Pattern avoidance seen in multiplicities of maximal weights of affine Lie algebra representations

Abstract

We prove that the multiplicities of certain maximal weights of <!-- MATH $\mathfrak{g}(A^{(1)}_{n})$ --> -modules are counted by pattern avoidance on words. This proves and generalizes a conjecture of Jayne-Misra. We also prove similar phenomena in types <!-- MATH $A^{(2)}_{2n}$ --> and <!-- MATH $D^{(2)}_{n+1}$ --> . Both proofs are applications of Kashiwara's crystal theory.