Sizes of Conjugacy Classes of GL(n, q)

5.2 Conjugacy Classes of GL(n, q)

5.2.2 Sizes of Conjugacy Classes of GL(n, q)

T C T

Table 5.1: Number of types of classes and number of classes ofGL(n, q)

n t(n) c(n, q)

1 1 q−1

2 4 q²−1

3 8 q³−q

4 22 q⁴−q

5 42 q⁵−(q²+q−1) 6 103 q⁶−q² 7 199 q⁷−(q³+q²−1)

Definition 5.2.2. Let cbe a conjugacy class given by ({f_i},{d_i},{z_i},{ν_i}) with lengthk, then 1. c is called primary class if and only if k= 1.

2. c is called regular classif and only if l(ν_i)≤1, ∀ 1≤i≤k.

3. c is called semisimple class if and only if l(ν_i⁰)≤1, ∀ 1≤i≤k.

4. c is called regular semisimple class if it is both regular and semisimple. Alternatively a class is regular semisimple if and only if νi = 1, ∀ 1≤i≤k.

Note 5.2.2. 1. The definition ofcbeing a primary class implies that forg∈c,the characteristic polynomial of g is f(t) = (t^d+ad−1t^d−1 +· · ·+ 1)^s for some s and hence ∂f = d and d|n.

In particular if f(t) = (t−1)ⁿ, then we call c a unipotent. Note that we have defined in Definition 3.1.5 the unipotency of an elementA∈GL(n, q) and of a subgroupH≤GL(n, q).

2. The definition ofcbeing a regular semisimple class ofGL(n, q) implies that any elementg∈c hasndistinct eigenvalues.

We classify all classes of GL(2, q), GL(3, q) and GL(4, q) according to Definition 5.2.2. These classes have been given in Tables 4.1, 5.3 and 6.10 respectively.

Chapter 5 — The Character Table of GL(n, q)

Table 5.2: Conjugacy classes of GL(2, q), GL(3, q) and GL(4, q)

n Primary Classes Unipotent Classes Regular Classes Semisimple Classes Regular Semisimple Classes 2 T⁽¹⁾, T⁽²⁾, T⁽⁴⁾ T⁽¹⁾, T⁽²⁾, α= 1 T⁽²⁾,T⁽³⁾, T⁽⁴⁾ T⁽³⁾, T⁽⁴⁾ T⁽³⁾, T⁽⁴⁾

3 T⁽¹⁾,T⁽²⁾, T⁽¹⁾, T⁽²⁾, T⁽³⁾, T⁽⁴⁾, T⁽⁶⁾, T⁽⁵⁾, T⁽⁶⁾ T⁽⁶⁾, T⁽⁷⁾ T⁽³⁾, T⁽⁸⁾ T⁽³⁾, α= 1 T⁽⁷⁾, T⁽⁸⁾ T⁽⁷⁾, T⁽⁸⁾ T⁽⁸⁾

T⁽¹⁾, T⁽²⁾ T⁽¹⁾, T⁽²⁾ T⁽⁵⁾, T⁽⁸⁾ T⁽¹⁾, T⁽⁶⁾ T⁽¹⁶⁾, T⁽¹⁷⁾ T⁽³⁾, T⁽⁴⁾ T⁽³⁾, T⁽⁴⁾ T⁽¹¹⁾, T⁽¹³⁾ T⁽⁹⁾, T⁽¹²⁾ T⁽¹⁸⁾, T⁽²¹⁾ 4 T⁽⁵⁾,T⁽¹⁹⁾ T⁽⁵⁾, α= 1 T⁽¹⁵⁾,T⁽¹⁶⁾ T⁽¹⁴⁾, T⁽¹⁶⁾ T⁽²²⁾

T⁽²⁰⁾,T⁽²²⁾ T⁽¹⁷⁾,T⁽¹⁸⁾ T⁽¹⁷⁾, T⁽¹⁸⁾ T⁽²⁰⁾,T⁽²¹⁾ T⁽¹⁹⁾, T⁽²¹⁾

T⁽²²⁾ T⁽²²⁾

Two modules (V,Ω) and (V⁰,Ω⁰) are said to be isomorphic orequivalent if and only if V ∼=V⁰ and Ω and Ω⁰ generate the same ring of endomorphisms ofV.

For anyn×nmatrix AoverFq (not necessarily invertible), we define the moduleV_Aof A to be V_A= (V(n, q), R),

whereR=hA,Fqi is the ring generated by A,together with scalars from Fq.That is R=

( _k X

i=0

aiAⁱ|ai ∈Fq

)

∼=Fq[t]

and Fq[t] operates onV(n, q) byt.v =Av, ∀v∈V(n, q).Note that t^j is the composition of ttaken j times. Thus any A∈Mn×n(Fq) defines anFq[t]−module, which we denote by V_A.

Definition 5.2.4. A function f :V_A −→ V_B is said to be an Fq[t]−isomorphism if it is homo- morphism and bijection. The modules VA and VB are called Fq[t]−isomorphic.

Lemma 5.2.3. Let V and W be vector spaces over Fq and let T :V −→V and S :W −→ W be linear transformations that determineFq[t]−modulesVT andVS respectively. A functionf :VT −→

V_S is an Fq[t]−homorphism if and only if

1. f is linear transformation of the vector spacesV and W, 2. f(T v) =S(f(v)), ∀v∈V.

PROOF. See Rotman [65].

T C T

Proposition 5.2.4. Two matricesAandB are similar if and only if the correspondingFq[t]−modules VA andVB are isomorphic.

PROOF. LetT, S :V −→V be linear transformations affordingAandB respectively and also letVT

and V_S be the correspondingFq[t]−modules defined byT and S respectively. Suppose that A and B are similar matrices. Hence there exists P ∈GL(n, q) such that B =P AP⁻¹. Iff :V −→V is the linear transformation corresponds toP,then we claim thatf is an Fq[t]−isomorphism between V_T andV_S.From Lemma 5.2.3, it suffices to show thatf(T v) =S(f(v)), ∀v∈V, i.e.,f T =Sf.In terms of matrices, this representsP A=BP,which we have. Thus VA∼=VB.

Conversely, suppose that f : V −→ V is an Fq[t]−isomorphism between V_T and V_S. By Lemma 5.2.3, we have Sf =f T.Since f is an isomorphism, it follows thatS =f T f⁻¹.IfP is the matrix corresponding to the linear transformation f, thenB =P AP⁻¹; that is A and B are similar ma-

trices. This completes the proof.

IfA, B∈GL(n, q) are in a conjugacy classc,then by Proposition 5.2.4 we haveV_A∼=V_B.It follows that we can writeVcin place ofVAwithout any ambiguity. Next we review some notions from the elementary Ring Theory to learn more about the structure of V_c.

We recall that aprincipal ideal domain Ris an integral domain such that all its ideals are principal ideals. That is if I is an ideal of R, then I =hai = aR fora ∈R. For any v ∈V, where V is an R−module, the annihilator Ann(v) is defined to be the set

Ann(v) ={r ∈R|rv= 0_V}.

It is not difficult to see thatAnn(v) E

|{z}

ideal

R. IfR is a principal ideal domain, then

Ann(v) ={ar|r∈R}=aR, for somea∈R.

Moreover, if p is an irreducible element of R (has no divisors in R except p and 1R), then an R−module V is called ap−primary if for all v∈V,

Ann(v) =hp^αi=p^αR, for someα∈N.

Theorem 5.2.5. LetR be a principal ideal domain andV be a finitely generatedR−module. Then V =

s

M

i=1

V_i,

where each Vi is cyclic submodule and isomorphic to either R or R/p^mR, for some irreducible element p of R. Moreover, the decomposition is unique up to the order of the factors.

Chapter 5 — The Character Table of GL(n, q)

PROOF. See Rotman [65].

We would like to apply the above discussion to the case R = Fq[t], which it can be shown that it is a principal ideal domain. Thus for the annihilator of v of the Fq[t]−modulesVc we take the fixed element to be a monic polynomial of smallest degree in the ideal, that is if v ∈ V_c, then Ann(v) =hfi,wheref is a monic polynomial such that∂f ≤∂g, ∀g∈Fq[t].Letfi ∈ F.ByVhf_ii, we mean thef_i−primary submodule ofV_c; that is the submodule consisting of allv ∈V_cannihilated by some power off_i.The submodules V_hf1i, V_hf2i,· · · , V_hfki of V_c are referred as thecharacteristic submodules since f1, f2,· · ·, f_k are the irreducible factors which appear in the characteristic poly- nomial of an element in the conjugacy classc.Thus givingV_hfiithe name characteristic submodule becomes more appropriate.

If c= ({f_i},{d_i},{z_i},{ν_i}),then by Theorem 5.2.5 we haveV_c=

k

M

i=1

V_hfii,where each V_hfii is of the form

Vhf_ii=

l(νi)

M

j=1

Fq[t]/hf_ii^νij

and νi ={ν_i1, νi2,· · · , νi_l(νi)}is the partition associated withfi inc.Therefore

V_c=

k

M

i=1

V_hfii=

k

M

i=1 l(νi)

M

j=1

Fq[t]/hf_ii^νij. (5.6)

It has been shown in Lemma 2.1 of Green [27] that if Aut(Vc) is the automorphism group of Vc, then

Aut(V_c) =

k

O

i=1

Aut(V_hfii). (5.7)

Now by equation (2.6) of MacDonald [50] we have

|Aut(V_hfii)|=q^{di(|νi|+2n(νi))}φνi(q^−di).

Consequently

|Aut(V_c)|=|

k

O

i=1

Aut(V_hfii)|=

k

Y

i=1

|Aut(V_hfii)|=

k

Y

i=1

q^{di(|νi|+2n(νi))}φ_νi(q^−di). (5.8)

The following Theorem is of great importance and is the main theorem of this subsection. It characterizesC_GL(n,q)(A).

Theorem 5.2.6. Let A∈GL(n, q) lies in a conjugacy class c.Then C_GL(n,q)(A) =Aut(V_c).

T C T

PROOF. Suppose that σ∈Aut(V_c).Thenσ :V_c−→V_cand

σ(ru+sv) =rσ(u) +sσ(v), ∀r, s∈R=hA,Fqi, u, v ∈V(n, q).

We know thatAut(V(n, q)) =GL(n, q) andσ(Au) =Aσ(u) (since A is regarded as a scalar from the ring R). Thus

σ(Au) =Aσ(u) =⇒ (σA)u= (Aσ)u, ∀u∈V(n, q)

=⇒ Aσ=σA

=⇒ σ ∈C_GL(n,q)(A)

=⇒ Aut(V_c)⊆C_GL(n,q)(A).

Conversely, ifσ ∈C_GL(n,q)(A),then

Aσ=σA =⇒ (Aσ)u= (σA)u, ∀u∈V(n, q),

=⇒ σ(Au) =Aσ(u)

=⇒ σ ∈Aut(Vc)

=⇒ C_GL(n,q)(A)⊆Aut(V_c).

Hence C_GL(n,q)(A) =Aut(V_c).

Now in terms of equation (5.8) and Theorem 5.2.6 ifA∈GL(n, q) lies inc= ({f_i},{d_i},{z_i},{ν_i}), then we deduce that

|C_GL(n,q)(A)|=

k

Y

i=1

q^{di(|νi|+2n(νi))}φνi(q^−di). (5.9) It follows that

|C_A|= (

n−1

Y

s=0

(qⁿ−q^s))/

k

Y

i=1

q^{di(|νi|+2n(νi))}φ_νi(q^−di). (5.10)

Sometimes we may writea_νi to denote q^{di(|νi|+2n(νi))}φ_νi(q^−di).That is|C_GL(n,q)(A)|=

k

Y

i=1

a_νi. Corollary 5.2.7. Two conjugacy classes of the same type have same size.

PROOF. Suppose thatc1 = ({f_i},{d_i},{z_i},{ν_i}) and c2 = ({f_i⁰},{d⁰_i},{z⁰_i},{ν_i⁰}) are two classes of the same type with length k (see Remark 5.2.2). It follows by (5.5) that there exists σ ∈S_k such thatz⁰_i =z_σ(i), d⁰_i=d_σ(i) andν_i⁰ =ν_σ(i), ∀1≤i≤k.IfA1 ∈c1 and A2 ∈c2,then by (5.9) we have

|C_GL(n,q)(A₂)| =

k

Y

i=1

q^d

0

i(|ν⁰_i|+2n(ν_i⁰))φ_ν⁰

i(q^−d

0 i) =

k

Y

i=1

q^{dσ(i)(|νσ(i)|+2n(νσ(i)))}φ_νσ(i)(q^−dσ(i))

=

k

Y

i=1

q^{di(|νi|+2n(νi))}φνi(q^−di) =|C_GL(n,q)(A1)|.

Chapter 5 — The Character Table of GL(n, q)

The result follows by (5.10).

Remark 5.2.2. Note thatkis the length of the data parameterizingc.The termlength of conjugacy class has another meaning.