Exercise set 2.5

Throughout, K denotes a field and I(n,m) the set of maps from [n] to [m]. There is a natural action of 𝔖_n on I(n,m).

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(Difficulty level 2) Consider a vector space over K of dimension m, say K^m. Consider (K^m)^⊗n = K^m ⊗⊗ K^m (n times). Let 𝔖_n act on this by permuting the factors σ(v₁ ⊗v_m) := v_σ^-11 ⊗⊗ v_σ^-1m. Show that K[I(n,m)] ≃ (K^m)^⊗n. Conclude that:

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(Difficulty level 3) Let R be the K-algebra M₂(K) × M₃(K). Let φ be the inclusion of R in M₁₁(K) given by

Let C denote the commutant in M₁₁(K) of the image φ(R). Show that C is a semisimple algebra. Determine its Wedderburn decomposition. How does the defining representation K¹¹ of M₁₁(K) decompose as a C-module? Determine C as a subset of M₁₁(K).

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(Difficulty level 2) Recall that an element of (K^m)^⊗n is said to be a symmetric n-tensor if it is invariant under ρ(w) for every w in 𝔖_n. Assuming K to be algebraically closed of characteristic 0, show that the space of symmetric tensors is a simple GL_m(K)-module of dimension equal to the number of weak compositions of n with at most m parts.

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(Difficulty level 2) Recall that an element of (K^m)^⊗n is said to be an alternating n-tensor if it is transforms under ρ(w) by the sign of w, for every w in 𝔖_n. Assuming K to be algebraically closed of characteristic 0, show that the space of alternating tensors is either zero (iff m < n) or is a simple GL_m(K)-module of dimension equal to (m n) .

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(Difficulty level 3) Let K be algebraically closed of characteristic 0. Determine the partitions λ with at most m parts for which the corresponding irreducible GL_m(K)-module W_λ is one dimensional. Show that the non-negative integral powers of the determinant are the only 1-dimensional polynomial representations of GL_m(K).

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(Difficulty level 2) Compute the dimensions and characters of the representation W_2,1 of GL₂(ℂ) and the representation W_2,1 of GL₃(ℂ).

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(Difficulty level 3) Show that the set of diagonalizable m×m complex matrices is dense (with respect to the usual topology) in the space of all m×m complex matrices. Deduce that the set of diagonalizable invertible m×m matrices is dense in the space of all invertible m×m complex matrices. State and prove an analogous statement over an arbitrary algebraically closed field with respect to the “Zariski topology”.

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(Difficulty level 3) Recall that the “convolution” product of two elements α and β of SK(m) is defined as follows:

Show that the order of integration in the above can be reversed, or in other words that:

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(Difficulty level 2) Express the element δ_I (where I stands for the m × m identity matrix) as a linear combination of the “standard basis” ϵ_i,j of SK(m) where (i,j) varies over a set of representatives of the 𝔖_n-orbits of I(n,m) × I(n,m).