Reflection group relations arising from cluster algebras

Seven, Ahmet

doi:10.1090/proc/13157

Reflection group relations arising from cluster algebras
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by Ahmet I. Seven PDF

Proc. Amer. Math. Soc. 144 (2016), 4641-4650 Request permission

Abstract:

There is a well-known analogy between cluster algebras and Kac-moody algebras: roughly speaking, Kac-Moody algebras are associated with symmetrizable generalized Cartan matrices while cluster algebras correspond to skew-symmetrizable matrices. In this paper, we study an interplay between these two classes of matrices. We obtain relations in the Weyl groups of Kac-Moody algebras that come from mutation classes of skew-symmetrizable matrices. More precisely, we establish a set of relations satisfied by the reflections of the so-called companion bases; these include c-vectors, which parametrize coefficients in a cluster algebra with principal coefficients. These relations generalize the relations obtained by Barot and Marsh for finite type. For affine type, we also show that the reflections of the companion bases satisfy the relations obtained by Felikson and Tumarkin. As an application, we obtain some combinatorial properties of the mutation classes of skew-symmetrizable matrices.

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