A conjecture of Procesi and the straightening algorithm of Rota.
Let R = Q[x1, x2,..., xn] and R* denote the quotient of R by the ideal generated by the elementary symmetric functions. R*, under the action of Sn, yields a graded version of the left regular representation. Procesi asked for a basis of R* consisting of homogeneous polynomials Gamma[S, C] indexed by pairs of tableaux, with S standard and C cocharge, that exhibits the decomposition of R* into its irreducible components. Procesi also suggested a way to construct the Gamma[S, C]. Using Rota's straightening algorithm, I show that certain polynomials [S, C] closely related to the Gamma[S, C] terms yield the desired basis. Parallel to the ring R* there is a family of Sn-modules R that have recently been studied by Garsia and Procesi. These modules have a graded character that is closely related to the q-Kostka-Foulkes polynomials Klambdamu(q). The [S, C] can be shown to yield also a basis when restricted to a given R. Through this connection the work reported here leads to an additional way of proving the charge interpretation for the polynomials Klambdamu(q).
Allen
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Department of Mathematics, University of California at San Diego, La Jolla, CA 92037, USA.