Paths and tableaux descriptions of Jacobi-Trudi determinant associated with quantum affine algebra of type C_n

TL;DR

This paper explores a combinatorial approach to expressing the q-character of certain finite-dimensional representations of quantum affine algebra of type C_n using paths and tableaux, extending previous work on type D_n.

Contribution

It introduces a path and tableau-based combinatorial formula for the Jacobi-Trudi determinant related to quantum affine algebra of type C_n, generalizing prior results.

Findings

Positive sum expression over tuples of paths

Translation into tableaux on skew diagrams

Extension of path method to type C_n

Abstract

We study the Jacobi-Trudi-type determinant which is conjectured to be the q-character of a certain, in many cases irreducible, finite-dimensional representation of the quantum affine algebra of type C_n. Like the D_n case studied by the authors recently, applying the Gessel-Viennot path method with an additional involution and a deformation of paths, we obtain a positive sum expression over a set of tuples of paths, which is naturally translated into the one over a set of tableaux on a skew diagram.