On a determinant formula for some real regular representations

TL;DR

This paper interprets a known formula for real regular representations of GL(n) as a matrix determinant, using the Lewis Carroll identity to derive new relations that connect to quantum affine algebra representations.

Contribution

It introduces a determinant-based interpretation of Lapid-Mínguez's formula and extends relations to quantum affine Schur-Weyl duality, generalizing T-systems.

Findings

New determinant relations for real regular representations

Generalization of Mukhin-Young's Extended T-systems

Connections between classical and quantum affine algebra representations

Abstract

We interpret a formula established by Lapid-M\'{\i}nguez on real regular representations of ${\rm GL}_n$ over a local non-archimedean field as a matrix determinant. We use the Lewis Carroll determinant identity to prove new relations between real regular representations. Through quantum affine Schur-Weyl duality, these relations generalize Mukhin-Young's Extended $T$-systems, for representations of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_k)$, which are themselves generalizations of the celebrated $T$-system relations.