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Representation type of higher level cyclotomic quiver Hecke algebras in affine type C

Susumu Ariki, Berta Hudak, Linliang Song, Qi Wang

TL;DR

This work classifies the representation type of cyclotomic quiver Hecke algebras for affine type $C^{(1)}_\ell$ across levels, establishing wildness in most higher-level cases and finitely many tame/finite exceptions. The authors develop a framework based on a connected quiver structure on the set of dominant maximal weights $\operatorname{max}^+(\Lambda)$, enabling systematic reductions to level-one results and new level-two analyses. They explicitly describe tame basic algebras (often Brauer graph algebras with straight-line graphs) and, in a few exceptional tamely behaved cases (notably $(t7)$ and $(t8)$), determine Morita equivalence classes via silting mutations and derived equivalences. The study connects with Brauer graph, silting theory, and derived-equivalence techniques to yield a comprehensive classification with potential applications to subcategories and dimension/decomposition problems in representation theory of cyclotomic KLR algebras. Overall, the paper provides a detailed, operational map from weight data to representation type, clarifying when tame or finite structures arise and illustrating how higher-level phenomena relate to well-understood level-one and Brauer-graph patterns.

Abstract

We determine the representation type of cyclotomic quiver Hecke algebras of affine type C. In the tame cases, we explicitly describe their basic algebras under the assumption $\text{ch}\ \mathbb{k}\ne2$, relying on the Morita invariance of cellularity.

Representation type of higher level cyclotomic quiver Hecke algebras in affine type C

TL;DR

This work classifies the representation type of cyclotomic quiver Hecke algebras for affine type across levels, establishing wildness in most higher-level cases and finitely many tame/finite exceptions. The authors develop a framework based on a connected quiver structure on the set of dominant maximal weights , enabling systematic reductions to level-one results and new level-two analyses. They explicitly describe tame basic algebras (often Brauer graph algebras with straight-line graphs) and, in a few exceptional tamely behaved cases (notably and ), determine Morita equivalence classes via silting mutations and derived equivalences. The study connects with Brauer graph, silting theory, and derived-equivalence techniques to yield a comprehensive classification with potential applications to subcategories and dimension/decomposition problems in representation theory of cyclotomic KLR algebras. Overall, the paper provides a detailed, operational map from weight data to representation type, clarifying when tame or finite structures arise and illustrating how higher-level phenomena relate to well-understood level-one and Brauer-graph patterns.

Abstract

We determine the representation type of cyclotomic quiver Hecke algebras of affine type C. In the tame cases, we explicitly describe their basic algebras under the assumption , relying on the Morita invariance of cellularity.
Paper Structure (82 sections, 71 theorems, 292 equations)

This paper contains 82 sections, 71 theorems, 292 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cartan datum in affine type C
  4. Dominant maximal weight
  5. Cyclotomic KLR algebra
  6. Some tame and wild algebras
  7. Brauer graph algebra
  8. Tilting mutation and derived equivalence
  9. The derived equivalence class of (t7)
  10. A connected quiver in affine type C
  11. A connected graph of $\operatorname{max}^+(\Lambda)$
  12. A connected quiver of $\operatorname{max}^+(\Lambda)$
  13. Comparison with previous level one results
  14. Proof strategy for the main theorem
  15. Results in level one cases
  16. ...and 67 more sections

Key Result

THEOREM 1

Let $R^\Lambda(\beta)$ be a cyclotomic KLR algebra of type $C^{(1)}_\ell$ and suppose that $R^\Lambda(\beta)$ is of finite representation type. If $\operatorname{char} \Bbbk\ne2$, then $R^\Lambda(\beta)$ is Morita equivalent to one of the following algebrasThese algebras already appeared in ASW-rep-

Theorems & Definitions (141)

  • THEOREM : finite cases
  • THEOREM : tame cases
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4: Ar-rep-type
  • Theorem 2.5: APS-type-C
  • Example 2.6
  • Example 2.7
  • Definition 2.8
  • ...and 131 more