Invariant and coinvariant spaces for the algebra of symmetric polynomials in non-commuting variables (Q612953)

Summary: We analyze the structure of the algebra \(\mathbb K\langle{\mathbf x}\rangle^{{\mathfrak S}_n}\) of symmetric polynomials in non-commuting variables in so far as it relates to \(\mathbb K[{\mathbf x}]^{{\mathfrak S}_n}\), its commutative counterpart. Using the ``place-action'' of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of \(\mathbb K\langle{\mathbf x}\rangle^{{\mathfrak S}_n}\) analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups.