¨Ubungen zu Darstellungstheorie — Blatt 8 TU Kaiserslautern Let K

Übungen zu Darstellungstheorie — Blatt 8
Jun.-Prof. Dr. Caroline Lassueur
TU Kaiserslautern
Dr. Mikaël Cavallin
Prof. Dr. Gunter Malle
Abgabetermin: Di, 20.12.2016, 18:00 Uhr
WS 2016/17
Let K be a field and G be a finite group.
Aufgabe 27.
Let U, V be KG-modules.
(a) Prove that ΦU,V : U∗ ⊗ V → HomK (U, V) defined by ΦU,V (ϕ ⊗ v)(u) := ϕ(u)v for ϕ ∈ U∗ ,
v ∈ V, u ∈ U, is a KG-isomorphism.
(b) Let TrV : V ∗ ⊗ V → K, ( f, v) 7→ f (v) be the trace map. Prove that TrV is a KGhomomorphism and that TrV ◦(ΦV,V )−1 coincides with the ordinary trace of matrices.
(c) Prove that V is a direct summand of V ⊗ V ∗ ⊗ V. Moreover, if Char(K) = p > 0 and
p | dimK V, then V ⊕ V is a direct summand of V ⊗ V ∗ ⊗ V.
Aufgabe 28.
Prove that if G has odd order, then for every g ∈ G, we have
X
ν(χ)χ(g) = 1.
χ∈Irr(G)
Aufgabe 29.
Let G = D2n be a dihedral group of order 2n, n ≥ 3. Prove that ν(χ) = 1 for all irreducible
characters χ of G. (Hint: Calculate the number of elements g ∈ G such that g2 = 1.)
Aufgabe 30.
Check that the distributivity of multiplication over addition holds in K0 (KG), and determine
whether or not K0 (KG) has a neutral element for the multiplication.

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