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Singularities of normalized R-matrices and E-invariants for Dynkin quivers

Ryo Fujita

Abstract

We study the singularities of normalized R-matrices between arbitrary simple modules over the quantum loop algebra of type ADE in Hernandez--Leclerc's level-one subcategory using equivariant perverse sheaves, following the previous works by Nakajima [Kyoto J. Math. 51(1), 2011] and Kimura--Qin [Adv. Math. 262, 2014]. We show that the pole orders of these R-matrices coincide with the dimensions of E-invariants between the corresponding decorated representations of Dynkin quivers. This result can be seen as a correspondence of numerical characteristics between additive and monoidal categorifications of cluster algebras of finite ADE type.

Singularities of normalized R-matrices and E-invariants for Dynkin quivers

Abstract

We study the singularities of normalized R-matrices between arbitrary simple modules over the quantum loop algebra of type ADE in Hernandez--Leclerc's level-one subcategory using equivariant perverse sheaves, following the previous works by Nakajima [Kyoto J. Math. 51(1), 2011] and Kimura--Qin [Adv. Math. 262, 2014]. We show that the pole orders of these R-matrices coincide with the dimensions of E-invariants between the corresponding decorated representations of Dynkin quivers. This result can be seen as a correspondence of numerical characteristics between additive and monoidal categorifications of cluster algebras of finite ADE type.

Paper Structure

This paper contains 26 sections, 20 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Algebraic preliminaries and main theorem
  4. Representations of quantum loop algebras
  5. Normalized R-matrices
  6. E-invariants
  7. Main theorem
  8. Cluster theoretical interpretation
  9. Cluster algebras
  10. Additive categorification
  11. Interpretation by injective copresentations
  12. Monoidal categorification
  13. Conclusion
  14. Geometric preliminaries
  15. Notation
  16. ...and 11 more sections

Key Result

Theorem 1.1

For any simple objects $L$ and $L'$ of $\mathscr{C}_1$, we have where $\mathcal{M}$ and $\mathcal{M}'$ are rigid decorated representations of $Q$ satisfying $\tilde{\chi}_q(L) = CC(\mathcal{M})$ and $\tilde{\chi}_q(L') = CC(\mathcal{M}')$ up to frozen factors.

Theorems & Definitions (37)

  • Theorem 1.1
  • Proposition 2.1: KKOP
  • Lemma 2.2: cf. KKKO
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6: Hernandez--Leclerc HLHL2
  • Theorem 2.7
  • Remark 2.8
  • Lemma 2.9
  • ...and 27 more