Jacobi-Trudi determinants and characters of minimal affinizations

TL;DR

This paper proves a conjecture relating Jacobi-Trudi determinants to characters of minimal affinizations in orthogonal and symplectic Lie algebras, confirming a key formula using symmetric function symmetries.

Contribution

It establishes the conjecture of Chari and Greenstein, showing that these characters are expressed by Jacobi-Trudi determinants, expanding understanding of representation characters.

Findings

Proved the conjecture on Jacobi-Trudi determinants for minimal affinizations.

Demonstrated characters are given by Jacobi-Trudi determinants in types B, C, D.

Utilized symmetries of symmetric functions to achieve the proof.

Abstract

In their study of characters of minimal affinizations of representations of orthogonal and symplectic Lie algebras, Chari and Greenstein conjectured that certain Jacobi-Trudi determinants satisfy an alternating sum formula. In this note, we prove their conjecture and slightly more. The proof relies on some symmetries of the ring of symmetric functions discovered by Koike and Terada. Using results of Hernandez, Mukhin-Young, and Naoi, this implies that the characters of minimal affinizations in types B, C, and D are given by a Jacobi-Trudi determinant.