Fall 2014 Problem 5. Let ρ be a representation of a finite group G on a vector space V and let v ∈ V.

(a) Show that averaging ρ_{g}(v) over G gives a vector \overline{v} ∈ V which is fixed by G.

(b) What can you say about this vector when ρ is an irreducible representation?

Fall 2012 Problem 6. Consider the dihedral group D_{4} = ⟨r, s : s^{2} = r^{4} = 1, rs = sr^{−1}⟩ of order 8.

(a) Find the conjugacy classes of D_{4}.

(b) Find the character table of D_{4}.

Fall 2006 Problem 5: Define ϕ_{A}(X) = AXA^{−1} if X ∈ ℂ^{2×2} and A ∈ GL_{2}(ℂ).

(1) Check that this defines a linear representation of GL_{2}(ℂ)

(2) Show that this representation is reducible.

(3) Write down one element of each orbit of the above action and in each case find the stabilizer of that element.

Winter 2007 Problem 3: Consider the following 5 by 5 character table for the finite group G in which a is a primitive 5th root of unity:

\begin{matrix}1 & 1 & 1 & 1 & 1 \\ 3 & −1 & 0 & a^{3} + a^{2} + 1 & a^{4} + a + 1 \\ 3 & −1 & 0 & a^{4} + a + 1 & a^{3} + a^{2} + 1 \\ 4 & 0 & 1 & −1 & −1 \\ 5 & 1 & −1 & 0 & 0 \end{matrix}

(1) Show that G is simple and of order 60 and find the orders of its conjugacy classes.

(2) Find the decomposition into irreducible representations of the tensor produt of the two 3-dimensional irreducible representations.

Spring 2013 Problem 3. The dihedral group D_{4} acts as the symmetries of a square in the plane ℝ^{2} with coordinates x and y. Suppose that the corners of this square are at (±1, ±1). Then D_{4} acts linearly and it therefore has an induced action on the vector space V_{n} of homogeneous polynomials in x and y of degree n. Find the character of V_{n} viewed as a representation of D_{4}. (Note: the character will depend on n.)

Fall 2007 Problem 1: Show that every matrix in the image of a complex representation of a finite group is diagonalizable.

Fall 2018 Problem 6. Let V be the subspace of ℂ^{3} spanned by v_{1}, v_{2} = (1, −1, 0), (0, 1, −1) which is an invariant subspace under the permutation action of S_{3}, and so gives a two dimensional representation ρ : S_{3} → GL(V).

(a) Write down the matrices of ρ(σ) in this basis.

(b) Describe a Hermitian inner product ⟨, ⟩ on V in the basis v_{1}, v_{2} which is G-invariant. i.e. ⟨ρ(g)u, ρ(g)v⟩ = ⟨v, u⟩.

(c) Describe the tensor product V ⊗_{R} V, as an R-module where R = ℂ[H] and H ⊂ S_{3} is the subgroup {1, (123), (132)}.

Spring 2015 Problem 6. Let G be a finite group and ρ : G → GL_{n}(ℂ) a representation.

(a) Show δ : G → ℂ, g ↦ det(ρ(g)) is linear character of G (i.e. a group homomorphism to the multiplicative group).

(b) Show: If δ(g) = −1 for some g ∈ G then G has a normal subgroup of index 2.

(c) Show: If G has order 2k, k odd, then G has a normal subgroupu of index 2.

(d) Let χ(g) = tr(ρ(g)) and g ∈ G an involution. Show: (i) χ(g) is an integer; (ii) χ(g) ≡ χ(1) mod 2; (iii) if G has no normal subgroup of index 2, then χ(g) ≡ χ(1) mod 4.

Spring 2014 Problem 5: Let V be a linear representation of a group G over the field ℚ(\sqrt{2}) with character χ : G → ℚ(\sqrt{2}). Consider V as a vector space Vℚ over the subfield ℚ with the same action of G. Find the character χℚ of this new representation in terms of the original character χ.