Koszul duality and Soergel bimodules for dihedral groups

Abstract:

Every Coxeter system $(W,S)$ gives rise to a Hecke algebra $\textbf {H} _{(W,S)}$ which can be categorified by the additive monoidal category of Soergel bimodules $\mathcal S\mathcal B$. Under this isomorphism the Kazhdan-Lusztig basis $\{\underline {H}_x\}_{x\in W}$ corresponds to certain indecomposable Soergel bimodules $\{B_x\}_{x\in W}$ (up to shift). In this thesis we study the structure of the endomorphism algebra (of maps of all degrees) $\mathcal A:= \operatorname {End}^\bullet _{\mathcal S\mathcal B}\left (\bigoplus _{x\in W} B_x \right )\otimes _R\mathbb {R}$. Via category $\mathcal {O}$ it has been proven for all Weyl groups that $\mathcal A$ is a self-dual Koszul algebra. We extend this result to all dihedral groups by purely algebraic methods using representation theory of quivers and Soergel calculus.