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On dominant $\ell$--weights and maps between Weyl modules for quantum affine $A_n$

Matheus Brito, Vyjayanthi Chari

TL;DR

This work analyzes dominant $\ell$-weights and morphisms between Weyl modules for the quantum affine algebra of type $A_n$ within the Hernandez–Leclerc subcategory $\mathscr F_n$. It builds a combinatorial framework using intervals $[i,j]$ and closure operations $\bar{\bold s}[n]$, governed by maps $\tau_{m,\ell}$ and Nil-Hecke relations, to classify when $W(\omega_{\mathbf{s}'})$ embeds into $W(\omega_{\mathbf{s}})$ and to determine all possible $\ell$-weights in $\operatorname{wt}_\ell^+ W(\omega_{\mathbf{s}})$. The authors prove a strong bound $\dim\operatorname{Hom}_{\widehat{\bold U}_n}(W(\omega_{\mathbf{s}'}), W(\omega_{\mathbf{s}})) \le 1$, with equality iff $\mathbf{s}'\in\bar{\mathbf{s}}[n]$, and show the socle of a Weyl module is a direct sum of Weyl modules corresponding to closed elements, becoming simple for large $n$. They apply these results to mixed Weyl modules, providing inclusions and head/socle decompositions, and to the construction of tensor subcategories generalizing HL's $\mathscr C_\ell$, along with an Ext-vanishing criterion. A lifting argument then extends the large-$n$ conclusions to all $n$, delivering a coherent picture of $\,\ell$-weight structure, morphisms, and extensions in this category.

Abstract

We determine the set of dominant $\ell$--weights in the Weyl (or standard) modules for quantum affine $A_n$. We then prove that the space of homomorphisms between standard modules is at most one-dimensional and give a necessary and sufficient condition for equality to hold. We also describe the socle of the standard module and prove that the socle is simple for large $n$. Finally, we give applications of our results to mixed Weyl modules, calculating extensions in the category and identify new families of tensor subcategories of finite dimensional representations.

On dominant $\ell$--weights and maps between Weyl modules for quantum affine $A_n$

TL;DR

This work analyzes dominant -weights and morphisms between Weyl modules for the quantum affine algebra of type within the Hernandez–Leclerc subcategory . It builds a combinatorial framework using intervals and closure operations , governed by maps and Nil-Hecke relations, to classify when embeds into and to determine all possible -weights in . The authors prove a strong bound , with equality iff , and show the socle of a Weyl module is a direct sum of Weyl modules corresponding to closed elements, becoming simple for large . They apply these results to mixed Weyl modules, providing inclusions and head/socle decompositions, and to the construction of tensor subcategories generalizing HL's , along with an Ext-vanishing criterion. A lifting argument then extends the large- conclusions to all , delivering a coherent picture of -weight structure, morphisms, and extensions in this category.

Abstract

We determine the set of dominant --weights in the Weyl (or standard) modules for quantum affine . We then prove that the space of homomorphisms between standard modules is at most one-dimensional and give a necessary and sufficient condition for equality to hold. We also describe the socle of the standard module and prove that the socle is simple for large . Finally, we give applications of our results to mixed Weyl modules, calculating extensions in the category and identify new families of tensor subcategories of finite dimensional representations.
Paper Structure (21 sections, 23 theorems, 124 equations)

This paper contains 21 sections, 23 theorems, 124 equations.

Table of Contents

  1. Main Results
  2. The algebra $\widehat{\bold U}_n$ and the category $\mathscr F_n$
  3. The group $\mathcal{I}_n$
  4. $\Sigma_r$-orbits, connected pairs, the maps $\tau_{m,\ell}$ and closed subsets of $\mathbb I_n^r$
  5. Dominant $\ell$--weights
  6. Maps between Weyl Modules and the Socle of a Weyl module
  7. Some Applications
  8. The maps $\tau_{m,\ell}$ and the Proof of Proposition \ref{['closedelement']}
  9. Proof of Proposition \ref{['closedelement']}
  10. Proof of Theorem \ref{['weylmaps']}
  11. Proof of Theorem \ref{['weylmaps']}(i)
  12. Proof of Theorem \ref{['weylmaps']}(ii)
  13. Proof of Theorem \ref{['socleirr']}
  14. $\ell$--roots and a partial order on $\mathcal{I}_n^+$
  15. The case $n>n(\bold s)$
  16. ...and 6 more sections

Key Result

Proposition 1

Suppose that $\bold s\in\mathbb I_{n,+}^r$ with $n\ge n(\bold s)$. Then there exists $\bold s_0\in\bar{\bold s}[n]$ which is closed and the set of all closed elements in $\bar{\bold s}[n]$ is contained in $\Sigma_r\bold s_0$.

Theorems & Definitions (27)

  • Proposition
  • Theorem 1
  • Corollary
  • Remark
  • Theorem 2
  • Proposition
  • Lemma
  • Remark
  • Proposition
  • Corollary
  • ...and 17 more