Alexandre Esterle (Amiens)

Bilinear forms for self-dual W-graphs.

Abstract:

W-graphs were first introduced by Kazhdan and Lusztig in 1979. In their paper, they showed how to construct W-graphs affording irreducible representations in type A. Their method generalizes to many types and in 1981, Gyoja proved that any irreducible representation of an Iwahori-Hecke algebra could be afforded by a W-graph. In this talk, we will explain how W-graphs give us representations of Hecke algebras and then explain representation properties which can be deduced from those W-graphs. In particular, the dual representation of a representation afforded by a W-graph is afforded by a W-graph which can be determined in a combinatoric way. We will look at properties which should naturally be verified by W-graphs affording self-dual representations and see they are not always verified. We will then show how to find W-graphs verifying those properties for a given self-dual representation. We will then use those properties to look at the image of Artin as subgroups of the invertible elements of Hecke algebras in type H4.