St Andrews

Tropicalization is a process by which we replace algebraic geometry with the geometry of piecewise linear (tropical) objects. One of the central questions in the field is when this process can be reversed: that is, when can we realize a tropical object with an honest algebraic one. In this talk, I'll discuss some recent work on the tropical to Lagrangian correspondence, and state under what conditions homological mirror symmetry allows us to transfer Lagrangian realizations into algebraic ones.

In this talk we present a new construction of a Wedderburn basis for

the generic q-Schur algebra using the Du-Kazhdan-Lusztig basis. We show

that this gives rise to a new view on the Du-Lusztig homomorphism to the

asymptotic algebra. At the end we explain a potential plan for an attack

on James' conjecture using a reformulation by Meinolf Geck.

The talk starts with a gentle recollection of facts about

Iwahori-Hecke-Algebras of type A and q-Schur algebras and aims to be

accessible to people who are not (yet) experts in the representation

theory of q-Schur algebras.

All this is joint work with Olivier Brunat (Bochum).