A cyclage poset structure for Littlewood-Richardson tableaux

TL;DR

This paper introduces a new graded poset structure for Littlewood-Richardson tableaux that generalizes existing concepts and connects combinatorial enumeration with representation theory of GL(n).

Contribution

It defines a novel cyclage poset for LR tableaux and links its grading to Poincare polynomials of modules related to nilpotent conjugacy classes.

Findings

The poset generalizes the cyclage structure on column-strict tableaux.

Enumeration polynomials match Poincare polynomials of certain modules.

The grading function extends the charge statistic.

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.