A COMBINATORIAL FORMULA FOR GRADED MULTIPLICITIES IN EXCELLENT FILTRATIONS

Abstract

A filtration of a representation whose successive quotients are isomorphic to Demazure modules is called an excellent filtration. In this paper we study graded multiplicities in excellent filtrations of fusion products for the current algebra $$ {\mathfrak{sl}}_2\left[t\right] $$ . We give a combinatorial formula for the polynomials encoding these multiplicities in terms of two-dimensional lattice paths. Corollaries to our main theorem include a combinatorial interpretation of various objects such as the coefficients of Ramanujan’s fifth order mock theta functions ϕ0, ϕ1, ψ0, ψ1, Kostka–Foulkes polynomials for hook partitions and quotients of Chebyshev polynomials. Moreover, a combinatorial interpretation of the graded multiplicities in a level one flag of a local Weyl module associated to the simple Lie algebras of type Bn and G2 is obtained.