Multiplicities of maximal weights of the sℓ ̂(n)-module V (kΛ0)

Abstract

Consider the affine Lie algebra [Formula: see text] with null root [Formula: see text], weight lattice [Formula: see text] and set of dominant weights [Formula: see text]. Let [Formula: see text] [Formula: see text] denote the integrable highest weight [Formula: see text]-module with level [Formula: see text] highest weight [Formula: see text]. Let [Formula: see text] denote the set of weights of [Formula: see text]. A weight [Formula: see text] is a maximal weight if [Formula: see text]. Let [Formula: see text] denote the set of maximal dominant weights which is known to be a finite set. The explicit description of the weights in the set [Formula: see text] is known [R. L. Jayne and K. C. Misra, On multiplicities of maximal dominant weights of [Formula: see text]-modules, Algebr. Represent. Theory 17 (2014) 1303–1321]. In papers [R. L. Jayne and K. C. Misra, Lattice paths, Young tableaux, and weight multiplicities, Ann. Comb. 22 (2018) 147–156; R. L. Jayne and K. C. Misra, Multiplicities of some maximal dominant weights of the [Formula: see text]-modules [Formula: see text], Algebr. Represent. Theory 25 (2022) 477–490], the multiplicities of certain subsets of [Formula: see text] were given in terms of some pattern-avoiding permutations using the associated crystal base theory. In this paper the multiplicity of all the maximal dominant weights of the [Formula: see text]-module [Formula: see text] are given generalizing the results in [R. L. Jayne and K. C. Misra, Lattice paths, Young tableaux, and weight multiplicities, Ann. Comb. 22 (2018) 147–156; R. L. Jayne and K. C. Misra, Multiplicities of some maximal dominant weights of the [Formula: see text]-modules [Formula: see text], Algebr. Represent. Theory 25 (2022) 477–490].