both sides yields the following facts: kþk1 leaves P as well as Q invariant; also ðkk1ÞðPÞJQ and ðkk1ÞðQÞHP. We consider h¼1jPl1j

Q and

h0¼ 1j

Pl1jQ. Thenh andh0 are both involutions which conjugatek tok1 and

they have determinantsð1Þmandð1Þmþ1 respectively.

Case3. In this case, we have the characteristic polynomialwkðtÞ ¼ ðt2zÞm

wherem

is odd. SinceW is cyclic, there is a vectoruAW such thatE¼ fu;ku;. . .;k2m1ugis a basis. By substitutingkmu¼ ywe get

E¼ fkmy;. . .;y;. . .;km1yg:

We consider the basis

B¼ fy;ðkzk1Þy;ðk2þz2k2Þy;ðk3z3k3Þy;. . .;ðkm1þzm1kmþ1Þy;

ðkþzk1Þy;ðk2z2k2Þy;ðk3þz3k3Þy;. . .;ðkmþzmkmÞyg:

We denote the subspace generated by the first m vectors of B by P and the latter m vectors by Q. This time kzk1 leaves P as well as Q invariant. Also

ðkþzk1ÞðPÞJQandðkþzk1ÞðQÞJP. Defineh¼1j

Pl1jQ and observe that

(17)

Step 2. Write g¼0

igi, as above; the construction outlined in Step 1 yields a

reversing involution, h¼0

ihi, where hi is a reversing involution for gi. Whenever

n22ðmod 4Þorqis even, it is easy to see that the construction in Step 1 yields an

involution h which has determinant 1. We must deal with the remaining situation: whenn12ðmod 4Þandq11ðmod 4Þ.

Suppose that one of the submodulesVi has odd dimension. Then we can choosehi

to have determinant 1 or1; this ensures that we can choosehto have determinant 1 and we are done.

Suppose that all submodulesVihave even dimension, and that the reversing

invo-lution h¼0

ihi constructed in Step 1 has determinant 1. Clearly one of the

sub-modules,Vjsay, must have dimensionnj12ðmod 4Þ. Now Lemma 4.3 implies that

there is anaAFqsuch thatanj ¼ 1. Then we can define

h0¼ 0 n i¼1;i0j hi lhjðaIÞ:

Clearlyh0is a reversing element forgand deth0¼1 as required. r

Corollary 4.5.If g is real(resp.z-real)inGLnðqÞthen g is strongly real(resp. strongly

z-real) in hSLnðqÞ;ki for k an element of determinant 1. If, furthermore,

n22ðmod 4Þor q is even,then an involution h exists inSLnðqÞsuch that hgh¼g1.

Proof.Takegreal (resp.z-real) in GLnðqÞ. Note that in the proof of Proposition 4.4,

we did not use the fact that g is contained in SLnðqÞ and, in all cases, we found a

reversing element of determinant 1 or 1. In fact, in the odd characteristic case, we always found a reversing element which was an involution; this proves the first state-ment when the characteristic is odd.

Forn22ðmod 4Þandqodd, we were able to do better: we found an involution

in SLnðqÞwhich reversesg; this proves the second statement when the characteristic

is odd. The only thing left to prove is the following: if g is real in GLnðqÞ with q

even, then g is strongly real in GLnðqÞ. (This statement is strong enough because,

when q is even, there are no z-real elements, and all involutions of GLnðqÞ lie in

SLnðqÞ.)

Take g real in GLnðqÞwith q even. We refer to Proposition 3.5 and writeg as a

block matrix as follows:

0 p0~pp ðJl;plJl;~ppÞ l 0 p¼pp~ Jl;p :

We will consider two cases: (1) g¼Jr;plJr;~pp;

(18)

In both cases p is an irreducible polynomial. Clearly if we can find an involution which reverses g in both of these cases, then we can build an involution which re-versesgin general.

Consider the first case. Thengis conjugate to a matrix of form

g1¼

B 0 0 B1

;

whereBis a squaredd matrix withd ¼degðpÞ. Thushg1h1¼g11 where

h¼ 0 I

I 0

:

Sinceh2¼1, we conclude thatg

1is strongly reversible and thus so isg.

Now consider the second case. Ifp¼tþ1 thengis an involution and we are done. Thus we assume that phas even degree,d. Letg¼gsgube the Jordan decomposition.

Since g is real, Lemma 3.1 implies that gs is real. NowCGLnðqÞðgsÞGGLrðq

dÞ. We

denote byxthe element which reversesg; it acts as an involutory field automorphism on GLrðqdÞ.

We can think ofg as lying inside GLrðqdÞ; then g is conjugate to an elementg1 where g1¼ a 1 . . . . . . . . . 1 a 0 B B B B @ 1 C C C C A; g 1 1 ¼ a1 a2 a3 . . . . . . . . . . . . . . . a3 . . . a2 a1 0 B B B B B B B @ 1 C C C C C C C A ;

where a is an element ofFqd which satisfiesaq d=2 ¼a1. Theng 1 is reversed by an elementhswhere h¼ 1 0 0 a2 a3 a4 a4 2a5 3a6 a6 3a7 . . . 0 B B B B B B @ 1 C C C C C C A ; i:e: hij¼ ð1Þ jþ1 j2 i2 aiþj2; jdi; 0; j<i:

Here we define k0 ¼1 and k1 ¼ 1 for k a positive integer (of course, since the characteristic is even, hij¼0 for many i, j). Now ðhxÞ2 ¼hhx acts trivially on

GLrðqdÞ. But this means thatðhxÞ2AZðGLrðqdÞÞ. SinceZðGLrðqdÞÞhas odd order

we conclude thathxcan be chosen so that ðhxÞ2¼1. Henceg1 is strongly real and thus so isg. r

(19)

We summarize our results for real elements with the following theorem:

Theorem 4.6. Suppose that n22ðmod 4Þ or q23ðmod 4Þ. Then the total number

of real conjugacy classes in SLnðqÞ is equal to Pjnj¼nhnsln where n¼ fn1;. . .;nrg,

hn¼ ðq1;n1;. . .;nrÞand the value of slnis given in Proposition4.1. Furthermore,if

n22ðmod 4Þor q is even,this is the same as the total number of strongly real conju-gacy classes inSLnðqÞ.

5 SLn(q),n12 (mod 4) andq13 (mod 4)

In this section we assume thatn12ðmod 4Þandq13ðmod 4Þ, and we takez¼ 1,

a non-square. The formula given in Theorem 4.6 gives the number of conjugacy classes in SLnðqÞ which are real in GLnðqÞ. Thus, in order to count the number

of real conjugacy classes in SLnðqÞ, we take this formula and count how many

GLnðqÞ-real conjugacy classes in SLnðqÞ fail to be real in SLnðqÞ. Our analysis is

based on the following:

Lemma 5.1.Let gASLnðqÞ. Then g is real(resp.z-real)inSLnðqÞif and only if there

exists a reversing element hAGLnðqÞsuch thatdeth is a square inFq.

Proof.Clearly ifgis real (resp.z-real) in SLnðqÞthen there exists a reversing element

hin SLnðqÞand dethis a square inFq.

Now for the converse: suppose that there exists a reversing element hAGLnðqÞ

such that deth is a square. For a positive integercwe have dethc¼ ðdethÞc and, if

cis odd, thenhcghc¼hgh1 and sohcis a reversing element. Since dethc¼1 for some odd integerc, we are done. r

Now let g be a real (resp. z-real) element in GLnðqÞ with reversing element

hAGLnðqÞ. In view of Lemma 5.1 we will be interested in determining whether

dethis a square or a non-square inFq. We will use two commutative diagrams:

GLaðFqÞ ! i GLaðFqÞ det ? ? ? y ? ? ? ydet Fq ! i Fq GLaðqdÞ ! i GLadðqÞ det ? ? ? y ? ? ? ydet Fqd ! N F q:

The mapidenotes a natural inclusion map. The mapNis the norm map defined as follows:

N:Cqd1!Cq1; x7!xq

d1þþqþ1

:

The commutativity of the first diagram is obvious. The commutativity of the second is explained by the following fact: Ndet:GLaðqdÞ !Fq, when viewed as a map

(20)

from a subgroup of GLadðqÞ, is multilinear, alternating (on columns) and satisfies

ðNdetÞðIÞ ¼1; in other words it is a determinant and so must coincide with deti:GLaðqdÞ !Fq by [5, Proposition 4.6, p. 514].

Thanks to the given inclusion maps, there are several determinant maps applicable to any given matrix. In what follows we write detjkjto specify the fieldkin which our

image lies. Note that the form ofNimplies that, foryAGLaðqdÞ, detqðyÞis square if

and only if detqdðyÞis square.

We will build a reversing element forg, in a similar way to the proof of Corollary 4.5, by considering three basic cases. Let V be theFq½t-module associated with V

(see Section 3.1) and letwgðtÞbe the characteristic polynomial ofg. (1) Vis a cyclic module forgwithwgðtÞ ¼ pðtÞ ¼ ðtG1Þa.

(2) Vis a cyclic module forgwithwgðtÞ ¼ pðtÞa, wherepðtÞis a self-reciprocal (orz -self-reciprocal) irreducible polynomial of even degreed.

(3) V¼WplWq, a module for g, such that Wp andWq are cyclic modules with

characteristic polynomials pðtÞa and pp~ðtÞa (or pðtÞa andppðtÞa) such that pðtÞis irreducible.

Lemma 5.2. Let V be a cyclic module associated with g such that wgðtÞ ¼ ðtG1Þa.

Suppose that h reverses g.

(1) If a is odd,thendeth may be a square or a non-square inGLaðqÞ.

(2) If a10ðmod 4Þ,thendeth is a square inGLaðqÞ.

(3) If a12ðmod 4Þ,thendeth is a non-square inGLaðqÞ.

Proof.Writegin upper triangular form. Takehsuch thath1gh¼g1; thenhmust be upper triangular with diagonalða;a;a;a;. . .Þ. The result follows by consider-ing the determinant. r

Lemma 5.3.Let V be a cyclic module associated with g such thatwgðtÞ ¼ pðtÞa where pðtÞis an irreducible polynomial of even degree d. Suppose that h reverses g. If a is odd,

thendeth may be a square or non-square inGLadðqÞwhile,if a is even,thendeth is a

square inGLadðqÞ.

Proof.Writeg¼gsgufor the Jordan decomposition ofg. In GLdðqÞ,gsis centralized

by GLaðqdÞand sogscan only be reversed by a field automorphism of GLaðqdÞ. Let

xbe such a field automorphism; thenh¼yxwhere yis an element of GLaðqdÞwhich

satisfiesðguÞ1¼ ðyxÞðguÞðyxÞ1 (see Lemma 3.1).

Ifais odd then there is a central element in GLaðqdÞwhich has determinant a

non-square inFqd. Hence dethmay be a square or a non-square in GLadðqÞ.

Assume thatais even. Proposition 4.4 implies that there exists a reversing element

hin GLadðqÞsuch that dethis a square. Any other reversing element in GLadðqÞmust

equalhzwherezACGL

(21)

thatzmust be an element in GLaðqdÞwhich centralizesgu. We writeguas an element of GLaðqdÞ: g¼ 1 a . . . . . . . . . a 1 0 B B B B @ 1 C C C C A; whereaAF

qd. Thenzmust have the form

z¼ b1 b2 .. . . . . . . . . . . b2 b1 0 B B B B B @ 1 C C C C C A :

Hence, since a is even, detqdz is a square. Thus any reversing element for g in GLadðqÞhas determinant a square. r

Lemma 5.4.Let V ¼WplWq be the module associated with g. Suppose that Wpand

Wq are cyclic modules with characteristic polynomials pðtÞaand ~ppðtÞa(orppðtÞaÞ.

Sup-pose that pðtÞis irreducible of degree d,and that h reverses g. If a is odd thendetðhÞ

may be square or non-square while,if a is even,thendetðhÞis square.

Proof.We know thatgis conjugate in GL2adðqÞto either

g1 ¼ B 0 0 B1 or g2 ¼ B 0 0 zB1 :

By Lemma 4.2 we can assume thatg is equal tog1 org2. Ifa is odd then GLaðqdÞ

contains a central element with non-square determinant inFqd; sincegis central in a group isomorphic to GLaðqdÞ GLaðqdÞwe conclude that detðhÞmay be square or

non-square.

Suppose thatais even. Ifhpreserves blocks then p¼pp~(orp¼pp) and Lemma 5.3 implies that detðhÞis a square. Otherwisehreverses blocks and

h¼ 0 X

Y 0

for someX andY in GLadðqÞ. Thus

0 X Y 0 B 0 0 C 0 X Y 0 1 ¼ C 0 0 B ;

(22)

where C¼B in the real case, and C¼zB1 in the z-real case. This implies that

XCX1¼CandYBY1¼B; nowCGLadðqÞðBÞ ¼CGLadðqÞðCÞ, hence we need to ex-amine the centralizer ofBin GLadðqÞ.

Clearly detqh¼ ð1ÞadðdetXÞðdetYÞ ¼ ðdetXÞðdetYÞ since a is even. Then X

must lie in the centralizer ofgujWq and we have seen the form of such a centralizer in Lemma 5.3; we know that the determinant is always a square inFqd hence is a square inFq. r

This concludes our treatment of the three specific cases. We use the lemmas to build up the picture for generalV.

Proposition 5.5.Suppose that g lies in SLnðqÞwith n12ðmod 4Þand q13ðmod 4Þ.

Suppose that the corresponding module V for g splits into r cyclic submodules with cor-responding polynomials p1ðtÞa1;. . .;prðtÞar. If g is real in GLnðqÞ, then g is real in

SLnðqÞ if and only if ai is odd for some i. If g is z-real in GLnðqÞ then g is z-real

inSLnðqÞ.

Proof. Suppose first that g is real in GLnðqÞ. Leth be any element of GLnðqÞ such

thathgh1 ¼g1. We can breakVup into submodules of the three types listed above (call these h-minimal). If ai is odd for some i then considerW, the h-minimal

sub-module corresponding toai. Lemmas 5.2 to 5.4 imply thatgjW is conjugate to its

in-verse by elements of determinantþ1 and1 in GLðWÞ; this property will then hold forg.

Conversely if allaiare even then our above calculations show that, restricted to any

h-minimal submodule W, gjW is conjugate to its inverse by an element of

determi-nantþ1 or1 in GLðWÞ, but not both. In fact, of the three types listed above, this determinant is1 only whenWis cyclic andðpiðtÞÞai ¼ ðtG1Þaiwithai12ðmod 4Þ

(this is also the only time when dimW20ðmod 4Þ).

Sincen12ðmod 4Þthere will be an odd number of these1h-minimal

submod-ules, thereby ensuring thatgis only conjugate to its inverse by an element of determi-nant1.

Now suppose thatgisz-real and allaiare even. This implies that the dimension of

allh-minimal submodules is divisible by 4. Sincen12ðmod 4Þthis is a

contradic-tion. Thus ai is odd for some i. Let W be the h-minimal submodule

correspond-ing toai. Once again, Lemmas 5.2 to 5.4 imply thatgjW is conjugate to its inverse

by elements of determinant þ1 and 1 in GLðWÞ; this property will then hold forg. r

Corollary 5.6.Suppose that gASLnðqÞis of typen¼1n12n2. . .and is real inGLnðqÞ.

Then g is real inSLnðqÞif and only if ni>0for some odd i.

Proof. We simply need to convert the criterion given by Proposition 5.5 into the language of Macdonald, as described in Section 3.2. r

(23)

Theorem 5.7.Suppose that n12ðmod 4Þand q13ðmod 4Þ. Then the total number

of real conjugacy classes inSLnðqÞis equal to

X jnj¼n hnsln X jmj¼n hmslm

wheren¼ ð1n12n2. . .Þ,m¼ ð2d24d4. . .Þand the values of sl

nand slmare given in

Prop-osition4.1.

6 Strongly real conjugacy classes inSLnðqÞ

Theorem 6.1.Let g be an element ofSLnðqÞwhich is real inGLnðqÞ. Let g be of type

n¼1n12n2. . .,with associated self-reciprocal polynomials u

iof degree ni.

(1) If n22ðmod 4Þor if q is even then g is real as well as strongly real inSLnðqÞ.

(2) If n12ðmod 4Þand q is odd then,g is strongly real inSLnðqÞif and only if there is

an odd i for whichG1appears as a root of uiðtÞ.

Proof. The first statement follows directly from Corollary 4.5. Now suppose that

n12ðmod 4Þandqis odd.

Takeg a strongly real element in GLnðqÞ. We use the same notation as the

previ-ous section except that this time we require thath2¼1. LetW be ah-minimal sub-module ofV;W will have one of the same three types as before.

Suppose that W¼WplWq where Wp and Wq are cyclic with corresponding

characteristic polynomials pðtÞa and ~ppðtÞa such that pðtÞand pp~ðtÞ are distinct irre-ducible polynomials of degreed. Then

hjW ¼ 0 X

Y 0

for someXandYin GLadðqÞand dethjW ¼ ð1Þ adð

detXÞðdetYÞ. But, sinceh2¼1, we haveX ¼Y1and dethjW ¼ ð1Þad.

Suppose thatWis cyclic andd >1. Write the corresponding characteristic polyno-mial as pðtÞawhere pðtÞhas even degreed. We can considerWn

FqFq. ThengjW is

clearly real in GLðVn

FqFqÞbut is reducible. In fact there are pairings of blocks. As we have already seen, this implies that we have determinant ð1Þb over these pair-ings, wherebis the size of the block. Thushmust have determinantð1Þad=2.

Suppose thatW is cyclic and pðtÞ ¼tG1. It is easy to check that

(1) ifais odd then dethjW may be 1 or1, (2) ifa10ðmod 4Þthen dethj

W ¼1,

(3) ifa12ðmod 4Þthen dethj

(24)

We have now treated the three types. IfV does not have a cyclic submodule corre-sponding to a polynomial ðtG1Þa wherea is odd, then dethj

W ¼ ð1Þ

ðdimWÞ=2 and so deth¼ ð1Þn=2¼ 1; in particularg is not strongly real in SLnðqÞ. On the other

hand if V does have a cyclic submodule corresponding to a polynomial ðtG1Þa

whereais odd, then we can choosehto have deth¼1.

The proof is completed once we observe that V has a cyclic submodule corre-sponding to a polynomialðtG1Þawhere ais odd if and only if there is an oddifor whichG1 is a root ofuiðtÞ. r

Acknowledgement. Part of this work was completed when both authors were post-doctoral fellows at IMSc, Chennai. The first author is grateful to Ian Short, and others at the National University of Ireland, Maynooth, who took an interest in this work. Both authors would like to thank Rod Gow, Bill Kantor, and Amritanshu Pra-sad for helpful comments, and in addition the authors owe a large debt of gratitude to the anonymous referee; their insight has been invaluable.

References

[1] N. Gill and A. Singh. Real and strongly real classes in PGLnðqÞand quasi-simple covers

of PSLnðqÞ.J. Group Theory

[2] R. Gow. The number of equivalence classes of nondegenerate bilinear and sesquilinear forms over a finite field.Linear Algebra Appl.41(1981), 175–181.

[3] N. Jacobson.Lectures in abstract algebra, vol. 2.Linear algebra(van Nostrand, 1953). [4] P. Kleidman and M. Liebeck.The subgroup structure of the finite simple groups. London

Math. Soc. Lecture Note Ser. 129 (Cambridge University Press, 1990).

[5] S. Lang.Algebra, 3rd edn. Graduate Texts in Math. 211 (Springer-Verlag, 2002). [6] I. G. Macdonald. Numbers of conjugacy classes in some finite classical groups. Bull.

Austral. Math. Soc.23(1981), 23–48.

[7] A Singh and M. Thakur. Reality properties of conjugacy classes in algebraic groups.

Israel J. Math.145(2005), 157–192.

[8] T. A. Springer. Linear algebraic groups. In Algebraic Geometry IV, Encyclopaedia of Math. Sci. 55 (Springer-Verlag, 1994), pp. 1–121.

[9] P. H. Tiep and A. E. Zalesski. Real conjugacy classes in algebraic groups and finite groups of Lie type.J. Group Theory8(2005), 291–315.

[10] M. J. Wonenburger. Transformations which are products of two involutions.J. Math.

Mech.16(1966), 327–338.

Received 20 January, 2009; revised 12 June, 2010

Nick Gill, Department of Mathematics, University Walk, Bristol, BS8 1TW, United Kingdom E-mail: nickgill@cantab.net

Anupam Singh, IISER, Central Tower, Sai Trinity Building, near Garware Circle, Pashan, Pune 411021, India

References

Related documents

All of the participants were faculty members, currently working in a higher education setting, teaching adapted physical activity / education courses and, finally, were

South European welfare regimes had the largest health inequalities (with an exception of a smaller rate difference for limiting longstanding illness), while countries with

An analysis of the economic contribution of the software industry examined the effect of software activity on the Lebanese economy by measuring it in terms of output and value

According to the findings on objective three, the statutory protection to the right to privacy against mobile phone usage does not provide direct clue as majority of the

This suggest that developed countries, such as the UK and US, have superior payment systems which facilitate greater digital finance usage through electronic payments compared to

For the broad samples of men and women in East and West Germany every first graph in Fig- ure 1 to 4 (Appendix) shows considerable and well-determined locking-in effects during the

○ If BP elevated, think primary aldosteronism, Cushing’s, renal artery stenosis, ○ If BP normal, think hypomagnesemia, severe hypoK, Bartter’s, NaHCO3,

The goal of the proposed model is to help engineering students develop an understanding of the effect of manufacturing technologies and transportation modes on the carbon footprint