Exercise set 2.6

(1)

(Difficulty level 2) Show that the matrix of the character table of a finite group (over the complex numbers) is invertible. More precisely, show that the absolute value of the determinant of the character table is ∏ _g, where the product is over a set of representatives g of the conjugacy classes and z_g is the cardinality of the centralizer of g.

(2)

(Difficulty level 2) Show that the restriction to an invariant subspace of a diagonalizable linear operator on a finite dimensional vector space is diagonalizable. Suppose that a representation of a group breaks up into a direct sum of 1-dimensional representations. Show that any invariant subspace also breaks up into a direct sum of 1-dimensional representations.

(3)

(Difficulty level 3) Given a weak composition λ of n with m parts, we have naturally associated to it a 1-dimensional representation homogeneous of degre n of the torus T_m(K) by mapping Δ(x₁,…,x_n) to the monomial corresponding to λ. Show that this is a bijective correspondence (between such weak compositions and such 1-dimensional representions).

(4)

(Difficulty level 2) Let m be a positive integer and let λ be a partition with at most m parts. Consider the irreducible polynomial representation W_λ of GL_m(ℂ). Its character is s_λ(x₁,…,x_m), where s_λ is the Schur function corresponding to λ. Given a weak composition μ of n with m parts, let mon_μ be the corresponding monomial of degree n in the variables x₁, …, x_m. We call μ a weight of λ if the coefficient of mon_μ is s_λ(x₁,…,x_m) is not zero. The coefficient itself (in case it is not zero) is called the multiplicity of the weight μ in W_λ.

• Let λ be a partition of n with at most 2 parts. Determine the weights and their multiplicities in the GL₂(ℂ) module W_λ.
• Determine the weights and their multiplicities of the GL_m(ℂ)-modules W_(n) and W_(1ⁿ).

(5)

(Difficulty level 3) Let V = (K^m)^⊗n, where K is a field. Let ρ be the representation of 𝔖_n on V , and θ be the representation of GL_m(K) on V . Let w be in 𝔖_n. Show that the trace of ρ(w) ⋅ θ(Δ(x₁,…,x_m)) equals ∑ _λχ_λ(w)s_λ(x₁,…,x_m), as the sum varies over all partitions λ of n.

Hint: This is Lemma 6.5.1 in Amri’s book. We have V ≃ K[I(n,m)] (as 𝔖_n-modules) where I(n,m) = I denotes the set of all functions from [n] to [m], for the standard basis of (K^m)^⊗n is indexed by I. Note that the standard basis elements form common eigenvectors for the action of the torus (diagonal subgroup of GL_m), and are permuted by elements of 𝔖_n. Let W be the set of weak compositions of n with m parts. There is a “type map” I → W, and two elements of I belong to the same orbit of 𝔖_n if and only if their types are the same. So we may write K[I] = ⊕_τK[I_τ], where τ varies over W and I_τ is the fibre over τ of the type map. On each K[I_τ], each element of the torus acts like a scalar, like the monomial corresponding to τ.

From W to the set P of partitions of n we have a natural map: put the constituents of the weak composition in weakly decreasing order. For a partition μ of n, the 𝔖_n-representation K[X_τ] is isomorphic to K[X_μ] for all τ in the fibre in W over μ. Let us fix μ in P and consider ∑ _τ∈W,τμK[I_τ]. The trace of ρ(w) ⋅ θ(Δ(x₁,…,x_m)) on this is Trace(w,K[X_μ]) ⋅ m_μ(x₁,…,x_m).

Now K[X_μ] = ∑ _νV _ν^⊕K_νμ. So the required trace is

	∑ μ ∑ νKνμχν(w)mμ(x1,…,xm)
=	∑ νχν(w)
=	∑ νχν(w) ⋅ sν(x1,…,xm)