Regular Semisimple Elements and Primary Classes of GL(n, q)

5.2 Conjugacy Classes of GL(n, q)

5.2.3 Regular Semisimple Elements and Primary Classes of GL(n, q)

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.

T C T

c₁ = ({f_i},{d_i},{1}_{k times},{1}_ktimes). Therefore we have

k

X

i=1

d_i = n, that is (d₁, d₂,· · ·, d_k) ` n.

If c₂ = ({f_i⁰},{d⁰_i},{1}_ktimes,{1}_ktimes) is any other regular semisimple class of same type of c1, then d⁰_i = d_σ(i), for some σ ∈ Sk. Thus c1 and c2 determine the same partition. Hence any type of regular semisimple classes determines a partition of n. Conversely, any partition λ = (λ1, λ2,· · · , λk) ` n defines a type of regular semisimple classes, where a typical class c will have the form c= ({f_i},{λ_i},{1}_{k times},{1}_ktimes),1≤i≤k.Note that for anyλi ∈N,there exists an irreducible polynomial of degreeλ_i overFq.Hence types of regular semisimple classes are in one to one correspondence with the partitions ofnas claimed.

It turns out that we may denote any type of regular semisimple classes of GL(n, q) by T^λ and a typical class by c^λ without any ambiguity.

To count the number of regular semisimple conjugacy classes contained in each typeT^λ, λ`n,we put into our consideration the repetition of the parts ofλ.Therefore if we letr_i be the multiplicity of the integer i, then we can write λin the form λ= 1^r12^r2· · ·n^rn,where ri ∈ N∪ {0}. We have the following lemma.

Lemma 5.2.9. Let f(t) =

m

X

i=0

a_itⁱ∈Fq[t], a_m = 1.If α is a root of f, then α^q, α^q2,· · ·, α^qm−1 are the other roots.

PROOF. The Galois group Γ = Γ(Fq^m : Fq) is a cyclic group of order m and is generated by the Frobenius automorphism σ_q : a 7−→ a^q, ∀a ∈ Fq^m. We can see clearly that σ_q^j =σ_qj.Now given that α is a root of f, then

m

X

i=0

a_iαⁱ = 0. Acting by elements of Γ on both sides of the preceding equality we get, for allj= 0,1,· · · , m−1,

σ_q^j(

m

X

i=0

aiαⁱ) =σ_q^j(0) ⇐⇒

m

X

i=0

σ_q^j(aiαⁱ) = 0⇐⇒

m

X

i=0

aiσ_q^j(αⁱ) = 0

⇐⇒

m

X

i=0

a_iα^qji = 0⇐⇒

m

X

i=0

a_i(α^qj)ⁱ = 0.

The last equality tells thatα^qj is a root off whenever α is.

Proposition 5.2.10. The number of regular semisimple classes of type λ, which we denote by F(λ),is given by

F(λ) =

n

Y

i=1 ri−1

Y

s=0

(Ii(q)−s)

n

Y

i=1

ri! ,

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

• I_i(q) = ¹_i X

d|i

µ(d)q^di is the number of irreducible polynomials of degreei over Fq,

• if r_i−1<0, then the term

ri−1

Y

s=0

(I_i(q)−s) is neglected.

PROOF. Let f(t) be an irreducible polynomial of degree i over Fq. It is known by Gauss Lemma that the number of such polynomials is given by Ii(q) = ¹_i X

d|i

µ(d)q^di. Now let α1 be a root of f(t). It follows by Lemma 5.2.9 that the other roots of f(t) are α^q₁, α^q₁²,· · · , α^q₁ⁱ⁻¹. Thus if we choose α₁ as an eigenvalue of a representative of a regular semisimple class, thenα₁ together with the former set of powers of α1 form a complete set of eigenvalues of the Jordan block of size i.

Since each 1 ≤ i ≤ n appears ri times in the partition λ, it follows that we can choose α1 in I_i(q) ways, α₂ in I_i(q)−1 ways, and so forth till the α_ri,which we can choose in I_i(q)−(r_i−1) ways. We recall that a conjugacy class is unaltered by the arrangement of the eigenvalues in the Jordan block. Thus we divide by the number of all possible arrangements, which isr_i!. Repeating this for all 1≤i≤n,we get the required number mentioned in the statement of the Proposition.

For any positive integer n, two partitions namely, (1,1,· · · ,1)

| {z }

ntimes

` n and (n) ` n are of particu- lar interest.

Corollary 5.2.11. With q > n, then corresponding to the partitions λ = (1,1,· · · ,1)

| {z }

n times

` n and σ= (n)`n, we have F(λ) = (q−1)(q−2)···(q−n)

n! and F(σ) = _n¹X

d|n

µ(d)q^nd.

PROOF. Immediate from Proposition 5.2.10.

Remark 5.2.3. Note that by Propositions 5.2.8 and 5.2.10 the number of regular semisimple classes ofGL(n, q) is given by X

λ`n

F(λ). For fixed n= 1,2,3,4,5 one can calculate X

λ`n

F(λ) from Table 6.12 and compare withReg(G¹_n(q)).We can see thatX

λ`n

F(λ) =Reg(G¹_n(q)).

Finally we count the number of regular semisimple elements contained in classc^λ.

Proposition 5.2.12. Let c^λ be a regular semisimple class, where λ= (λ1, λ2,· · · , λk)`n. Then

|c^λ|=

n−1

Y

s=0

(qⁿ−q^s)

k

Y

i=1

(q^λi−1)

. (5.12)

T C T

PROOF. Let g ∈ c^λ = ({f_i},{λ_i},{1}_ktimes,{1}_ktimes). Since ν_i = 1, ∀1 ≤ i ≤ k, we obtain by substituting in equation (5.9) that

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

k

Y

i=1

q^λiφ₁(q^−λi) =

k

Y

i=1

q^λi

q^λi−1 q^λi

=

k

Y

i=1

(q^λi−1).

The result follows by (5.10).

Now we give the main theorem of this subsection which counts the number of regular semisimple elements of GL(n, q).

Theorem 5.2.13. Withλ= (λ₁, λ₂,· · ·, λ_k)≡1^r12^r2· · ·n^rn forr_i ∈N∪{0},the number of regular semisimple elements of GL(n, q) is given by

X

λ`n n−1

Y

s=0

(qⁿ−q^s)

n

Y

i=1 ri−1

Y

s=0

(Ii(q)−s)

k

Y

i=1

(q^λi −1)

n

Y

i=1

r_i!

. (5.13)

PROOF. Direct result from Propositions 5.2.8 and 5.2.10, together with equation (5.12).

Example 5.2.2. Let us consider the regular semisimple classes of type T^(2,2) of GL(4, q). Each classc^(2,2) will have the formc^(2,2) = ({f₁, f2},{2,2},{1,1},{1,1}) wheref1(t) =t²+a1t+a0 and f₂(t) =t²+b₁t+b₀ are two distinct irreducible polynomials overFq.To count the number of classes of this type, we follow Proposition 5.2.10. Thus we may write the partition (2,2) in the form 2², that is r1 =r3=r4 = 0 andr2= 2.Therefore

F(2²) =

4

Y

i=1 ri−1

Y

s=0

(I_i(q)−s)

n

Y

i=1

ri!

=

1

Y

s=0

(I2(q)−s) 0! 2! 0! 0!

= I₂(q)(I₂(q)−1)

2 = 1

2 q²−q

2

q²−q−2

2 = q(q²−1)(q−2)

8 .

Now applying equation (5.12) to any class of this type, we get

|c^(2,2)| =

3

Y

s=0

(q⁴−q^s)

2

Y

i=1

(q^λi−1)

= (q⁴−1)(q⁴−q)(q⁴−q²)(q⁴−q³) (q²−1)(q²−1)

= q⁶(q−1)(q²+ 1)(q³−1).

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

q⁶(q−1)(q²+ 1)(q³−1)q(q²−1)(q−2)

8 = q⁷(q⁴−1)(q³−1)(q−1)(q−2) 8

regular semisimple elements of type (2,2).Repeating the above work for the other four partitions of 4, we get forGL(4, q),a total number of regular semisimple elements given by

q¹⁶−2q¹⁵+q¹³+q¹²−2q¹⁰−q⁹−q⁸+ 2q⁷+q⁶.

For example the groupGL(4,5),which is of order 116064000000 has 9299587000 regular semisimple elements.

In the Appendix we list the number of types, conjugacy classes, elements in each conjugacy class of regular semisimple elements of GL(n, q) for n= 1,2,3,4,5.

Recall that a classc= ({f_i},{d_i},{z_i},{ν_i}) ofGL(n, q) with lengthkis called primary if and only ifk= 1.The next Proposition counts the number of primary classes of GL(n, q).

Proposition 5.2.14. The number of primary classes ofGL(n, q) is given by X

d|n

|P(n

d)| ·I_d(q). (5.14)

PROOF. By definition a conjugacy class c of GL(n, q) is primary if and only if c = (f, d,ⁿ_d, ν) for some f ∈ F with degree d, d|n and ν ` <sup>n</sup><sub>d</sub>. For fixed d and any ν ` ⁿ_d we have I_d(q) irreducible polynomials f of degree d that defines a primary class. It follows that there are |P(ⁿ_d)| ·I_d(q) conjugacy classes defined by the fixed integerdand partitions of ⁿ_d.The result follows by lettingd

runs over all divisors of n.

Corollary 5.2.15. There are exactly In(q) = X

d|n

µ(d)q^nd primary regular semisimple classes of GL(n, q).

PROOF. A classcofGL(n, q) is primary and regular semisimple if and only ifc= (f, n,1,1) for some f ∈ F with degree n. It follows by Proposition 5.2.8 that c is a regular semisimple class of type λ= (n)`n.Hence by Corollary 5.2.11 we haveF((n)) =I_n(q) =X

d|n

µ(d)q^nd.In particular ifn=p⁰ is a prime integer (whether p⁰ =p, the characteristic of Fq or not), then there are I_p⁰(q) = ^qp

0

−q p⁰

primary regular semisimple classes of GL(p⁰, q).

Corollary 5.2.16. The groupGL(p⁰, q)has exactly(q−1)|P(p⁰)|+^qp

0

−q

p⁰ primary conjugacy classes.

T C T

PROOF. By Proposition 5.2.14 the number of primary conjugacy classes of GL(p⁰, q) is given by X

d|p⁰

|P(p⁰

d)| ·I_d(q).We haved∈ {1, p⁰}.Ifd= 1,then there are|P(p⁰)|types of primary classes each consists ofq−1 conjugacy classes. On the other hand ifd=p⁰,thenc= (f, p⁰,1,1) for somef ∈ F with prime degree p⁰.Therefore by Corollary 5.2.15 we haveF((p⁰)) = ^qp

0

−q

p⁰ .Hence the result.

Example 5.2.3. See Table 5.2.

Orders of regular semisimple elements of GL(n, q)

Suppose that g is a regular semisimple element of GL(n, q) in a conjugacy class c^λ, where λ = (λ₁, λ₂,· · · , λ_k) ` n. Assume that F^∗_qλi = hε_ii, ∀1 ≤ i ≤ k and for each group F^∗_qλi, we fix one generator εi, that is if λj = λi, for some j and i, then we identify εj with εi. It follows that the eigenvalues ofg are

ε^j₁¹, ε^j₁^1q,· · ·, ε^j₁^1qλ1−1; ε^j₂², ε^j₂^2q,· · · , ε^j₂^2qλ2−1; · · · ;ε^j_k^k, ε^j_k^kq,· · ·, ε^j_k^kqλk−1 (5.15) for some integers j₁, j₂,· · ·, j_k.

Theorem 5.2.17. With the above, the order of g is given by o(g) =lcm

q^λ1−1

gcd(j₁, q^λ1 −1), q^λ1 −1

gcd(j₁q, q^λ1 −1),· · ·, q^λ1 −1

gcd(j₁q^λ1−1, q^λ1 −1), q^λ2−1

gcd(j2, q^λ2 −1), q^λ2 −1

gcd(j2q, q^λ2 −1),· · ·, q^λ2 −1

gcd(j2q^λ2−1, q^λ2 −1), ... q^λk−1

gcd(jk, q^λk−1), q^λk−1

gcd(jkq, q^λk−1),· · · , q^λk−1 gcd(jkq^λk−1, q^λk−1)

.

PROOF. For 1 ≤ l ≤k, 0 ≤r ≤ λl−1, let h denotes the row of eigenvalues of g given by (5.15).

Also let tdenotes the row o(ε^j_l^lqr) for 1≤l≤k, 0≤r≤λ_l−1.

We know thatg∼D=diag(h).Assume thato(g) =t. Then

g^t∼D^t = (diag(h))^t=diag(h^t) =In⇐⇒ε^j_l^lqrt= 1, ∀1≤l≤k, ∀0≤r≤λl−1

⇐⇒ o(ε^j_l^lqr)|t, ∀1≤l≤k, ∀0≤r≤λl−1

⇐⇒ lcm(t)|t. (5.16)

Let o(ε^j_l^lqr) =tlr for each 1 ≤l ≤k, 0 ≤ r ≤λl−1 and let d= gcd(t). Now for each 1≤ m ≤ k, 0≤j≤λm−1 we have

lcm(t) = Y

r,l

t_lr d =tmj



 Y

(l,r)6=(m,j)

tlr

d



.

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

g^lcm(t)∼D^lcm(t)=diag

(ε^j₁¹)^lcm(t),(ε^j₁^1q)^lcm(t),· · · ,(ε^j₁^1qλ1−1)^lcm(t) (ε^j₂²)^lcm(t),(ε^j₂^2q)^lcm(t),· · · ,(ε^j₂^2qλ2−1)^lcm(t) ... (ε^j_k^k)^lcm(t),(ε^j_k^kq)^lcm(t),· · ·,(ε^j_k^kqλk−1)^lcm(t)

=In,

where lcm(t) in each diagonal entry (ε^jm^mqrλm−1)^lcm(t) is replaced by tmj



 Y

(l,r)6=(m,j)

tlr

d



. This implies that t|lcm(t), since o(g) = t. From equation (5.16) we have lcm(t)|t. Hence t = o(g) = lcm(t). Now the result follows from elementary group theory, where we know that o(ε^j_l^lqr) =

(q^λl−1)/(gcd(jlq^r, q^λl−1)).

As a corollary of Theorem 5.2.17 we show the existence of an element of GL(n, q) of order qⁿ−1 (this has been mentioned in Darafasheh [15] without proof).

Corollary 5.2.18. The groupGL(n, q), n >1 has at leastq^n(n−1)2

n−2

Y

s=0

(qⁿ⁻¹−q^s) elements of order qⁿ−1 and at least twice of the previous number if q is even.

PROOF. Letg, h∈GL(n, q) such that{ε_n, ε^qn,· · · , ε^qnⁿ⁻¹}and{ε²_n, ε^2qn,· · · , ε^2qnⁿ⁻¹}are the eigenvalues of gand h respectively. It follows from Theorem 5.2.17 that

o(g) = lcm

qⁿ−1

gcd(1, qⁿ−1), qⁿ−1

gcd(q, qⁿ−1),· · ·, qⁿ−1 gcd(qⁿ⁻¹, qⁿ−1)

= lcm(qⁿ−1, qⁿ−1,· · ·, qⁿ−1) =qⁿ−1.

Ifq is even, thenqⁿ−1 is odd and hence gcd(2, qⁿ−1) = 1,which yields that gcd(2q^m, qⁿ−1) = 1, ∀ 0 ≤ m ≤n−1. Hence o(h) = qⁿ−1 by similar argument used for o(g). The result follows

since all conjugate elements have the same order.