Question

I need help on #5 and #7

Exercise 3.5. Let p: G the fixed subspace GL(V) be a representation of a finite group G. Define Vα εV | 99υ = υ,Vgε G. 1. Show that ve is a G-invariant subspace. 2. Show that 1 ΕΣ ε να |G hEG for all v e V. 3. Show that if v E V, then 1 Σ \GI hEG 4. Conclude dim VG is the rank of the operator 1 P = |G| Σ hEG 5. Show that P2 = P. 6. Conclude Tr(P) is the rank of P. 7. Conclude - αΣΤφ). dim VG |G hEG

Answer 1

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