Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Monoidal categorification from alternating snakes

Matheus Brito, Vyjayanthi Chari

TL;DR

The paper develops a combinatorial and categorical framework linking alternating snake modules for the quantum affine algebra of type $A_n$ to monoidal categorification of cluster algebras of type $A_N$. It defines a canonical monoidal subcategory $oldsymbol{ C}_n(oldsymbol{s})$ from a prime alternating snake $oldsymbol{s}$, proves finiteness of prime objects, and establishes a precise factorization theory governed by the KKOP invariant $oldsymbol{ rak d}$. It then proves isomorphisms of Grothendieck rings across height-function realizations and shows that the constructed category monoidally categorifies a cluster algebra of type $A_N$, with explicit $N$ determined by the snake data. The results connect the $oldsymbol{ C}$-categories to the height-function framework of Hernandez–Leclerc and BC19a, yielding a robust monoidal categorification of cluster algebras via alternating snakes and providing a combinatorial toolset for tensor product factorizations in these categories. Overall, the work unifies alternating snakes, prime factorization, and monoidal categorification, enriching the link between quantum affine representation theory and cluster algebras in type $A$.

Abstract

In a recent paper, the authors introduced the notion of an alternating snake and a corresponding family of finite dimensional modules for the quantum affine algebra associated to $A_n$. We prove that under some restrictions, an alternating snake defines a canonical monoidal category. We prove that this category has finitely many prime objects. As a consequence we prove that the Grothendieck ring is isomorphic to the Grothendieck ring of the category $\mathscr C_ξ$ for a suitable height function. In particular it follows that the special family of alternating snakes provides a monoidal categorification of a cluster algebra of type $A_N$ for a suitable value of $N$.

Monoidal categorification from alternating snakes

TL;DR

The paper develops a combinatorial and categorical framework linking alternating snake modules for the quantum affine algebra of type to monoidal categorification of cluster algebras of type . It defines a canonical monoidal subcategory from a prime alternating snake , proves finiteness of prime objects, and establishes a precise factorization theory governed by the KKOP invariant . It then proves isomorphisms of Grothendieck rings across height-function realizations and shows that the constructed category monoidally categorifies a cluster algebra of type , with explicit determined by the snake data. The results connect the -categories to the height-function framework of Hernandez–Leclerc and BC19a, yielding a robust monoidal categorification of cluster algebras via alternating snakes and providing a combinatorial toolset for tensor product factorizations in these categories. Overall, the work unifies alternating snakes, prime factorization, and monoidal categorification, enriching the link between quantum affine representation theory and cluster algebras in type .

Abstract

In a recent paper, the authors introduced the notion of an alternating snake and a corresponding family of finite dimensional modules for the quantum affine algebra associated to . We prove that under some restrictions, an alternating snake defines a canonical monoidal category. We prove that this category has finitely many prime objects. As a consequence we prove that the Grothendieck ring is isomorphic to the Grothendieck ring of the category for a suitable height function. In particular it follows that the special family of alternating snakes provides a monoidal categorification of a cluster algebra of type for a suitable value of .
Paper Structure (20 sections, 20 theorems, 108 equations)

This paper contains 20 sections, 20 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. The algebra $\widehat{\bold U}_n$ and the category $\mathscr F_n$
  4. An alternative formulation
  5. Alternating snakes
  6. Sink Source type
  7. Connected and Prime
  8. Prime factor
  9. The set $\mathbb I_n(\bold s)$ and $\bold p\bold r(\bold s)$
  10. The main result
  11. Proof of Theorem \ref{['mainthm']} (i),(ii)
  12. An enumeration
  13. The set $\mathbb I_n(\bold s)$
  14. Proof of Theorem \ref{['mainthm']}(i)
  15. Proof of Theorem \ref{['mainthm']}(ii)
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\bold s\in \bold S_{{\rm alt}}^s$ be prime.

Theorems & Definitions (27)

  • Definition
  • Definition
  • Theorem 1
  • Lemma
  • Remark
  • Lemma
  • Remark
  • Proposition
  • Proposition
  • Lemma
  • ...and 17 more