arXiv subjects
Publications and source records attributed to Ivan Marin.
In this short note we provide an alternative proof of a theorem of Kapovich and Millson about the Malcev completion of an arbitrary Artin group, and determine the Malcev completion of the braid group of an irreducible finite complex reflection group.
We construct new geometric realizations of simplicial and pre-simplicial sets where the standard $n$-simplex, viewed as the space of probability measures on $n+1$ elements, is replaced by the space of $(n+1)$-valued random variables, with the topology of probability convergence. We prove that the map which associates to a random variable its probability law is an homotopy equivalence from these new geometric realizations to the classical ones. Finally, we prove that this realization provides a new Quillen equivalence between simplicial sets and topological spaces.
In this paper we introduce a class of `parabolic' subgroups for the generalized braid group associated to an arbitrary irreducible complex reflection group, which maps onto the collection of parabolic subgroups of the reflection group. Except for one case, which is proven separately elsewhere, we prove that this collection forms a lattice, so that intersections of parabolic subgroups are parabolic subgroups. In particular, every element admits a parabolic closure, which is the smallest parabolic subgroup containing it. We furthermore prove that it provides a simplicial complex which generalizes the curve complex of the usual braid group. In the case of real reflection groups, this complex generalizes the one previously introduced by Cumplido, Gebhardt, Gonz\'alez-Meneses and Wiest for Artin groups of spherical type. We show that it shares similar properties, and similarly conjecture its hyperbolicity, with a few additional results in this direction.
We investigate the rational cohomology of the quotient of (generalized) braid groups by the commutator subgroup of the pure braid groups. We provide a combinatorial description of it using isomorphism classes of certain families of graphs. We establish Poincar\'e dualities for them and prove a stabilization property for the infinite series of reflection groups.
Let $W_0$ be a reflection subgroup of a finite complex reflection group $W$, and let $B_0$ and $B$ be their respective braid groups. In order to construct a Hecke algebra $\widetilde{H}_0$ for the normalizer $N_W(W_0)$, one first considers a natural subquotient $\widetilde{B}_0$ of $B$ which is an extension of $N_W(W_0)/W_0$ by $B_0$. We prove that this extension is split when $W$ is a Coxeter group, and deduce a standard basis for the Hecke algebra $\widetilde{H}_0$. We also give classes of both split and non-split examples in the non-Coxeter case.
In the context of Hecke algebras of complex reflection groups, we prove that the generalized Hecke algebras of normalizers of parabolic subgroups are semidirect products, under suitable conditions on the parameters involved in their definition.
We introduce a new `geometric realization' of an (abstract) simplicial complex, inspired by probability theory. This space (and its completion) is a metric space, which has the right (weak) homotopy type, and which can be compared with the usual geometric realization through a natural map, which has probabilistic meaning : it associates to a random variable its probability mass function. This `probability map' function is proved to be a (Serre) fibration and a (weak) homotopy equivalence.
In this technical note, we complete the PhD work of A. Esterle about determining the image of any Artin group of finite Coxeter type inside the associated Hecke algebra over a finite field, when the latter is semisimple. The only remaining case was the 48-dimensional irreducible representation in type $H_4$, for which the image is proven here to be $Ω_{48}^+$.
We introduce a distance function between simplicial complexes and study several of its properties.
The Brauer-Chen algebra is a generalization of the algebra of Brauer diagrams to arbitrary complex reflection groups, that admits a natural monodromic deformation. We determine the generic representation theory of the first non trivial quotient of this algebra. We also define natural extensions of this algebra and prove that they similarly admit natural monodromic deformations.
We study a quotient of the group algebra of the braid group in which the Artin generators satisfy a cubic relation. This quotient is maximal among the ones satisfying such a cubic relation. It is finite-dimensional for at least n at most 5 and we investigate its module structure in this range. We also investigate the proper quotients of it that appear in the realm of quantum groups, and describe another maximal quotient related to the usual Hecke algebras. Finally, we describe the connection between this algebra and a quotient of the algebra of horizontal chord diagrams introduced by Vogel. We prove that these two are isomorphic for n at most 5.
This article extends the works of Gonçalves, Guaschi, Ocampo [GGO] and Marin [MAR2] on finite subgroups of the quotients of generalized braid groups by the derived subgroup of their pure braid group. We get explicit criteria for subgroups of the (complex) reflection group to lift to subgroups of this quotient. In the specific case of the classical braid group, this enables us to describe all its finite subgroups : we show that every odd-order finite group can be embedded in it, when the number of strands goes to infinity. We also determine a complete list of the irreducible reflection groups for which this quotient is a Bieberbach group.
We construct classifying spaces for discrete and compact Lie groups, with the property that they are topological groups and complete metric spaces in a natural way. We sketch a program in view of extending these constructions.
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.
We prove two new cases of the Broué-Malle-Rouquier freeness conjecture for the Hecke algebras associated to complex reflection groups. These two cases are the complex reflection groups of rank 2 called $G_{20}$ and $G_{21}$ in the Shephard and Todd classification. This reduces the number of remaining unproven cases to 3.
We attach to every Coxeter system (W,S) an extension C_W of the corresponding Iwahori-Hecke algebra. We construct a 1-parameter family of (generically surjective) morphisms from the group algebra of the corresponding Artin group onto C_W. When W is finite, we prove that this algebra is a free module of finite rank which is generically semisimple. When W is the Weyl group of a Chevalley group, C_W naturally maps to the associated Yokonuma-Hecke algebra. When W = S_n this algebra can be identified with a diagram algebra called the algebra of `braids and ties'. The image of the usual braid group in this case is investigated. Finally, we generalize our construction to finite complex reflection groups, thus extending the Broue-Malle-Rouquier construction of a generalized Hecke algebra attached to these groups.
We complete the computation of the integral homology of the generalized braid group $B$ associated to an arbitrary irreducible complex reflection group $W$ of exceptional type. In order to do this we explicitely computed the recursively-defined differential of a resolution of $\mathbf{Z}$ as a $\mathbf{Z} B$-module, using parallel computing. We also deduce from this general computation the rational homology of the Milnor fiber of the singularity attached to most of these reflection groups.
We prove the freeness conjecture of Broue, Malle and Rouquier for the Hecke algebras associated to the primitive complex 2-reflection groups with a single conjugacy class of reflections.