Leonardo Patimo: Atomic Decompositions in Hecke Algebras and Crystal Graphs

Abstract:

The Kazhdan–Lusztig bases of the Hecke algebra have remarkable positivity properties, some of which remain to be fully explored. In the case of type A spherical Hecke algebras, the KL basis is positive with respect to the atomic basis, whose elements record all weights of some irreducible representation with multiplicity one.

A counterpart in representation theory of this positivity is the atomic decomposition of crystal graphs. Such a decomposition is obtained by considering the closure of Weyl group orbits under an extremal crystal operator. The grading in KL polynomials here corresponds to the charge statistics on semistandard tableaux defined by Lascoux and Schuetzenberger.

In the talk, we will present refinements of these decompositions, both in the Hecke algebra and in crystal graphs, which conjecturally agree on regular weights. We will also discuss what happens beyond type A and how this approach can help to find charge statistics in general types.