# Mathematical Sciences Research Institute

Home » Positivity and greedy bases in rank 2 cluster algebras

# Seminar

Positivity and greedy bases in rank 2 cluster algebras September 19, 2012 (03:30 PM PDT - 04:30 PM PDT)
Location: MSRI: Simons Auditorium
Speaker(s) Andrei Zelevinsky
Description No Description
Video
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.