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Journal of the European Mathematical Society

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Volume 18, Issue 5, 2016, pp. 1113–1159
DOI: 10.4171/JEMS/609

Published online: 2016-03-26

A cluster algebra approach to $q$-characters of Kirillov–Reshetikhin modules

David Hernandez[1] and Bernard Leclerc[2]

(1) Université Paris Diderot – Paris 7, Paris Rive Gauche, France
(2) Université de Caen, France

We describe a cluster algebra algorithm for calculating $q$-characters of Kirillov–Reshetikhin modules for any untwisted quantum affine algebra $U_q(\widehat{\mathfrak{g}})$. This yields a geometric $q$-character formula for tensor products of Kirillov–Reshetikhin modules. When $\mathfrak g$ is of type $A, D, E$, this formula extends Nakajima's formula for $q$-characters of standard modules in terms of homology of graded quiver varieties.

Keywords: Quantum affine algebra, cluster algebras, $q$-characters, Kirillov–Reshetikhin modules, geometric character formula

Hernandez David, Leclerc Bernard: A cluster algebra approach to $q$-characters of Kirillov–Reshetikhin modules. J. Eur. Math. Soc. 18 (2016), 1113-1159. doi: 10.4171/JEMS/609