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Affine cluster monomials are generalized minors

Part of: Linear algebraic groups and related topics Arithmetic rings and other special rings

Published online by Cambridge University Press:  14 June 2019

Dylan Rupel ,
Salvatore Stella  and
Harold Williams
Dylan Rupel
Affiliation:
University of Notre Dame, Department of Mathematics, Notre Dame, IN 46556, USA email [email protected]
Salvatore Stella
Affiliation:
Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK email [email protected]
Harold Williams
Affiliation:
University of California, Davis, Department of Mathematics, Davis, CA 95616, USA email [email protected]
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Abstract

We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac–Moody groups. We prove that all cluster monomials with $\mathbf{g}$-vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero–Chapoton description via quiver representations. In type $A_{1}^{(1)}$, we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac–Moody algebras.

Keywords

cluster algebrasKac–Moody groups

MSC classification

Primary: 13F60: Cluster algebras 20G44: Kac-Moody groups
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 7 , July 2019 , pp. 1301 - 1326
Copyright
© The Authors 2019 

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