A Conjecture of Procesi and a New Basis for the Decomposition of the Graded Left Regular Representation of Sn

AbstractLet R=Q[x1,x2 ,...,xn] be the ring of polynomials in the variables x1,x2,...xn and let R* denote the quotient of R by the ideal generated by the elementary symmetric functions. It is well known that R*, under the action of Sn, yields a graded version of the left regular representation. Several years ago Procesi asked for a basis of R* consisting of homogeneous polynomials Γ[S, C] indexed by pairs of tableaux, with S standard and C cocharge, which exhibits the decomposition of R* into its irreducible components. Procesi also suggested a way in which such polynomials Γ[S, C] could be constructed. Using one of the versions of Rota′s straightening algorithm, we show that certain polynomials Π[S, C] closely related to the Γ[S, C]′s yield the desired basis. Our basis appears to have some remarkable properties. Parallel to the ring R* there is a family of Sn-modules Rμ which have been recently studied by Garsia and Procesi. These modules are important since they yield a graded character which is closely related to the q-Kostka-Foulkes polynomials Kλμ(q). The polynomials Π[S, C] can be shown to yield also a basis when restricted to a given Rμ. Through this connection our work leads to an entirely new way of proving the charge interpretation for the polynomials Kλμ(q). Indeed, computer data strongly suggest a conjecture that implies the Lascoux- Schützenberger result