Equivariant multiplicities via representations of quantum affine algebras

Duration: 1 hour 4 mins

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Description:	Dr Elie Casbi (Max-Planck-Institut für Mathematik, Bonn) 22nd November 2021 \| 16:00 - 17:00


Created:	2021-11-23 15:06
Collection:	Cluster algebras and representation theory
Publisher:	Isaac Newton Institute for Mathematical Sciences
Copyright:	Dr Elie Casbi
Language:	eng (English)


Abstract:	Equivariant multiplicities via representations of quantum affine algebras Elie Casbi For any simply-laced type simple Lie algebra g and any height function ξ adapted to an orientation Q of the Dynkin diagram of g, Hernandez-Leclerc introduced a certain category C⁄ξ of representations of the quantum affine algebra Uqppgq, as well as a subcategory CQ of C⁄ξ whose complexified Grothendieck ring is isomorphic to the coordinate ring CrNs of a maximal unipotent subgroup. In this talk, I will present our construction of an algebraic morphism Drξ on a torus Y⁄ξ containing the image of K0pC⁄ξq under the truncated q-character morphism. We prove that the restriction of Drξ to K0pCQq coincides with the morphism D recently introduced by Baumann-Kamnitzer-Knutson in their study of equivariant multiplicities of Mirković-Vilonen cycles. I will begin by showing how the cluster structure of CrNs played a key role in the proof of this result. I will also explain how this alternative description of D allowed us to prove a conjecture from an earlier work of mine on the distinguished values of D on the flag minors of CrNs. Finally I will conclude with some applications of our results and some perspectives of further developments. This is a joint work with Jianrong LI.

Abstract:

Equivariant multiplicities via representations of quantum affine algebras
Elie Casbi
For any simply-laced type simple Lie algebra g and any height function ξ adapted to an
orientation Q of the Dynkin diagram of g, Hernandez-Leclerc introduced a certain category
C⁄ξ of representations of the quantum affine algebra Uqppgq, as well as a subcategory CQ of
C⁄ξ whose complexified Grothendieck ring is isomorphic to the coordinate ring CrNs of a
maximal unipotent subgroup. In this talk, I will present our construction of an algebraic morphism Drξ on a torus Y⁄ξ containing the image of K0pC⁄ξq under the truncated q-character
morphism. We prove that the restriction of Drξ to K0pCQq coincides with the morphism D recently introduced by Baumann-Kamnitzer-Knutson in their study of equivariant multiplicities
of Mirković-Vilonen cycles. I will begin by showing how the cluster structure of CrNs played
a key role in the proof of this result. I will also explain how this alternative description of D
allowed us to prove a conjecture from an earlier work of mine on the distinguished values of
D on the flag minors of CrNs. Finally I will conclude with some applications of our results
and some perspectives of further developments.
This is a joint work with Jianrong LI.

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