Prove that the following assertions

RepresentationTheory— ExerciseSheet7 TU Kaiserslautern

Jun.-Prof. Dr. CarolineLassueur FB Mathematik

BernhardB ¨ohmler

Due date:– WS 2020/21

Throughout,Gdenotes a finite group and (F,O,k) is ap-modular system such thatFcon- tains an exp(G)-th root of unity. Furthermore, all modules considered are assumed to be leftmodules and finitely generated.

Exercise1.

LetU,V,Wbe non-zerokG-modules. Prove that the following assertions.

(a) If 0 U V W 0 is a s.e.s. ofkG-modules, then ϕ_V =ϕ_U+ϕ_W.

(b) If the composition factors ofUareS₁, . . . ,S_m (m∈Z^≥1) with multiplicitiesn₁, . . . ,n_m respectively, then

ϕ_U =n₁ϕ_S1 +. . .+n_mϕ_Sm.

In particular, if twokG-modules have isomorphic composition factors, counting mul- tiplicities, then they have the same Brauer character.

(c) ϕ_U⊕V=ϕ_U+ϕ_Vandϕ_U⊗_kV =ϕ_U·ϕ_V. Exercise2.

Prove that two kG-modules afford the same Brauer character if and only if they have isomorphic composition factors (including multiplicities).

Exercise3.

LetHbe ap⁰-subgroup of a finite groupG. Prove that the characterΦk is a constituent of the trivialF-character ofHinduced toG.

Exercise4.

Letϕ, λ∈IBr_p(G) and assume thatλis linear. Prove thatλϕ∈IBr_p(G) andλΦ_ϕ= Φ_λϕ. Exercise5.

LetGbe a finite group and letρregdenote the regularF-character ofG. Prove that:

ρreg = X

ϕ∈IBrp(G)

ϕ(1)Φϕ and (ρreg)|

G_p⁰ = X

ϕ∈IBrp(G)

Φϕ(1)ϕ .

Exercise6.

Prove that:

(a) the inverse of the Cartan matrix ofkGisC⁻¹=(hϕ, ψi

p⁰)_ϕ,ψ∈IBrp(G); and (b) |G|_p|Φϕ(1) for everyϕ∈IBrp(G).

Exercise7.

LetUbe akG-module and letPbe a PIM ofkG. Prove that dim_kHom_kG(P,U)= 1

|G| X

g∈G_p⁰

ϕ_P(g⁻¹)ϕ_U(g)

Exercise8.

Let G := A₅, the alternating group on 5 letters. Calculate the Brauer character table, the Cartan matrix and the decomposition matrix ofGforp=3.

[Hints. (1.) Use the ordinary character table ofA₅and reduction modulop. (2.) A simple group does not have any irreducible Brauer character of degree 2.]

Exercise9.

Deduce from Remark 35.12 of the Lecture Notes that column orthogonality relations for the Brauer characters take the formΠ^trΦ =B, i.e. giveng,h∈G_p⁰ we have

X

φ∈IBrp(G)

φ(g)Φ_ϕ(h⁻¹)=











|C_G(g)| ifgandhareG-conjugate,

0 otherwise.