A Cyclage Poset Structure for Littlewood-Richardson Tableaux

A graded poset structure is defined for the sets of LittlewoodRichardson (LR) tableaux that count the multiplicity of an irreducible gl(n)module in the tensor product of irreducible gl(n)-modules corresponding to rectangular partitions. This poset generalizes the cyclage poset on columnstrict tableaux defined by Lascoux and Schützenberger, 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 none other than the Poincaré polynomials of isotypic components of the certain modules supported in the closure of a nilpotent conjugacy class. In particular explicit tableau formulas are obtained for the special cases of these Poincaré polynomials given by Kostka-Foulkes polynomials, the coefficient polynomials of two-column Macdonald-Kostka polynomials, and the Poincaré polynomials of isotypic components of coordinate rings of closures of conjugacy classes of nilpotent matrices. These q-analogues conjecturally coincide with q-analogues of the number of certain sets of rigged configurations and the q-analogues of LR coefficients defined by the spin-weight generating functions of ribbon tableaux of Lascoux, Leclerc, and Thibon.