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Endomorphism Algebras in Coxeter Categorifications and HarishChandra 2Categories
 West, Benjamin
 Advisor(s): Rouquier, Raphael
Abstract
Given the data of a Coxeter system (W,S), a Coxeter categorification is a 2category in which the objects are subsets of S, the generating 1morphisms categorify induction and restriction functors associated to parabolic subgroups, and the generating 2morphisms impose certain coherence conditions and structural properties among the 1morphisms. Of particular interest is the structure of the 2homomorphism spaces of these 1morphisms. Furthermore, given a connected, reductive, algebraic group G over an algebraically closed field k, a chosen Frobenius endomorphism F on G determines a parameter q in k, and the Weyl group of G gives rise to a Coxeter system. When this system is of rank 1, we construct by generators and relations an extension of the Coxeter categorification, independent of q, where the 2homomorphism spaces are free modules of finite rank over the ring of Laurent polynomials with integer coefficients. An explicit description of the 2homomorphism spaces between generating 1morphisms is given, along with an algorithm lifting these descriptions to the 2homomorphism spaces of arbitrary 1morphisms. Then a nontrivial 2functor from this 2category is constructed into the 2category of bimodules. Some conjectural constructions are given in the case that W has arbitrary finite rank, in particular a proposal for the endomorphism ring of the generating 1morphism from the empty set to S that is an extension of an algebra introduced by Marin.
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