A combinatorial formula for graded multiplicities in excellent filtrations

TL;DR

This paper provides a combinatorial formula for graded multiplicities in excellent filtrations of fusion products for the current algebra sl_2[t], linking algebraic structures to lattice path combinatorics and special functions.

Contribution

It introduces a novel combinatorial formula for graded multiplicities in excellent filtrations, connecting representation theory with lattice path enumeration and special functions.

Findings

Derived a combinatorial formula using two-dimensional lattice paths.

Connected graded multiplicities to Ramanujan's mock theta functions and Kostka polynomials.

Provided interpretations for multiplicities in type B_n and G_2 Lie algebras.

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 $\mathfrak{sl}_2[t]$. 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 coeffficients of Ramanujan's fifth order mock theta functions $\phi_0, \phi_1, \psi_0, \psi_1$, Kostka polynomials for hook partitions and quotients of Chebyshev polynomials. We also get 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 $B_n \text{ and } G_2$.