Invariants of Weyl group action and $q$-characters of quantum affine algebras

TL;DR

This paper studies the invariants under Weyl group actions on certain rational function fields related to quantum affine algebras, providing detailed descriptions of subfields fixed by reflections, extending previous work on $q$-characters at roots of unity.

Contribution

It offers a detailed analysis of the Weyl group invariant subfields within the rational function fields associated with quantum affine algebras, focusing on invariants under simple reflections.

Findings

Description of $r_i$-invariant subfields for each simple reflection

Extension of previous results on $W$-invariants at roots of unity

Deeper understanding of symmetries in quantum affine algebra representations

Abstract

Let $W$ be the Weyl group corresponding to a finite dimensional simple Lie algebra $\mathfrak{g}$ of rank $\ell$ and let $m>1$ be an integer. In [I21], by applying cluster mutations, a $W$-action on $\mathcal{Y}_m$ was constructed. Here $\mathcal{Y}_m$ is the rational function field on $cm\ell$ commuting variables, where $c \in \{ 1, 2, 3 \}$ depends on $\mathfrak{g}$. This was motivated by the $q$-character map $\chi_q$ of the category of finite dimensional representations of quantum affine algebra $U_q(\hat{\mathfrak{g}})$. We showed in [I21] that when $q$ is a root of unity, $\mathrm{Im} \chi_q$ is a subring of the $W$-invariant subfield $\mathcal{Y}_m^W$ of $\mathcal{Y}_m$. In this paper, we give more detailed study on $\mathcal{Y}_m^W$; for each reflection $r_i \in W$ associated to the $i$th simple root, we describe the $r_i$-invariant subfield $\mathcal{Y}_m^{r_i}$ of $\mathcal{Y}_m$.