Path description for $q$-characters of fundamental modules in type $C$

TL;DR

This work develops a type $C_n$ analogue of the Mukhin–Young path description for $q$-characters by introducing a combinatorial path model in $\mathbb{R}^2$ and an associated monomial map $\mathsf{m}$. For fundamental modules $L(Y_{i,k})$, the $q$-character is given by $\chi_q(L(Y_{i,k})) = \sum_{p \in \overline{\mathscr{P}}_{i,k}} \mathsf{m}(p)$, where $\overline{\mathscr{P}}_{i,k}$ consists of admissible paths with a parity-refined condition, and coefficients are all equal to $1$ (the thin property). The paper also analyzes the connected components of the path set and provides an outline of the proof using screening operators, linking combinatorics to the algebraic structure of $\chi_q$. These results offer a concrete combinatorial toolkit for understanding fundamental representations in type $C$ and motivate future work on extending path descriptions to higher-level modules and other Lie types.

Abstract

In this paper, we investigate the behavior of monomials in the $q$-characters of the fundamental modules over a quantum affine algebra of untwisted type C. As a result, we give simple closed formulae for the $q$-characters of the fundamental modules in terms of sequences of vertices in $\mathbb{R}^2$, so-called paths, with an admissible condition. This may be viewed as a type C analog of the path description of $q$-characters in types A and B due to Mukhin--Young.

Theorems & Definitions (6)

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