Speaker: Samuel Heard
University of Notre Dame
Will give a Graduate Student Seminar entitled:
Double Bruhat Cells and Total Positivity
Abstract:
The theory of total positivity was developed in the 1930's to study matrices whose minors were all nonnegative. This notion was generalized in the 1990s when Lustzig introduced the notion of a totally nonnegative variety G>0 in a reductive group G. Fomin and Zelevinsky further studied G>0 for G a simply connected complex Lie group with Weyl Group W. In this setting, we can describe G as a disjoint union of varieties Gu,v indexed by u,v ε W x W.
In this presentation, we will cover three results of Fomin and Zelevinsky concerning special properties of Gu,v. We will first show that Gu,v is a smooth variety, and describe it as an open subset of affine space. Then, we will give natural coordinates on Gu,v coming from Lie Theory. Finally, we will use these coordinates to describe G>0 in G.
Date: 11-07-2024
Time: 2:00 pm
Location: 258 Hurley Bldg
Originally published at math.nd.edu.