Lattice Paths, Young Tableaux, and Weight Multiplicities

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 shape ${\lambda \vdash \ell}$ with ${l(\lambda) \leq k}$, which is also the number of (k + 1)k··· 21-avoiding permutations in ${S_\ell}$. Finally, we apply this result to the representation theory of the affine Lie algebra ${\widehat{sl}(n)}$ and show that this gives the multiplicity of certain maximal dominant weights in the irreducible highest weight ${\widehat{sl}(n)}$-module ${V(k \Lambda_0)}$.