Monday, December 3, 2018 at 1:00pm to 2:00pm
Kaprielian Hall (KAP), 414
3620 South Vermont Avenue, Los Angeles, CA 90089
Matthew Hogancamp, USC
Abstract: The Hecke algebra H_n is a deformation of the group algebra of the symmetric group, and has important connections to representation theory of quantum sl_n, low-dimensional topology (via the HOMFLY-PT link polynomial and sl_n Reshetikhin-Turaev invariants), and the algebraic combinatorics of symmetric functions. There is a categorfiication of this story, which replaces the Hecke algebra with the tensor category of Soergel bimodules (often called the Hecke category). The Hecke category if of central importance in geometric representation theory and the construction of Khovanov-Rozansky homology theories for knots and links. At the categorical level, the relation between Hecke algebras and symmetric functions manifests as a deep connection between the Hecke category and the algebraic geometry of Hilbert schemes of points in the plane. This connection, conjectured by Gorsky-Negut-Rasmussen, is still quite mysterious. In this talk I will survey this story, including results of mine and others on the Hecke category and newly emerging details on its relation to Hilbert schemes. Recently Eugene Gorsky and I made significant progress in understanding this relationship. In our work we define the curved Hecke category, generated by certain curved complexes of Soergel bimodules, and construct an explicit functor to sheaves on the Hilbert scheme.