Abstracts
for the Seminar

Fall 2015

**Speaker: **Evgeny Smirnov, HSE, National Research University, Moscow

**Title: **Spherical double flag varieties

**Time:** 2:30 PM, Monday, November 2, 2015

**Place:** Malott 206

**Abstract:** Classical Schubert calculus deals with orbits of a Borel subgroup in
GL(V) acting on a Grassmann variety Gr(k,V) of k-planes in a
finite-dimensional vector space V. These orbits (Schubert cells) and
their closures (Schubert varieties) are very well studied both from
the combinatorial and the geometric points of view.
One can go one step farther, considering the direct product of two
Grassmannians and a Borel subgroup in GL(V) in acting diagonally on
this variety. In this case, the number of orbits still remains finite,
but their combinatorics and geometry of their closures become much
more involved. However, something still can be said about them. I will
explain how to index the closures of a Borel subgroup in
Gr(k,V)xGr(l,V) combinatorially and construct the resolutions of their
singularities, which are similar to Bott-Samelson resolutions for
ordinary Schubert varieties. I will also speak about the analogues of
these results for direct products of partial flag varieties for
reductive groups of types different from A_n, due to P.Achinger,
N.Perrin, and myself.

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