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

TL;DR

This paper presents a new formula for the energy function of a level 1 perfect crystal of type C_n^{(1)} and uses it to derive character formulas as generating functions for partitions, confirming a conjecture and proposing a generalization.

Contribution

It introduces a simple energy function formula and connects character formulas to partition identities, proving a conjecture for all level 1 modules and suggesting a broader generalization.

Findings

Confirmed the conjecture for all level 1 standard modules.

Derived new character formulas as generating functions for partitions.

Proposed a non-specialised generalization of the conjecture.

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.