Weighted words at degree two, II: flat partitions, regular partitions, and application to level one perfect crystals

TL;DR

This paper extends bijections for colored partitions to degree two, enabling derivation of character formulas for certain affine Lie algebra modules and linking combinatorics with representation theory.

Contribution

It introduces flat and regular partitions and generalizes existing bijections to degree two, connecting combinatorial partition theory with affine Lie algebra representations.

Findings

Derived character formulas for level one modules of specific affine Lie algebras.

Established a combinatorial framework linking partitions to representation theory.

Extended bijections to colored partitions of degree two.

Abstract

In a recent work, Keith and Xiong gave a refinement of Glaisher's theorem by using a Sylvester-style bijection. In this paper, we introduce two families of colored partitions, flat and regular partitions, and generalize the bijection of Keith and Xiong to these partitions. We then state two results, the first at degree one, where partitions have parts with primary colors, and the second result at degree two for secondary-colored partitions, using the result of the first paper of this series. These results allow us to easily retrieve the Frenkel-Kac character formulas of level one standard modules for the type $A_{2n}^{(2)}, D_{n+1}^{(2)}$ and $B_n^{(1)}$, and also to make the connection between the result stated in paper one and the representation theory.