Cyclic sieving phenomenon on dominant maximal weights over affine Kac-Moody algebras

TL;DR

This paper establishes a cyclic sieving phenomenon on dominant maximal weights in affine Kac-Moody algebra modules, introducing a novel evaluation method and revealing dualities and formulas for weight counts.

Contribution

It introduces an S-evaluation and generalizes Sagan's action to affine Kac-Moody algebras, enabling a type-independent cyclic sieving construction.

Findings

Constructed a cyclic sieving phenomenon for dominant maximal weights.

Derived closed and recursive formulas for weight counts.

Discovered level-rank duality in the cardinalities.

Abstract

We construct a (bi)cyclic sieving phenomenon on the union of dominant maximal weights for level $\ell$ highest weight modules over an affine Kac-Moody algebra with exactly one highest weight being taken for each equivalence class, in a way not depending on types, ranks and levels. In order to do that, we introduce $\textbf{\textit{S}}$-evaluation on the set of dominant maximal weights for each highest modules, and generalize Sagan's action by considering the datum on each affine Kac-Moody algebra. As consequences, we obtain closed and recursive formulae for cardinality of the number of dominant maximal weights for every highest weight module and observe level-rank duality on the cardinalities.