Discrete Geometry and Combinatorics Seminar

Kirill ZaynullinUniversity of Ottawa
Oriented cohomology of a linear algebraic group vs localization in duoidal categories

Monday, November 13, 2023 - 2:30pm
Malott 206

The Chow ring CH(G) of a split semi-simple linear algebraic group G is one of the key geometric invariants in the theory of linear algebraic groups, torsors, motives of twisted flag varieties. Starting from pioneering works by Grothendieck and Borel, it has been studied for decades and computed for all simple groups (see e.g. Kac 1985, Duan 2015's). In the present talk we explain how to describe (and, hence, to compute) an oriented cohomology (Borel-Moore homology) functor A(G) using the localization techniques of Kostant-Kumar and the techniques of duoidal categories of Aguiar-Mahajan: we show that the natural Hopf-algebra structure on A(G) can be lifted to a 'bi-Hopf' structure on the T-equivariant cohomology AT(G/B) of the complete flag variety. More generally, we prove that the structure algebra of a Bruhat moment graph of a root system is a Hopf algebroid with respect to the right Hecke and left Brion-Knutson-Tymoczko actions. As an application, we obtain an effective combinatorial way to compute the coproduct on A(G). This is a joint work with Martina Lanini and Rui Xiong.