4.6 General Linear Groups 4.7 Steinbergs Unipotent Characters Farid - - PowerPoint PPT Presentation

4.6 General Linear Groups 4.7 Steinberg’s Unipotent Characters

Farid Aliniaeifard

York University <http://math.yorku.ca/~faridanf/>

July 3, 2015

Overview

4.6 General Linear Groups 4.7 Steinberg’s Unipotent Characters

Recall

◮ Gn = GLn = GLn(Fq) and A = A(G∗) := A(GL) ◮ A(GL) is a PSH with PSH basis Σ and following Product and

coproduct m := IndGLi+j

Pi,j

InflPi,j

GLi×GLj

∆ :=

• ResGLi+j

Pi,j

(−) Ki,j where Pi,j = gi l gj

• gi ∈ GLi, gj ∈ GLj, l ∈ Fi×j

q

Ki,j = Ii l Ij

4.6 General linear Groups

◮ Goal: Finding the cardinality of C = Σ ∩ p, where Σ is the

PSH basis for A(GL) and p is the set of primitive elements of A(GL). Why C is important? Theorem 3.12: Any PSH A has a canonical tensor product decomposition A =

• ρ∈C

A(ρ) with A(ρ) a PSH, and ρ the only primitive element in its PSH basis Σ(ρ) = {σ ∈ Σ : there exsits n ≥ 1 s.t (σn, ρ) = 0}.

◮ Also we know from Theorem 3.18 A(ρ) ∼

= Λ.

Definition:

◮ Call an irreducible representation ρ of GLn cuspidal for n ≥ 1

if it lies in C.

◮ Given an irreducible character σ of GLn, say that d(σ) = n. ◮ Let Cn := {ρ ∈ C : d(ρ) = n} for n ≥ 1 denote the subset of

cuspidal characters of GLn.

◮ Let F denote the set of all nonconstant monic irreducible

polynomials f (x) = x in Fq[x] with nonzero constant term.

Proposition 4.40

The number |Cn| of cuspidal characters of GLn(Fq) is the number

• f |Fn| of irreducible monic degree n polynomials f (x) = x in

Fq[x] with nonzero constant term. Proof: Using induction on n:

◮ For n = 1, GL1(Fq) = F∗ q and F1 = {f (x) = x − c : c ∈ F∗ q} ◮ The number of complex characters of GLn(Fq) = The number

• f conjugacy classes of GLn(Fq)

◮ These conjugacy classes are uniquely represented by rational

canonical forms, which are parametrized by functions λ : F → Par with the property that

f ∈F deg(f )|λ(f )| = n. ◮ (4.32) tells us that |Σn| is similarly parametrized by the

functions λ : C → Par having the property that

• ρ∈C deg(ρ)|λ(ρ)| = n.

F = ⊔n≥1Fn and C = ⊔n≥1Cn We have for an equality for all n ≥ 1

• 

 C

λ

→ Par :

• ρ∈C

deg(ρ)|λ(ρ)| = n   

• = |Σn|

=

• 

 F

λ

→ Par :

• ρ∈F

deg(f )|λ(f )| = n   

• Since there is only one partition λ having |λ| = 1:

|Cn| = |Σn| −

• 

 ⊔n−1

i=1 Ci λ

→ Par :

• ρ∈C

deg(ρ)|λ(ρ)| = n   

• |Fn| = |Σn| −
• 

 ⊔n−1

i=1 Fi λ

→ Par :

• ρ∈F

deg(f )|λ(f )| = n   

Example 4.41

The sets Fn of monic irreducible polynomials f (x) = x in F2[x] of degree n for n ≤ 3, so that we know how many cuspidal characters

• f GLn(Fq) in Cn to expect:

F1 = {x + 1} F2 = {x2 + x + 1} F3 = {x3 + x + 1, x3 + x2 + 1} Thus we expect

◮ one cuspidal character of GL1(F2), namely ρ1(= 1GL1(F2) ◮ one cuspidal character ρ2 of GL2(F2) ◮ two cuspidal character ρ3, ρ

′

3 of GL3(F2).

Steinberg’s Unipotent Characters

Define i := 1GL1 of GL1(Fq) Let P(1n) = B. We have in = IndGLn

B

1B = C[GLn/B] for some n. Definition: An irreducible character σ of GLn appearing as a constituent of IndGLn

B

1B = C[GLn/B] is called a unipotent character.

◮ In particular, 1GLn is a unipotent character of GLn for each n.

Proposition: One can choose Λ ∼ = A(GL)(i) in Theorem 3.20(g) so that hn → 1GLn. Proof:

◮ Theorem 3.18(a) ⇒ i2 = IndGL2 B

1B = 1GL2 + St2.

◮ Choose the isomorphism so as to send h2 → 1GL2.

◮ Claim: St⊥ 2 (1GLn) = 0 for every n ≥ 2.

Ex 4.25(e) ⇒ ∆(1GLn) =

• i+j=n
• ResGn

Pi,j1GLn

Ki,j =

• i+j=n

1GLi⊗1GLj so that St⊥

2 (1GLn) = (St2, 1GL2)1GLn−2 = 0 since St2 = 1GL2 ◮ Then hn → 1GLn.

Steinberg’s Unipotent Characters

◮ This subalgebra Λ ∼

= A(GL)(i) and the unipotent characters χλ

p corresponding under this isomorphism to the Schur

functions sλ was introduced by Steinberg.

◮ He wrote down χλ p by using Jacobi-Trudi determinant

expression for sλ = det(hλi−i+j).

◮ The Steinberg’s unipotent characters Stn, which is the

unipotent character corresponding under the isomorphism in last Proposition to en = S(1n), can be defined by the virtual sum Stn := χ(1n)

q

=

• α

(−1)n−l(α)IndGLn

Pα 1Pα

in which sum run through all compositions α of n.

References

Grinberg and Reiner Hopf Algerba in Combinatorics Bruce Sagan (2000) The Symmetric Group

The End