Characters of level $1$ standard modules of $C_n^{(1)}$ as generating functions for generalised partitions

Abstract

We give a new simple formula for the energy function of a level $1$ perfect crystal of type $C_n^{(1)}$ introduced by Kang, Kashiwara and Misra. We use this to give several expressions for the characters of level $1$ standard modules as generating functions for different types of partitions. We then relate one of these formulas to the difference conditions in the conjectural partition identity of Capparelli, Meurman, Primc and Primc, and prove that their conjecture is true for all level $1$ standard modules. Finally, we propose a non-specialised generalisation of their conjecture.