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Monoidal categorification on open Richardson varieties

Yingjin Bi

TL;DR

The paper proves that in Dynkin types with v ≤ w, the subcategory \\mathscr{C}_{w,v} of modules over quiver Hecke algebras provides a monoidal categorification of the coordinate ring \\mathbb{C}[\\mathcal{R}_{w,v}] after inverting frozen cluster variables. It achieves this by employing Lusztig parameterizations to identify simple objects, constructing an initial monoidal seed from mutations of seeds in A_q(N(w)), and showing that cluster monomials correspond to simple objects in \\mathscr{C}_{w,v}. The approach unifies the preprojective- and KLR-algebra perspectives, linking determinantal modules to unipotent and Richardson-variety coordinates and leveraging delta-/\\Lusztig data to control mutations. This work extends known categorifications of unipotent coordinate rings to open Richardson varieties, enabling a representation-theoretic realization of their (quantum) cluster structures with potential impacts on geometric representation theory and algebraic combinatorics.

Abstract

In this paper, we show that the subcategory $\mathscr{C}_{w,v}$ of modules over quiver Hecke algebras is a monoidal categorification of the coordinate ring of any open Richardson variety of Dynkin types after inverting the frozen cluster variables.

Monoidal categorification on open Richardson varieties

TL;DR

The paper proves that in Dynkin types with v ≤ w, the subcategory \\mathscr{C}_{w,v} of modules over quiver Hecke algebras provides a monoidal categorification of the coordinate ring \\mathbb{C}[\\mathcal{R}_{w,v}] after inverting frozen cluster variables. It achieves this by employing Lusztig parameterizations to identify simple objects, constructing an initial monoidal seed from mutations of seeds in A_q(N(w)), and showing that cluster monomials correspond to simple objects in \\mathscr{C}_{w,v}. The approach unifies the preprojective- and KLR-algebra perspectives, linking determinantal modules to unipotent and Richardson-variety coordinates and leveraging delta-/\\Lusztig data to control mutations. This work extends known categorifications of unipotent coordinate rings to open Richardson varieties, enabling a representation-theoretic realization of their (quantum) cluster structures with potential impacts on geometric representation theory and algebraic combinatorics.

Abstract

In this paper, we show that the subcategory of modules over quiver Hecke algebras is a monoidal categorification of the coordinate ring of any open Richardson variety of Dynkin types after inverting the frozen cluster variables.
Paper Structure (22 sections, 29 theorems, 123 equations, 1 figure)

This paper contains 22 sections, 29 theorems, 123 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Coordinate rings of open Richardson varieties
  3. Richardson Varieties
  4. Quantum Coordinate Ring
  5. Unipotent Quantum Coordinate Ring
  6. Cluster Algebras
  7. Cluster Structure on the Coordinate Rings of Unipotent Subgroups
  8. Lusztig's Parameterizations of Determinantial Minors
  9. Cluster Structure on the Coordinate Ring of the Open Richardson Variety
  10. $\dot{v}$-Lusztig Parameterization of $D\{0,k]$
  11. Mutation Sequences
  12. Categories of Modules Over Preprojective Algebras
  13. Cluster Categories Over Preprojective Algebras
  14. Determinantal Modules over Preprojective Algebras
  15. Delta-Vectors
  16. ...and 7 more sections

Key Result

Theorem 1.1

In the Dynkin case, for $v \leq w \in W$, the category $\mathscr{C}_{w,v}$ is a monoidal categorification of $\mathbb{C}[\mathcal{R}_{w,v}]$ after inverting the frozen cluster variables. In particular, every cluster monomial corresponds to a simple module in the category $\mathscr{C}_{w,v}$.

Figures (1)

  • Figure 1: The quiver $Q_{\overline{w}}$

Theorems & Definitions (55)

  • Theorem 1.1: Theorem \ref{['theo_categorification']}
  • Theorem 2.1: Kashiwara1993global, Proposition 7.2.2
  • Definition 2.2
  • Proposition 2.3: geiss2013cluster
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Example 2.8
  • Theorem 2.9
  • ...and 45 more