A bijection between Littlewood-Richardson tableaux and rigged configurations

Abstract.

We define a bijection from Littlewood-Richardson tableaux to rigged configurations and show that it preserves the appropriate statistics. This proves in particular a quasi-particle expression for the generalized Kostka polynomials $K_{\lambda R}(q)$ labeled by a partition $\lambda$ and a sequence of rectangles R. The generalized Kostka polynomials are q-analogues of multiplicities of the irreducible $GL(n, \mathbb{C})$-module $V^\lambda$ of highest weight $\lambda$ in the tensor product $V^{R_1} \otimes \cdots \otimes V^{R_L}$.