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The product of simple modules over KLR algebras and quiver Grassmannians

Yingjin Bi

Abstract

In this paper, we study the product of two simple modules over KLR algebras using the quiver Grassmannians for Dynkin quivers. More precisely, we establish a bridge between the Induction functor on the category of modules of KLR algebras and the irreducible components of quiver Grassmannians for Dynkin quivers via a sort of extension varieties, which is an analogue of the extension group in Hall algebras. As a result, we give a necessary condition when the product of two simple modules over a KLR algebra is simple using the set of irreducible components of quiver Grassmannians. In particular, in some special cases, we provide a proof for the conjecture recently proposed by Lapid and Minguez.

The product of simple modules over KLR algebras and quiver Grassmannians

Abstract

In this paper, we study the product of two simple modules over KLR algebras using the quiver Grassmannians for Dynkin quivers. More precisely, we establish a bridge between the Induction functor on the category of modules of KLR algebras and the irreducible components of quiver Grassmannians for Dynkin quivers via a sort of extension varieties, which is an analogue of the extension group in Hall algebras. As a result, we give a necessary condition when the product of two simple modules over a KLR algebra is simple using the set of irreducible components of quiver Grassmannians. In particular, in some special cases, we provide a proof for the conjecture recently proposed by Lapid and Minguez.

Paper Structure

This paper contains 33 sections, 39 theorems, 229 equations.

Table of Contents

  1. Introduction
  2. Premise
  3. Notations
  4. Representations of Dynkin quivers
  5. Kostant Partitions
  6. Extension of Kostant partitions
  7. Hall maps for Representation spaces
  8. Induction maps
  9. Restriction maps
  10. The orbits and restriction maps
  11. Quiver Grassmannians and Extension varieties
  12. Quiver Grassmannians
  13. Irreducible components of quiver Grassmannians
  14. Graded quiver varieties
  15. Repetition quivers
  16. ...and 18 more sections

Key Result

Lemma 1.3

[Cerulli Irelli and Reineke] Under the above assumption, the set of irreducible components of ${\operatorname{Gr}}_\beta(M_\lambda)$ is identified with the following set

Theorems & Definitions (99)

  • Conjecture 1.1
  • Lemma 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Example 2.1
  • Definition 2.2
  • Example 2.3
  • ...and 89 more