Lattice extensions of Hecke algebras

Abstract

We investigate the extensions of the Hecke algebras of finite (complex) reflection groups by lattices of reflection subgroups that we introduced, for some of them, in our previous work on the Yokonuma-Hecke algebras and their connections with Artin groups. When the Hecke algebra is attached to the symmetric group, and the lattice contains all reflection subgroups, then these algebras are the diagram algebras of braids and ties of Aicardi and Juyumaya. We prove a stucture theorem for these algebras, generalizing a result of Espinoza and Ryom-Hansen from the case of the symmetric group to the general case. We prove that these algebras are symmetric algebras at least when W is a Coxeter group, and in general under the trace conjecture of Broué, Malle and Michel.

Truncations and extensions of the Brauer-Chen algebra

“…Our second main result (see theorem 2.6) is the following one. We proved in [17,18] that Hecke algebras admit natural extensions by the Möbius algebra L of the lattice L of the reflection subgroups of W , and that these algebras are monodromic deformation of W ⋉ L in the same way as the Hecke algebra is a monodromic deformation of W . Here we prove that the same phenomenon occurs for the Brauer-Chen algebra.…”

Braid groups of normalizers of reflection subgroups

“…The goal of this section is to recall the definitions referred to in the introduction, in particular of the group B0 , the short exact sequence (1.2), and the Hecke algebra H0 . For more details, see [3,9].…”

“…Recall from [9] that the Hecke algebra H0 associated to N W (W 0 ) is defined as the quotient of B0 by the same Hecke relations as in the definition of H 0 . If the short exact sequence (1.2) splits, then B0 is a semidirect product of B 0 with the group N W (W 0 )/W 0 , and consequently H0 is a semidirect product of H 0 with the group N W (W 0 )/W 0 .…”

“…One main reason for considering these questions, which was in fact the original motivation, is the study of the Hecke algebra H0 associated to N W (W 0 ), which was defined in [9] as a certain quotient of the group algebra of B0 . A splitting of (1.2) implies a semidirect product decomposition of H0 .…”

The principal block of a $\mathbb{Z}_\ell$-spets and Yokonuma type algebras

“…We show that the freeness conjecture holds for Coxeter groups (Theorem 5.12) and for most of the generalised Grassmannians (Theorem 5.13). We also discuss the relationship of Y to the algebra considered by Marin in [30,31]. Section 6 deals with the specialised algebras Y ψ and contains the proof of Theorem 1 (Theorem 6.5) and Corollary 2.…”