The Andrews-Olsson identity and Bessenrodt insertion algorithm on Young walls

TL;DR

This paper extends the Andrews-Olsson identity to two-colored partitions related to Young walls of quantum affine algebras, providing simple generating function formulas and crystal structures via generalized algorithms.

Contribution

It introduces an extension of the Andrews-Olsson identity to two-colored partitions and generalizes Bessenrodt's algorithms for Young walls of quantum affine algebras.

Findings

Derived simple product formulas for generating functions of reduced Young walls.

Established crystal structures on subsets of strict partitions.

Provided an alternative proof of the extended identity using generalized algorithms.

Abstract

We extend the Andrews-Olsson identity to two-colored partitions. Regarding the sets of proper Young walls of quantum affine algebras $\g_n=A^{(2)}_{2n}$, $A^{(2)}_{2n-1}$, $B^{(1)}_{n}$, $D^{(1)}_{n}$ and $D^{(2)}_{n+1}$ as the sets of two-colored partitions, the extended Andrews-Olsson identity implies that the generating functions of the sets of reduced Young walls have very simple formulae: \begin{center} $\prod^{\infty}_{i=1}(1+t^i)^{\kappa_i}$ where $\kappa_i=0$, $1$ or $2$, and $\kappa_i$ varies periodically. \end{center} Moreover, we generalize the Bessenrodt's algorithms to prove the extended Andrews-Olsson identity in an alternative way. From these algorithms, we can give crystal structures on certain subsets of pair of strict partitions which are isomorphic to the crystal bases $B(\Lambda)$ of the level $1$ highest weight modules $V(\Lambda)$ over $U_q(\g_n)$.