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Cédric Bonnafé

Publications and source records attributed to Cédric Bonnafé.

At least 55 records · Page 3Linked to original sources
arXiv2009

Automorphisms of Coxeter groups and Lusztig's conjectures for Hecke algebras with unequal parameters

Let $(W,S)$ be a Coxeter system, let $G$ be a finite solvable group of automorphisms of $(W,S)$ and let $φ$ be a weight function which is invariant under $G$. Let $φ_G$ denote the weight function on $W^G$ obtained by restriction from $φ$. The aim of this paper is to compare the ${\mathbf{a}}$-function, the set of Duflo involutions and the Kazhdan-Lusztig cells associated to $(W,φ)$ and to $(W^G,φ_G)$, provided that Lusztig's Conjectures hold.

math.RT↗
arXiv2009

On Kazhdan-Lusztig cells in type B

We prove that, for any choice of parameters, the Kazhdan-Lusztig cells of a Weyl group of type $B$ are unions of combinatorial cells (defined using the domino insertion algorithm).

math.RT↗
arXiv2008

Compactification des variétés de Deligne-Lusztig

We construct explicitly the normalisation of the Bott-Samelson-Demazure-Hansen compactification of Deligne-Lusztig varieties $X(w)$ in their covering $Y(w)$: we retrieve a result by Deligne-Lusztig about the local monodromy around the divisors of the compactification.

math.AG↗
arXiv2008

Topologie sur l'ensemble des parties positives d'un réseau

We define a notion of {\it positive part} of a lattice $Λ$ and we endow the set of such positive parts with a topology. We then study some properties of this topology, by comparing it with the one of $V^*/\RM_{> 0}$, where $V^*$ is the dual vector space of $\RM \otimes_\ZM Λ$.

math.GN↗
arXiv2008

Semidirect product decomposition of Coxeter groups

Let $(W,S)$ be a Coxeter system, let $S=I \dot{\cup} J$ be a partition of $S$ such that no element of $I$ is conjugate to an element of $J$, let $\widetilde{J}$ be the set of $W_I$-conjugates of elements of $J$ and let $\widetilde{W}$ be the subgroup of $W$ generated by $\widetilde{J}$. We show that $W=\widetilde{W} \rtimes W_I$ and that $(\widetilde{W},\widetilde{J})$ is a Coxeter system.

math.GR↗
arXiv2008

Around Solomon's descent algebras

We study different problems related to the Solomon's descent algebra $Σ(W)$ of a finite Coxeter group $(W,S)$: positive elements, morphisms between descent algebras, Loewy length... One of the main result is that, if $W$ is irreducible and if the longest element is central, then the Loewy length of $Σ(W)$ is equal to $\lceil |S| / 2 \rceil$.

math.RT↗
arXiv2008

Semi-continuité des cellules de Kazhdan-Lusztig

Computations in small Coxeter groups and infinite dihedral groups suggest that Kazhdan-Lusztig cells for unequal parameters obey to some "semicontinuity" phenomenon (as the parameter vary). The aim of this paper is to provide a rigorous theoretical background that allows to state some precise conjectures.

math.RT↗
arXiv2008

Cellular structures on Hecke algebras of type B

The aim of this paper is to gather and (try to) unify several approaches for the modular representation theory of Hecke algebras of type $B$. We attempt to explain the connections between Geck's cellular structures (coming from Kazhdan-Lusztig theory with unequal parameters) and Ariki's Theorem on the canonical basis of the Fock spaces.

math.RT↗
arXiv2007

Affineness of Deligne-Lusztig varieties for minimal length elements

We prove that the Deligne-Lusztig varieties associated to elements of the Weyl group which are of minimal length in their twisted class are affine. Our proof differs from the proof of He and Orlik-Rapoport and it is inspired by the case of regular elements, which correspond to the varieties involved in Broué's conjectures.

math.AG↗
arXiv2007

On domino insertion and Kazhdan--Lusztig cells in type $B_n$

Based on empirical evidence obtained using the {\sf CHEVIE} computer algebra system, we present a series of conjectures concerning the combinatorial description of the Kazhdan--Lusztig cells for type $B_n$ with unequal parameters. These conjectures form a far-reaching extension of the results of Bonnafé and Iancu obtained earlier in the so-called ``asymptotic case''. We give some partial results in support of our conjectures.

math.RT↗
arXiv2006

Two-sided cells in type $B$ (asymptotic case)

We compute two-sided cells of Weyl groups of type $B$ for the "asymptotic" choice of parameters. We also obtain some partial results concerning Kazhdan-Lusztig conjectures in this particular case.

math.RT↗
arXiv2006

On the irreducibility of Deligne-Lusztig varieties

Let $G$ be a connected reductive algebraic group defined over an algebraic closure of a finite field and let $F : G \to G$ be an endomorphism such that $F^d$ is a Frobenius endomorphism for some $d \geq 1$. Let $P$ be a parabolic subgroup of $G$ admitting an $F$-stable Levi subgroup. We prove that the Deligne-Lusztig variety $\{gP | g^{-1}F(g)\in P\cdot F(P)\}$ is irreducible if and only if $P$ is not contained in a proper $F$-stable parabolic subgroup of $G$.

math.GR↗
arXiv2006

On the character ring of a finite group

Let $G$ be a finite group and let $k$ be a sufficiently large finite field. Let $R(G)$ denote the character ring of $G$ (i.e. the Grothendieck ring of the category of ${\mathbb{C}}G$-modules). We study the structure and the representations of the commutative algebra $k \otimes\_{\mathbb{\ZM}} R(G)$

math.RT↗
arXiv2006

Coxeter orbits and modular representations

We study the modular representations of finite groups of Lie type arising in the cohomology of certain quotients of Deligne-Lusztig varieties associated with Coxeter elements. These quotients are related to Gelfand-Graev representations and we present a conjecture on the Deligne-Lusztig restriction of Gelfand-Graev representations. We prove the conjecture for restriction to a Coxeter torus. We deduce a proof of Broué's conjecture on equivalences of derived categories arising from Deligne-Lusztig varieties, for a split group of type $A\_n$ and a Coxeter element. Our study is based on Lusztig's work in characteristic 0.

math.RT↗
arXiv2006

Twisted invariant theory for reflection groups

Let $G$ be a reflection group acting on a vector space $V$ and let $γ$ be an automorphism of $V$ normalising $G$. We study how $γ$ acts on invariants and covariants (for various representations) of $G$, and properties of its eigenspaces.

math.GR↗
arXiv2005

A note on the Grothendieck ring of the symmetric group

Let $p$ be a prime number and let $n$ be a non-zero natural number. We compute the descending Loewy series of the algebra $R\_n/pR\_n$, where $R\_n$ denotes the ring of virtual ordinary characters of the symmetric group $S\_n$.

math.RT↗