A Cyclage Poset Structure for Littlewood–Richardson Tableaux

Abstract

A graded poset structure is defined for sets of Littlewood–Richardson (LR) tableaux whose cardinalities are the multiplicities of irreducible gl(n)-modules in the tensor product of several irreducible gl(n)-modules indexed by rectangular partitions. This is a generalization of the cyclage poset on tableaux defined by Lascoux and Schützenberger. This combinatorial construction is of independent interest and may be understood without reference to the underlying algebraic geometry. It is shown that the polynomials obtained by enumerating LR tableaux by shape and a generalized charge statistic, are the graded multiplicities of irreducibles in certain graded gl(n)-modules supported in the closure of a nilpotent conjugacy class. In particular, explicit tableau formulas are obtained for the special cases of the Kostka–Foulkes polynomials, the coefficient polynomials of two-column Macdonald–Kostka polynomials, and the graded multiplicities of coordinate rings of closures of conjugacy classes of nilpotent matrices. These polynomials coincide with the q -enumeration of rigged configurations and conjecturally coincide with the q -analogues of LR coefficients defined by the spin–weight generating functions of ribbon tableaux introduced by Lascoux, Leclerc, and Thibon.