Combinatorics Seminar: GL(n)-Orbits On Two Complete Flag Varieties and A Line

Abstract: Let G = GL(n, C) be the n × n complex general linear group, and let B be the flag variety of G. It is well-known that G acts diagonally on the triple product B × B × P n−1 with finitely many orbits. These orbits are of interest in geometric representation theory and have been an object of study for some time. In this talk, we describe a new approach to understand the geometric and combinatorial properties of these orbits. In particular, we show that to each orbit we can associate a pair of Weyl group elements and the closure relations between the orbits can be described using a variation on the product of Bruhat orders on W × W. If time permits, we will also discuss various combinatorial models for these orbits and how they can be used to compute the exponential generating function for the sequence {|G\(B × B × P n−1 )|}n≥1 and related counting problems.

Host: Martha Precup