Seminar on Combinatorics, Lie Theory, ​and Topology

Singular Light Leaves: A Diagrammatic Basis for Bott-Samelson Bimodules

by Leonardo Patimo (Università di Pisa)

Aula Magna (Department of Mathematics)

Aula Magna

Department of Mathematics


Bott-Samelson bimodules correspond geometrically to resolutions of singularities of Schubert varieties. In the nonsingular case, i.e., for Schubert varieties in full flag varieties, Libedinsky introduced a basis (called light leaves) between Bott-Samelson bimodules which has been extensively used to compute the decomposition behavior in the Hecke category, for example enabling the discovery of counterexamples to Lusztig's conjecture on representations in characteristic $p$.
In this talk, we will provide a thorough generalization to the singular setting, which corresponds geometrically to partial flag varieties. We will describe how to construct the basis of singular light leaves using the language of diagrammatic calculus. This construction has concrete applications in computing intersection forms, as well as more theoretical implications. This is a joint project with B. Elias, H. Ko, and N. Libedinsky.