Some crystal Rogers-Ramanujan type identities

TL;DR

This paper derives new Rogers-Ramanujan type identities for colored partitions using crystal base character formulas and Weyl-Kac characters, linking combinatorics with crystal theory.

Contribution

It introduces novel Rogers-Ramanujan type identities for colored partitions based on crystal base character formulas for affine Lie algebra modules.

Findings

New combinatorial identities for colored partitions

Connection between crystal bases and partition difference conditions

Comparison with identities from vertex operator constructions

Abstract

By using the Kang-Kashiwara-Misra-Miwa-Nakashima-Nakayashiki crystal base character formula for the basic $A_2^{(1)}$-module, and the principally specialized Weyl-Kac character formula, we obtain a Rogers-Ramanujan type combinatorial identity for colored partitions. The difference conditions between parts are given by the energy function of certain perfect $A_2^{(1)}$-crystal. We also recall some other identities for this type of colored partitions, but coming from the vertex operator constructions and with no apparent connection to the crystal base theory.