The quasiinvariants of the symmetric group

For m a non-negative integer and G a Coxeter group, we denote by QI _m(G) the ring of m-quasiinvariants of G, as defined by Chalykh, Feigin, and Veselov. These form a nested series of rings, with QI ₀(G) the whole polynomial ring, and the limit QI _∞ (G) the usual ring of invariants. Remarkably, the ring QI _m(G) is freely generated over the ideal generated by the invariants of G without constant term, and the quotient is isomorphic to the left regular representation of G. However, even in the case of the symmetric group, no basis for QI _m(G) is known. We provide a new description of QI _m(S _n), and use this to give a basis for the isotypic component of QI _m(S _n) indexed by the shape [n - 1, 1].