Jacobi–Trudi determinants and characters of minimal affinizations

Steven V Sam

Vol. 272 (2014), No. 1, 237–244

DOI: 10.2140/pjm.2014.272.237

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 and Young, and Naoi, this implies that the characters of minimal affinizations in types B, C, and D are given by a Jacobi–Trudi determinant.

Keywords

minimal affinizations, classical Lie algebras, symmetric functions

Mathematical Subject Classification 2010

Primary: 05E05

Secondary: 17B10

Milestones

Received: 16 August 2013

Revised: 15 October 2013

Accepted: 21 October 2013

Published: 9 October 2014