Student Number Theory Seminar 2019-20
3/23/20: Andy Hardt -- Iwahori-Hecke algberas in multiple contexts
Abstract: We define and explore (Iwahori-)Hecke algebras, which Iwahori defined as spaces of functions on reductive p-adic groups. Originally used as a tool for simplifying p-adic representation theory, they have a nice presentation as deformations of Weyl group algebras, and have found applications in disparate areas such as Schubert varieties, knot theory, and quantum groups. We won't have time for all of this, but we'll take a look at the p-adic representation definition of Hecke algebras, the "deformation" definition, and then Jimbo's use of Hecke algebras in his quantum Schur-Weyl duality. To relate this talk to other recent material, tecke algebra starred in John O'Brien's SNT talks the two weeks before Spring Break, and the quantum groups interpretation may be of particular interest to anyone in Ben Brubaker's Topics in Algebra course.