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Special Colloquium: Combinatorics of total positivity and cluster algebras

Special Events

Speaker: Melissa Sherman-Bennett, MIT
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Location: 1147 MSB
Start time: Mon, Jan 9 2023, 3:10PM

Classically, a matrix is totally nonnegative (TNN) if all determinants of submatrices are nonnegative. Lusztig generalized this notion, defining the TNN part of any semisimple Lie group G and partial flag variety G/P; Postnikov independently defined the TNN Grassmannian. The rich combinatorics of these TNN spaces led Fomin and Zelevinsky to define cluster algebras, a class of commutative ring intended to provide a combinatorial and algebraic framework for total nonnegativity. I'll give an overview of these topics, and will then discuss some recent developments in the combinatorics of total nonnegativity and cluster algebras, focusing on the amplituhedron (a generalization of the TNN Grassmannian related to high energy physics) and on the cluster structure of open Schubert and open Richardson varieties.