arXiv subjects

Anatole Castella

Publications and source records attributed to Anatole Castella.

4 recordsLinked to original sources

arXiv2017

Twisted Lawrence-Krammer representations

Lawrence-Krammer representations are an important family of linear representations of Artin-Tits groups of small type, which are known, under some assumptions on the parameters, to be faithful when the type is spherical (or more generally when they are restricted to the Artin-Tits monoid) and irreducible when the type is connected. Here, we investigate an analogue of these representations --- introduced by Digne in the spherical cases --- for every Artin-Tits monoid that appears as the submonoid of fixed points of an Artin-Tits monoid of small type under a group of graph automorphisms, and for the corresponding Artin-Tits group. Under the same assumptions on the parameters as in the small type cases, we first show that these so-called "twisted Lawrence-Krammer representations" are faithful, and we then prove, by computing their formulas when the group of graph automorphisms is of order two or three, their irreducibility in all the spherical and connected cases but one.

math.GR↗

arXiv2008

On (twisted) Lawrence-Krammer representations

Lawrence-Krammer representations (LK-representations for short) are linear representations of Artin-Tits groups of small type, which are of importance since they are known to be faithful when the type is spherical, or more generally when restricted to the monoid. If the construction is essentially unique for a given small and spherical type, the structure of the set of LK-representations for a given small type is not understood in general. Another important question is to ask if there exists an analogue of this construction in the non-small cases ; a first answer is given in [Digne, On the linearity of Artin Braid groups. J. Algebra 268, (2003) 39-57], where is constructed a faithful ``twisted'' LK-representation for the spherical, non-small and crystallographic types. The aim of this paper is to continue the investigations on those two topics. Regarding the first one, we classify the LK-representations of the Artin-Tits monoids and groups of small and affine type. Concerning the second one, we generalize the construction of op.cit. to any Artin-Tits monoid that appears as the submonoid of fixed points of an Artin-Tits monoid of small type under the action of graph automorphisms.

math.GR↗

arXiv2007

Flat modules over valuation rings

Let $R$ be a valuation ring and let $Q$ be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if $Q$ is maximal (respectively artinian). It is shown that each singly projective module is a content module if and only if any non-unit of $R$ is a zero-divisor and that each singly projective module is locally projective if and only if $R$ is self injective. Moreover, $R$ is maximal if and only if each singly projective module is separable, if and only if any flat content module is locally projective. Necessary and sufficient conditions are given for a valuation ring with non-zero zero-divisors to be strongly coherent or $π$-coherent. A complete characterization of semihereditary commutative rings which are $π$-coherent is given. When $R$ is a commutative ring with a self FP-injective quotient ring $Q$, it is proved that each flat $R$-module is finitely projective if and only if $Q$ is perfect.

math.RA↗

arXiv2004

Sur les automorphismes et la rigidite des groupes de Coxeter a angles droits

By underlying the commutation relation in a right-angled Coxeter group W, we recover the fact that right-angled Coxeter groups are rigid and we describe the second subgroup of Aut(W) that appears in the decomposition of Aut(W) into a semi-direct product established by J. Tits in "Sur le groupe des automorphismes de certains groupes de Coxeter" (Journal of Algebra 113, (1988), 346-357). ----- En mettant l'accent sur la relation de commutation dans un groupe de Coxeter a angles droits W, on redemontre le fait que les groupes de Coxeter a angles droits sont rigides et on decrit le second sous-groupe de Aut(W) apparaissant dans la decomposition en produit semi-direct de Aut(W) etablie par J. Tits dans "Sur le groupe des automorphismes de certains groupes de Coxeter"(Journal of Algebra 113, (1988), 346-357).

math.GR↗