no. 2
Singularities of normalized R-matrices and E-invariants for Dynkin quivers
Fujita, Ryo1 orcid
1Research Institute for Mathematical Sciences, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo, Kyoto, 606-8502, Japan
Annals of Representation Theory, Volume 3 (2026) no. 2, pp. 179-204

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$\hspace{1.111pt}$(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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/art.39
Classification: 13F60, 16G20, 17B37, 18M05
Keywords: quantum affine algebras, monoidal category, R-matrix, quiver representations, E-invariant, quiver varieties, cluster algebras

Fujita, Ryo  1

1 Research Institute for Mathematical Sciences, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo, Kyoto, 606-8502, Japan
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Fujita, Ryo. Singularities of normalized R-matrices and E-invariants for Dynkin quivers. Annals of Representation Theory, Volume 3 (2026) no. 2, pp. 179-204. doi: 10.5802/art.39
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