T1: Lectures 6-10

Question 1

State Maschke’s Theorem

Answer 1

Let (π,W) be a proper subrep of (π,V). There exists some (π,W’) such that

(π,V)=(π,W)⨁(π,W’)

Question 2

Define a projection onto a subspace

Answer 2

Consider W⊆V. A projection onto subspace W is the linear map T:V→V st

T(w)=w ∀w∈W, T(v)=w ∀v∈V

Question 3

Define the ‘group algebra’ C(G)

Answer 3

Formal linear combinations of the group elements of the form SUM_g in G a_g [g]

Question 4

What is the dimension of the group algebra?

Answer 4

dim(C(G)) = |G|

Question 5

How does rep (λ, C(G)) act on elements of G?

Answer 5

Left multiplies each element by g.

Question 6

How does Hom_G(C(G),V) relate to V?

Answer 6

Isomorphism: dim(V) = dim(Hom_g(C(G),V))

Question 7

Given a red rep (π,V) and irr rep (ρ,W), how many times does (ρ,W) appear into a decomp of (π,V) into irr?

Answer 7

dim(Hom_G(V,W))

Question 8

State the sum of squares formula

Answer 8

Sum over ρ∈Irr(G): dim(ρ)^2 = |G|

Question 9

Define the character χ_π for rep (π, V)

Answer 9

χ_π = tr(π(g))

tr = sum of eigenvalues counted w/multiplicity

Question 10

What is the character of the trivial rep (Id, C^n)

Answer 10

n

Question 11

For any 1d rep (π, V) what is the character?

Answer 11

χ_π = π(g)

Question 12

χ_(π ⊕ ρ) = ??

Answer 12

χ_(π) + χ_(ρ)

Question 13

χ_π (gh) = ??

Answer 13

χ_π (hg)

Question 14

χ_π (g^-1) = ??

Answer 14

χ_π (g)*

Question 15

What shape should a character table be?

Answer 15

Square: rows = columns

Question 16

What condition is imparted on the columns of a character table?

Answer 16

Sum of squares = |G|/|C|

where C indicates size of conjugacy class.

Question 17

Define the class function f: G → C

Answer 17

A function that is constant on conjugacy classes.

Question 18

Define the inner product of class functions/characters χ and ψ

Answer 18

1/|G| SUM_g (χ(g)*ψ(g))

= 1 if χ=ψ
=0 else

Note this is element g in G not conjugacy class

Question 19

What condition is there on different columns of a conjugacy table?

Answer 19

Inner product between two different columns must be zero.

Question 20

Given V and W are two representations of G with characters χ and ψ respectively, define ⟨χ, ψ⟩

Answer 20

⟨χ, ψ⟩ = dim Hom_G(V, W)

Question 21

Give the alternate definition of the representation (π,C^G). What is this isomorphic to?

Answer 21

(π(g)f)(h) = f(g^-1 h)

Isomorphic to the regular representation C[G].

Question 22

Give the inner product of characters from the same conjugacy class (i.e. ip of column with itself)

Answer 22

SUM_χ χ*(Ci)χ(Ci) = |G|/|C_i|

Question 23

What does the inner product of two reps tell us about the Hom space between them?

Answer 23

dim(Hom_g(V,W))

Question 24

Given W ⊂ V and π projects w to W, what can we say about V?

Answer 24

V = ker(π) ⊕ W

Question 25

How does multiplication for basis vectors of the group algebra behave?

Answer 25

[g][h] = [gh]

Question 26

How does a rep of G act on the group algebra?

Answer 26

Replace any basis vector with the rep action

Question 27

How do the characters of iso reps relate?

Answer 27

Characters are the same

Question 28

What condition do we impart on the rows of the character table?

Answer 28

Ip same row = 1 if reducible.

Ip diff rows = 0