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Extremal monomials of $q$-characters

Andrei Neguţ

TL;DR

The paper proves the Frenkel–Hernandez extremal monomial conjecture for finite-dimensional type 1 modules of the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ by showing that nonzero monomials in a $q$-character are confined to the Weyl-group cone intersections generated from the leading monomial under Chari’s braid action. The authors achieve this by embedding the coefficient data of $q$-characters into geometric stability loci of quiver representations, then extending Nakajima-type quiver varieties with generic stability parameters $\theta$ and constructing Weyl-reflection maps $\mathscr{S}_i$ between stability spaces. A key technical development is the explicit construction of $\mathscr{S}_i$ via linear-algebra on framed quivers, yielding dimension-preserving maps between moduli spaces that realize Weyl reflections. The work unifies geometric representation theory and quantum affine algebra combinatorics, generalizing Weyl invariance of classical characters and providing a geometric framework for the $q$-character coefficients $\mu_{\boldsymbol{x}}^{\boldsymbol{\psi}}$. It also outlines ungraded analogues and potential extensions within Nakajima’s framework.

Abstract

In this short paper, we prove a conjecture of Frenkel-Hernandez, which states that $q$-characters of finite-dimensional simple modules of the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ are bounded by the Weyl group orbit of the leading monomial under Chari's braid group action. This generalizes the Weyl group invariance of characters of finite-dimensional representations of $\mathfrak{g}$.

Extremal monomials of $q$-characters

TL;DR

The paper proves the Frenkel–Hernandez extremal monomial conjecture for finite-dimensional type 1 modules of the quantum affine algebra by showing that nonzero monomials in a -character are confined to the Weyl-group cone intersections generated from the leading monomial under Chari’s braid action. The authors achieve this by embedding the coefficient data of -characters into geometric stability loci of quiver representations, then extending Nakajima-type quiver varieties with generic stability parameters and constructing Weyl-reflection maps between stability spaces. A key technical development is the explicit construction of via linear-algebra on framed quivers, yielding dimension-preserving maps between moduli spaces that realize Weyl reflections. The work unifies geometric representation theory and quantum affine algebra combinatorics, generalizing Weyl invariance of classical characters and providing a geometric framework for the -character coefficients . It also outlines ungraded analogues and potential extensions within Nakajima’s framework.

Abstract

In this short paper, we prove a conjecture of Frenkel-Hernandez, which states that -characters of finite-dimensional simple modules of the quantum affine algebra are bounded by the Weyl group orbit of the leading monomial under Chari's braid group action. This generalizes the Weyl group invariance of characters of finite-dimensional representations of .

Paper Structure

This paper contains 11 sections, 5 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. The setting
  3. The extremal monomial conjecture
  4. Acknowledgements
  5. The proof
  6. Quiver Grassmannians
  7. Quiver varieties
  8. $\theta$-stability
  9. Reflections
  10. Constructing $\mathscr{S}_i$:
  11. The ungraded case

Key Result

Theorem 1.2

(HL) For any ${\boldsymbol{\psi}} = Y_{k,c}$ with $k \in I$ and $c \in {\mathbb{C}}^*$, we have where $N_{\boldsymbol{x},{\boldsymbol{\psi}}}^{\emph{stab}}$ is a certain variety that we will recall in Subsection sub:quiver variety old.

Theorems & Definitions (12)

  • Theorem 1.2
  • Theorem 1.4
  • Definition 2.3
  • Remark 2.4
  • Definition 2.6
  • Proposition 2.8
  • proof
  • Lemma 2.10
  • proof
  • Lemma 2.11
  • ...and 2 more