Lattice Paths, Young Tableaux, and Weight Multiplicities

TL;DR

This paper connects lattice path combinatorics, Young tableaux, and permutation patterns to representation theory, providing formulas for weight multiplicities in affine Lie algebra modules.

Contribution

It establishes a novel combinatorial interpretation of weight multiplicities using lattice paths, Young tableaux, and pattern-avoiding permutations.

Findings

Number of admissible lattice path sequences equals sum of squares of Young tableaux counts.

This quantity also counts certain pattern-avoiding permutations.

Application to affine Lie algebra representations yields explicit multiplicity formulas.

Abstract

For $\ell \geq 1$ and $k \geq 2$, we consider certain admissible sequences of $k-1$ lattice paths in a colored $\ell \times \ell$ square. We show that the number of such admissible sequences of lattice paths is given by the sum of squares of the number of standard Young tableaux of partitions of $\ell$ with height $\leq k$, which is also the number of $(k+1)k\cdots21$-avoiding permutations of $\{1, 2, \ldots, \ell\}$. Finally, we apply this result to the representation theory of the affine Lie algebra $\widehat{sl}(n)$ and show that this quantity gives the multiplicity of certain maximal dominant weights in the irreducible module $V(k\Lambda_0)$.