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Problem 1. Let G be a finite group, and let G b be the set of isomorphism classes of irreducible finite dimensional complex representations of G. For every σ ∈ G, we b choose a representative (V

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Algebra III/Introduction to Algebra III: Representation Theory Due: Please upload solutions to NUCT by Tuesday, June 16, 2020.

Problem 1. Let G be a finite group, and let G b be the set of isomorphism classes of irreducible finite dimensional complex representations of G. For every σ ∈ G, we b choose a representative (V

σ

, π

σ

) of the class σ. We define the Fourier transform of f ∈ C [G] to be the “function”

1

f b that to σ ∈ G b assigns the endomorphism

f b (σ) = X

g∈G

f (g)π

σ

(g) ∈ End

C

(V

σ

).

Prove the following statements:

(a) For all f

1

, f

2

∈ C [G] and σ ∈ G, b

f \

1

∗ f

2

(σ) = f b

1

(σ) ◦ f b

2

(σ),

where f

1

∗ f

2

∈ C [G] is the convolution product of f

1

and f

2

defined by (f

1

∗ f

2

)(g) = X

hk=g

f

1

(h)f

2

(k).

Here the sum ranges over pairs (h, k) ∈ G × G such that hk = g.

(b) For all f ∈ C [G], the Frobenius inversion formula f (g) = 1

|G|

X

σ∈Gb

n

σ

tr(π

σ

(g)

−1

◦ f b (σ))

holds. Here n

σ

= dim

C

(V

σ

) and we recall that |G| = P

σ∈Gb

n

2σ

.

1The Fourier transformfbis really not a function, but rather a section of a bundle, where the fiber ofσ is EndC(Vσ). Also, to avoid making the choice of (Vσ, πσ), one should work with the

∞-category of representations instead.

1

参照

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