A Littlewood-Richardson filtration at roots of 1 for multiparameter deformations of skew Schur modules.

Abstract

Let R be a commutative ring, q a unit of R and P a multiplicatively antisymmetric matrix with coefficients which are integers powers of q. Denote by SE(q,P) the multiparameter quantum matrix bialgebra associated to q and P.Slightly generalizing [Hashimoto-Hayashi,Tohoku Math.Tohoku Math.J. 44(1992)],we define a multiparameter deformation $L_{\l/\mu}V_P$ of the classical skew Schur module.In case R is a field and q is not a root of 1, arguments like those given in [H-H] show that $L_{\l/\mu}V_P$ is irreducible and its decomposition into irreducibles is $\sum_\nu c(\l/\mu;\nu)L_\nu V_P$ where the coefficients are the usual Littlewood-Richardson ones. When R is any ring and q is allowed to be a root of 1, we construct a filtration of $L_{\l/\mu}V_P$ as an SE(q,P)-comodule, such that its associated graded object is precisely $\sum_\nu c(\l/\mu;\nu)L_\nu V_P$.