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Positivity and greedy bases in rank 2 cluster algebras

Algebra & Discrete Mathematics

Speaker: Andrei Zelevinsky, Northeastern University and MSRI
Location: 1147 MSB
Start time: Mon, Oct 15 2012, 4:10PM

The original motivation for the study of cluster algebras was to design an algebraic framework for understanding total positivity and canonical bases. A lot of recent activity in the field has been directed towards various constructions of ``natural" bases in cluster algebras. One of the approaches to this problem was developed several years ago in a joint work with P.Sherman where it was shown that the indecomposable positive elements form a basis over integers in any rank 2 cluster algebra of finite or affine type. It is strongly suspected (but not proved) that this property does not extend beyond affine types. In a joint work with K.Lee and L.Li we go around this difficulty by constructing a new basis in any rank 2 cluster algebra, which we call the greedy basis. It consists of a special family of indecomposable positive elements that we call greedy elements. Inspired by a recent work of K.Lee - R.Schiffler and D.Rupel, we give an explicit combinatorial expression for greedy elements using the language of Dyck paths. The talk will be elementary and self-contained, not assuming any preliminary knowledge of the theory of cluster algebras.