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 $${V(k \Lambda_0)}$$ .