OAI.Problem346.canonical_foulkes_howe_surjective

The theorem states that, for every finite-dimensional complex vector space V and all natural numbers a and b with a ≥ 2 and a(a−1) ≤ b, there is a ℂ-linear map μ from Sym^b(Sym^a V) to Sym^a(Sym^b V) that satisfies the Foulkes condition, is surjective, and is the only linear map satisfying that condition. Here Sym^n V is the span, inside the symmetric algebra of V over ℂ, of the products v₀⋯v_{n−1} of n elements of V, and the symbol [v₀,…,v_{n−1}] denotes such a monomial regarded as an element of Sym^n V. A linear map μ satisfies the Foulkes condition when, for every array v of b rows and a columns of vectors v_{j,i} in V, μ sends the monomial [[v_{0,0},…,v_{0,a−1}], …, [v_{b−1,0},…,v_{b−1,a−1}]] in Sym^b(Sym^a V), whose inner monomials are the rows, to (1/(a!)^b) times the sum, over all b-tuples (σ₀,…,σ_{b−1}) of permutations of the a column positions, of the monomial in Sym^a(Sym^b V) whose i-th entry is the monomial [v_{0,σ₀(i)}, …, v_{b−1,σ_{b−1}(i)}], for i from 0 to a−1.