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

TL;DR

This paper investigates the singularities of normalized R-matrices in quantum loop algebras of type ADE, revealing a correspondence between pole orders and E-invariants in Dynkin quivers, linking additive and monoidal categorifications of cluster algebras.

Contribution

It establishes a precise link between the pole orders of R-matrices and E-invariants, advancing understanding of categorifications in cluster algebra theory.

Findings

Pole orders of R-matrices match E-invariant dimensions.

Connects singularities with representation theory of Dynkin quivers.

Bridges additive and monoidal categorifications of cluster algebras.

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.