arXiv subjects

Marc Cabanes

Publications and source records attributed to Marc Cabanes.

13 recordsLinked to original sources

arXiv2024

The McKay Conjecture on character degrees

We prove that for any prime $\ell$, any finite group has as many irreducible complex characters of degree prime to $\ell$ as the normalizers of its Sylow $\ell$-subgroups. This equality was conjectured by John McKay. The conjecture was reduced by Isaacs--Malle--Navarro (2007) to a conjecture on representations, linear and projective, of finite simple groups that we finish proving here using the classification of those groups. We study mainly characters of normalizers N$_{\mathbf G}({\mathbf S})^F$ of Sylow $d$-tori ${\mathbf S}$ ($d\geq 3$) in a simply-connected algebraic group ${\mathbf G}$ of type D$_l$ ($l\geq 4$) for which $F$ is a Frobenius endomorphism. We also introduce a certain class of $F$-stable reductive subgroups ${\mathbf M}\leq {\mathbf G}$ of maximal rank where ${\mathbf M}^\circ$ is of type some D$_{k}\times\ $D$_{l-k}$. The finite groups ${\mathbf M}^F$ are an efficient substitute for N$_{\mathbf G}({\mathbf S})^F$ or the $\ell$-local subgroups of ${\mathbf G}^F$ relevant to McKay's abstract statement. For a general class of those subgroups ${\mathbf M}^F$ we describe their characters and the action of Aut$({\mathbf G}^F)_{{\mathbf M}^F}$ on them, showing in particular that Irr$({\mathbf M}^F)$ and Irr$({\mathbf G}^F)$ share some key features in that regard.

math.RT↗

arXiv2022

On semisimple classes and component groups in type $\mathsf{D}$

In adjoint reductive groups $H$ of type $\mathsf{D}$ we show that for every semisimple element $s$, its centralizer splits over its connected component, i.e., $C_H(s) = C_H(s)^\circ \rtimes \check A$ for some complement $\check A$ with strong stability properties. We derive several consequences about the action of automorphisms on semisimple conjugacy classes. This helps to parametrize characters of the finite groups $\mathsf{D}_{l,\text{sc}}(q)$ and $^2\mathsf{D}_{l,\text{sc}}(q)$ and describe the action of automorphisms on them. It is also a contribution to the final proof of the McKay conjecture for the prime 3, see [S21], [S23].

math.GR↗

arXiv2020

On the Inductive Alperin-McKay Conditions in the Maximally Split Case

The Alperin-McKay conjecture relates height zero characters of an $\ell$-block with the ones of its Brauer correspondent. This conjecture has been reduced to the so-called inductive Alperin-McKay conditions about quasi-simple groups by the third author. Those conditions are still open for groups of Lie type. The present paper describes characters of height zero in $\ell$-blocks of groups of Lie type over a field with $q$ elements when $\ell$ divides $q-1$. We also give information about $\ell$-blocks and Brauer correspondents. As an application we show that quasi-simple groups of type $C$ over $\mathbb{F}_q$ satisfy the inductive Alperin-McKay conditions for primes $\ell\geq 5$ and dividing $q-1$. Some methods to that end are adapted from the work of Malle--Späth.

math.RT↗

arXiv2019

Descent equalities and the inductive McKay condition for types B and E

We establish the inductive McKay condition introduced by Isaacs-Malle-Navarro \cite{IMN} for finite simple groups of Lie types $\tB_l$ ($l\geq 2$), $\tE_6$, $^2\tE_6$ and $\tE_7$, thus leaving open only the types $\tD$ and $^2\tD$. We bring to the methods previously used by the authors for type $\tC$ \cite{CS17C} some descent arguments using Shintani's norm map. This provides for types different from $ \tA, \tD, {}^2\tD$ a uniform proof of the so-called global requirement of the criterion given by the second author in \cite[2.12]{S12}. The local requirements from that criterion are verified through a detailed study of the normalizers of relevant Levi subgroups and their characters.

math.RT↗

arXiv2017

Local methods for blocks of finite simple groups

This survey is about old and new results about the modular representation theory of finite reductive groups with a strong emphasis on local methods. This includes subpairs, Brauer's Main Theorems, fusion, Rickard equivalences. In the defining characteristic we describe the relation between $p$-local subgroups and parabolic subgroups, then give classical consequences on simple modules and blocks, including the Alperin weight conjecture in that case. In the non-defining characteristics, we sketch a picture of the local methods pioneered by Fong-Srinivasan in the determination of blocks and their ordinary characters. This includes the relationship with Lusztig's twisted induction and the determination of defect groups. We conclude with a survey of the results and methods by Bonnafé-Dat-Rouquier giving Morita equivalences between blocks that preserve defect groups and the local structures. The text grew out of the course and talks given by the author in July and September 2016 during the program "Local representation theory and simple groups" at CIB Lausanne. Written Oct 2017, to appear in a proceedings volume published by EMS.

math.RT↗

arXiv2016

Inductive McKay condition for finite simple groups of type C

We verify the inductive McKay condition for simple groups of Lie type C, namely finite projective symplectic groups. This contributes to the program of a complete proof of the McKay conjecture for all finite groups via the reduction theorem of Isaacs-Malle-Navarro and the classification of finite simple groups. In an important step we use a new counting argument to determine the stabilizers of irreducible characters of a finite symplectic group in its outer automorphism group. This is completed by analogous results on characters of normalizers of Sylow d-tori in those groups.

math.RT↗

arXiv2014

On the inductive Alperin-McKay condition for simple groups of type A

As a sequel to [CS13b], we verify the so-called inductive AM-condition introduced in [Sp12] for simple groups of type A and blocks with maximal defect. This is part of the program set up to verify the Alperin-McKay conjecture through its reduction to a problem on quasi-simple groups (see [Sp13]) but also the missing direction of Brauer's height zero conjecture (see [NS14])

math.RT↗

arXiv2013

Equivariant character correspondences and inductive McKay condition for type A

As a step to establish the McKay conjecture on character degrees of finite groups, we verify the inductive McKay condition introduced by Isaacs-Malle-Navarro for simple groups of Lie type $A_{n-1}$, split or twisted. Key to the proofs is the study of certain characters of SL$_n(q)$ and SU$_n(q)$ related to generalized Gelfand-Graev representations. As a by-product we can show that a Jordan decomposition for the characters of the latter groups is equivariant under outer automorphisms. Many ideas seem applicable to other Lie types.

math.RT↗

arXiv2012

Two remarks on the reduction of Alperin's weight conjecture

The so-called inductive McKay condition on finite simple groups, due to Isaacs-Malle-Navarro (2007), has been recently reformulated by Späth. We show that this reformulation applies to the reduction theorem for Alperin's weight conjecture, due to Navarro-Tiep (2011). This also simplifies the checking of the inductive condition for Alperin's weight conjecture in the case of simple groups of Lie type with regard to the defining prime.

math.RT↗

arXiv2012

On Ternary Quotients of Cubic Hecke Algebras

We prove that the quotients of the group algebra of the braid group introduced by L. Funar in Comm. Math. Phys., 1995, collapses in characteristic distinct from 2. In characteristic 2 we define several quotients of it, which are connected to the classical Hecke and Birman-Wenzl-Murakami quotients, but which admit in addition a symmetry of order 3. We also establish conditions on the possible Markov traces factorizing through it.

math.GT↗

arXiv2011

On Jordan Decomposition of Characters for SU(n,q)

As shown by Bonnafé, a step in proving a Jordan decomposition of characters of finite special linear groups is the parametrization of unipotent characters of centralizers of semi-simple elements in projective linear groups. We show the same kind of result in the case of finite special unitary groups. The proof leads to a mild adaptation of Bonnafé's methods expounded in [B99]. The outcome is a Jordan decomposition of characters compatible with Lusztig's twisted induction.

math.RT↗

arXiv2011

Odd Character Degrees for Sp(2n,2)

We check McKay conjecture on character degrees for the case of symplectic groups over the field with two elements Sp(2n,2) and the prime 2. Then we check the inductive McKay condition (Isaacs-Malle-Navarro 2007) for Sp(4,2^m) and all primes.

math.RT↗

arXiv2008

On Okuyama's theorems about Alvis-Curtis duality

The purpose of this paper is to report on the unpublished manuscript [O] by T. Okuyama where are proved some conjectures generalizing to homotopy categories the theorems of [CaRi] and [LS] holding in derived categories. We refer to the latter references and [CaEn]~§4 for a broader introduction to the subject. The main theme is the one of complexes related with the Coxeter complex and the action of parabolic subgroups on them, either for finite groups with BN-pairs or for finite dimensional Hecke algebras. Okuyama's contractions prove a quite efficient tool in a number of situations (see the proof of Solomon-Tits theorem in §~6). We often stray away from Okuyama's proofs when it allows simplifications.

math.RT↗