Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Alternating snake modules and a determinantal formula

Matheus Brito, Vyjayanthi Chari

TL;DR

The paper defines and analyzes alternating snake modules for the quantum loop algebra $\widehat{\bold U}_n$, establishing that they generalize snake and HL-cluster modules. It proves precise primeness criteria and a unique prime-factor decomposition, presents a Weyl-module presentation, and provides a determinantal formula expressing $[V(\omega_{\bold s})]$ as $\det A(\bold s)$ with a matrix built from Weyl-module classes; under mild hypotheses the determinant expansion has coefficients $\pm 1$. A robust toolkit, including KKOP invariants and $\ell$-weight theory, underpins these results and yields structural theorems for Weyl modules and exact sequences that realize $V(\omega_{\bold s})$ as quotients of Weyl modules. The framework is applied to category $\mathcal{O}(\mathfrak{gl}_r)$ via the Arakawa–Suzuki functor, producing explicit decompositions of certain (generally infinite-dimensional) irreducibles in terms of Verma modules, and identifying a large family of non-regular, non-dominant weights with Kazhdan–Lusztig coefficients in $\{\pm1\}$. Overall, the work bridges quantum affine representation theory, cluster-algebra phenomena, and classical category $\mathcal{O}$ questions through explicit combinatorial and determinantal constructions.

Abstract

We introduce a family of modules for the quantum affine algebra which include as very special cases both the snake modules and the modules arising from a monoidal categorification of cluster algebras. We give necessary and sufficient conditions for these modules to be prime and prove a unique factorization result. We also give an explicit formula expressing the module as an alternating sum of Weyl modules. Finally, we give an application of our results to classical questions in the category $\mathcal{ O}(\mathfrak{gl}_r)$. Specifically we apply our results to show that there are a large family of non-regular, non-dominant weights $μ$ for which the non-zero Kazhdan-Lusztig coefficients $c_{μ, ν}$ are $\pm 1$.

Alternating snake modules and a determinantal formula

TL;DR

The paper defines and analyzes alternating snake modules for the quantum loop algebra , establishing that they generalize snake and HL-cluster modules. It proves precise primeness criteria and a unique prime-factor decomposition, presents a Weyl-module presentation, and provides a determinantal formula expressing as with a matrix built from Weyl-module classes; under mild hypotheses the determinant expansion has coefficients . A robust toolkit, including KKOP invariants and -weight theory, underpins these results and yields structural theorems for Weyl modules and exact sequences that realize as quotients of Weyl modules. The framework is applied to category via the Arakawa–Suzuki functor, producing explicit decompositions of certain (generally infinite-dimensional) irreducibles in terms of Verma modules, and identifying a large family of non-regular, non-dominant weights with Kazhdan–Lusztig coefficients in . Overall, the work bridges quantum affine representation theory, cluster-algebra phenomena, and classical category questions through explicit combinatorial and determinantal constructions.

Abstract

We introduce a family of modules for the quantum affine algebra which include as very special cases both the snake modules and the modules arising from a monoidal categorification of cluster algebras. We give necessary and sufficient conditions for these modules to be prime and prove a unique factorization result. We also give an explicit formula expressing the module as an alternating sum of Weyl modules. Finally, we give an application of our results to classical questions in the category . Specifically we apply our results to show that there are a large family of non-regular, non-dominant weights for which the non-zero Kazhdan-Lusztig coefficients are .

Paper Structure

This paper contains 36 sections, 37 theorems, 184 equations.

Table of Contents

  1. Alternating Snake Modules: Main Results
  2. The algebra $\widehat{\bold U}_n$ and the category $\mathscr F_n$
  3. The group $\mathcal{I}_n$
  4. $\ell$--weights
  5. Alternating snakes
  6. Examples
  7. Prime factorizations
  8. Prime factors
  9. A presentation of $V({\hbox{\boldmath $\omega$}}_\bold s)$
  10. A determinantal formula
  11. The matrix $A(\bold s)$
  12. Examples
  13. Alternating snakes: further examples
  14. An application to category $\mathcal{O}(\mathfrak{gl}_r)$
  15. The Arakawa--Suzuki functor
  16. ...and 21 more sections

Key Result

Theorem 1

Suppose that $\bold s = ([i_1,j_1],\cdots, [i_r,j_r]) \in \bold S_{{\rm alt}}$, $r\ge 1$. In particular, $V({\hbox{\boldmath $\omega$}}_\bold s)$ is prime if and only if conn and ijneq hold.

Theorems & Definitions (42)

  • Definition
  • Theorem 1
  • Definition
  • Corollary
  • Theorem 2
  • Remark
  • Theorem 3
  • Lemma
  • Proposition
  • Remark
  • ...and 32 more