A cyclage poset structure for Littlewood-Richardson tableaux

Abstract

A graded poset structure is defined for the sets of Littlewood-Richardson (LR) tableaux that count the multiplicity of an irreducible GL(n)-module in the tensor product of irreducibles indexed by a sequence of rectangular partitions. This poset generalizes the cyclage poset on column-strict tableaux defined by Lascoux and Schutzenberger, and its grading function generalizes the charge statistic. It is shown that the polynomials obtained by enumerating LR tableaux by shape and the generalized charge, are the Poincare polynomials of isotypic components of the certain modules supported in the closure of a nilpotent conjugacy class.