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Register a new userHomological stability for Iwahori-Hecke algebras
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Authors
Richard Hepworth
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We show that the Iwahori-Hecke algebras H_n of type A_{n-1} satisfy homological stability, where homology is interpreted as an appropriate Tor group. Our result precisely recovers Nakaokas homological stability result for the symmetric groups in the case that the defining parameter is equal to 1. We believe that this paper, and our joint work with Boyd on Temperley-Lieb algebras, are the first time that the techniques of homological stability have been applied to algebras that are not group algebras.
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We prove that certain sequences of Artin monoids containing the braid monoid as a submonoid satisfy homological stability. When the $K(pi,1)$ conjecture holds for the associated family of Artin groups this establishes homological stability for these groups. In particular, this recovers and extends Arnolds proof of stability for the Artin groups of type $A$, $B$ and $D$.
This article gives a fairly self-contained treatment of the basic facts about the Iwahori-Hecke algebra of a split p-adic group, including Bernsteins presentation, Macdonalds formula, the Casselman-Shalika formula, and the Lusztig-Kato formula.
We prove that certain families of Coxeter groups and inclusions $W_1hookrightarrow W_2hookrightarrow...$ satisfy homological stability, meaning that in each degree the homology $H_ast(BW_n)$ is eventually independent of $n$. This gives a uniform treatment of homological stability for the families of Coxeter groups of type $A_n$, $B_n$ and $D_n$, recovering existing results in the first two cases, and giving a new result in the third. The key step in our proof is to show that a certain simplicial complex with $W_n$-action is highly connected. To do this we show that the barycentric subdivision is an instance of the basic construction, and then use Daviss description of the basic construction as an increasing union of chambers to deduce the required connectivity.
We state a conjecture that relates the derived category of smooth representations of a p-adic split reductive group with the derived category of (quasi-)coherent sheaves on a stack of L-parameters. We investigate the conjecture in the case of the principal block of GL_n by showing that the functor should be given by the derived tensor product with the family of representations interpolating the modified Langlands correspondence over the stack of L-parameters that is suggested by the work of Helm and Emerton-Helm.
Let $W$ be a Coxeter group. The goal of the paper is to construct new Hopf algebras that contain Hecke algebras $H_{bf q}(W)$ as (left) coideal subalgebras. Our Hecke-Hopf algebras ${bf H}(W)$ have a number of applications. In particular they provide new solutions of quantum Yang-Baxter equation and lead to a construction of a new family of endo-functors of the category of $H_{bf q}(W)$-modules. Hecke-Hopf algebras for the symmetric group are related to Fomin-Kirillov algebras, for an arbitrary Coxeter group $W$ the Demazure part of ${bf H}(W)$ is being acted upon by generalized braided derivatives which generate the corresponding (generalized) Nichols algebra.
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