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Symmetries of Grothendieck rings in representation theory

David Hernandez

TL;DR

This work surveys newly discovered symmetries of Grothendieck rings arising in representation theory, focusing on cluster algebra symmetries via monoidal categorification and Weyl-group-like actions on $q$-characters. It explains how cluster structures emerge in Grothendieck rings of quantum affine and shifted quantum affine algebras, and how these symmetries yield concrete results on simple representations and character formulas, including QQ-relations and Baxter-type relations. The authors present substantial results on monoidal categorifications, global structures, and the interplay between cluster algebras, $q$-characters, and prefundamental representations, culminating in conjectures that cluster monomials correspond to simple objects in shifted settings. The significance lies in connecting representation theory, quantum integrable models, and combinatorial structures via a unified framework for Grothendieck rings, enabling explicit computations and new structural insights.

Abstract

This is a written version of the invited lecture at the 9th European Congress of Mathematics in July 2024 in Sevilla. We review certain new symmetries of Grothendieck rings that have emerged in representation theory.

Symmetries of Grothendieck rings in representation theory

TL;DR

This work surveys newly discovered symmetries of Grothendieck rings arising in representation theory, focusing on cluster algebra symmetries via monoidal categorification and Weyl-group-like actions on -characters. It explains how cluster structures emerge in Grothendieck rings of quantum affine and shifted quantum affine algebras, and how these symmetries yield concrete results on simple representations and character formulas, including QQ-relations and Baxter-type relations. The authors present substantial results on monoidal categorifications, global structures, and the interplay between cluster algebras, -characters, and prefundamental representations, culminating in conjectures that cluster monomials correspond to simple objects in shifted settings. The significance lies in connecting representation theory, quantum integrable models, and combinatorial structures via a unified framework for Grothendieck rings, enabling explicit computations and new structural insights.

Abstract

This is a written version of the invited lecture at the 9th European Congress of Mathematics in July 2024 in Sevilla. We review certain new symmetries of Grothendieck rings that have emerged in representation theory.
Paper Structure (20 sections, 13 theorems, 93 equations)

This paper contains 20 sections, 13 theorems, 93 equations.

Table of Contents

  1. Grothendieck group
  2. Grothendieck group of an additive category
  3. Grothendieck groups in representation theory
  4. Monoidal categories
  5. Grothendieck ring
  6. Grothendieck ring of a monoidal category
  7. Grothendieck rings in representation theory
  8. Applications of Grothendieck rings in representation theory
  9. Cluster algebra symmetries
  10. A quick review on cluster algebras
  11. Monoidal categorification of cluster algebras
  12. Quantum affine algebras
  13. Monoidal categorification - examples
  14. Global structures
  15. Shifted quantum affine algebras
  16. ...and 5 more sections

Key Result

Theorem 1.2

[Jordan-Hölder series] Each object $M$ in $\mathcal{F}$ admits a series of subobjects with $M_i/M_{i+1}$ simple object. The number $n_V(M)$ of occurrence of a simple object $V$ (as a quotient $M_i/M_{i+1}$) depends only on $M$, and not on the series of subobjects.

Theorems & Definitions (39)

  • Definition 1.1
  • Theorem 1.2
  • proof
  • Remark 2.1
  • Proposition 3.1
  • Example 3.2
  • Remark 3.3
  • Remark 3.4
  • Example 3.5
  • Remark 3.6
  • ...and 29 more