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Steadied Khovanov-Lauda-Rouquier algebras and local models for blocks

Dinushi Munasinghe, Ben Webster

TL;DR

The paper develops steadied quotients $\mathcal{R}(\chi,\Lambda,\alpha)$ as tilted cyclotomic KLRW algebras to model blocks of Ariki–Koike algebras and RoCK blocks, unifying various Morita-equivalence phenomena via Weyl-group actions. It constructs a comprehensive framework including slope data, standardizations, and a categorical $\mathfrak{sl}_2$ action, then proves a central Morita equivalence: $\mathcal{R}(w^{-1}\cdot\chi,\Lambda,\alpha)\simeq^{\mathrm{Morita}}\mathcal{R}(\chi,\Lambda,w\bullet_{\Lambda}\alpha)$. The approach yields derived/perverse equivalences between tilted blocks, explains chamber-dependence through Scopes walls, and provides concrete local models for RoCK blocks by connecting to Turner doubles and EK2-type results. Overall, the work generalizes known RoCK-block local-models to all blocks of Ariki–Koike algebras, offering a unified, diagrammatic Morita-framework with robust categorical-action techniques and potential quiver-variety interpretations in characteristic zero.

Abstract

It's known that many different blocks of $\mathbb{F}_pS_n$ for different values of $n$ are equivalent as categories, though the corresponding block algebras are almost never isomorphic. Thus, it is a challenging problem to give one particularly nice representative of this Morita equivalence class of algebras. This has been accomplished for the case of RoCK blocks through work of Chuang--Kessar, Turner, and Evseev--Kleshchev. In this paper, we give a new perspective on this problem, applying not just to RoCK blocks of $S_n$, but also to all blocks of Ariki--Koike algebras. We do this by considering steadied quotients of KLRW algebras: these algebras are a natural generalization of cyclotomic quotients, already related to $S_n$ and Ariki--Koike algebras in work of Brundan--Kleshchev. These algebras are defined by ``tilting'' the cyclotomic relations so that we kill the two-sided ideal defined by certain configurations on the left and right sides of our diagrams. We show a Morita equivalence between these algebras and blocks of Ariki-Koike algebras generalizing the work discussed above.

Steadied Khovanov-Lauda-Rouquier algebras and local models for blocks

TL;DR

The paper develops steadied quotients as tilted cyclotomic KLRW algebras to model blocks of Ariki–Koike algebras and RoCK blocks, unifying various Morita-equivalence phenomena via Weyl-group actions. It constructs a comprehensive framework including slope data, standardizations, and a categorical action, then proves a central Morita equivalence: . The approach yields derived/perverse equivalences between tilted blocks, explains chamber-dependence through Scopes walls, and provides concrete local models for RoCK blocks by connecting to Turner doubles and EK2-type results. Overall, the work generalizes known RoCK-block local-models to all blocks of Ariki–Koike algebras, offering a unified, diagrammatic Morita-framework with robust categorical-action techniques and potential quiver-variety interpretations in characteristic zero.

Abstract

It's known that many different blocks of for different values of are equivalent as categories, though the corresponding block algebras are almost never isomorphic. Thus, it is a challenging problem to give one particularly nice representative of this Morita equivalence class of algebras. This has been accomplished for the case of RoCK blocks through work of Chuang--Kessar, Turner, and Evseev--Kleshchev. In this paper, we give a new perspective on this problem, applying not just to RoCK blocks of , but also to all blocks of Ariki--Koike algebras. We do this by considering steadied quotients of KLRW algebras: these algebras are a natural generalization of cyclotomic quotients, already related to and Ariki--Koike algebras in work of Brundan--Kleshchev. These algebras are defined by ``tilting'' the cyclotomic relations so that we kill the two-sided ideal defined by certain configurations on the left and right sides of our diagrams. We show a Morita equivalence between these algebras and blocks of Ariki-Koike algebras generalizing the work discussed above.

Paper Structure

This paper contains 19 sections, 31 theorems, 80 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definition of steadied quotients
  3. KLRW algebras
  4. Pressures
  5. Dependence on alcoves
  6. Actions of the Weyl group
  7. Standardizations
  8. Slope data
  9. Standard modules
  10. Finite-dimensionality
  11. The categorical action
  12. Induction and restriction functors
  13. Categorical $\mathfrak{sl}_2$-actions
  14. The property (KM3)
  15. Morita equivalence
  16. ...and 4 more sections

Key Result

Theorem 1.1

We have an isomorphism $\operatorname{AK}^{\Lambda}_{\alpha}\cong R^{\Lambda}_{\alpha}$ from the block algebra of the Ariki--Koike algebra to the cyclotomic quotient of the KLR algebra.

Figures (3)

  • Figure 1: An illustration of the definitions of $f_{\mathbf{i}}^+$. The dotted black line is the graph of $f_{\mathbf{i}}$, the graph of $f_{\mathbf{i}}^+$ is in purple.
  • Figure 2: An illustration of the definitions of $g_{\mathbf{i}}^{\pm}$. The dotted black line is the graph of $f_{\mathbf{i}}$, the graph of $g_{\mathbf{i}}^+$ is in red, that of $g_{\mathbf{i}}^-$ in blue. Note that these coincide for $x\geq k$, shown in purple.
  • Figure 3: The Kleshchev-Muth isomorphism

Theorems & Definitions (74)

  • Theorem 1.1: BK
  • Theorem A
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Lemma 2.7
  • Definition 2.8
  • ...and 64 more