Total Positivity in Twisted Flag Varieties

Lusztig’s theory of total positivity has led to remarkable connections between geometry, combinatorics, and representation theory. In this talk, I will discuss how this theory extends to twisted flag varieties for arbitrary Kac–Moody groups. This is based on a joint work with Kaitao Xie.

Our main result shows that the totally nonnegative part of a twisted flag variety admits a cell decomposition, and the closure of each cell is a regular CW complex. This generalizes earlier work on ordinary flag varieties and allows us to deduce similar structural results for the totally nonnegative double flag variety and for the link of in totally nonnegative double Bruhat cells, the latter answering a conjecture of Fomin and Zelevinsky. I will also briefly mention some connection to canonical bases of tensor product.

For information about the Richard P. Stanley Seminar in Combinatorics, visit… https://math.mit.edu/combin/