Schubert varieties are certain closed subvarieties of the flag variety of a complex algebraic group \(G\). They are indexed by elements of the Weyl group \(W\) of \(G\). In general they are singular and some of the structure of their singularities can be understood via the resolutions known as the Bott-Samelson resolutions and the resulting interplay between the intersection cohomology of the Schubert variety and the ordinary cohomology of the fibers of these resolutions.

This talk focuses on joint work with my student, Jennifer Koonz, on some generalizations of the Bott-Samelson resolutions, which provide some new information about the singularities of Schubert varieties. Special cases of these resolutions have already appeared in work by Polo, Wolper, Ryan, and Zelevinsky in the case where \(G\) is the general linear group \(\operatorname{GL}(n)\) and \(W\) is the symmetric group \(S_n\). I will explain the Bott-Samelson resolutions and these new resolutions in some small examples for the general linear group and perhaps finish with a couple of propositions that are valid for arbitrary \(G\).