Correspondence functors and duality

S. Bouc; J. Thevenaz

A correspondence functor is a functor from the category of finite sets and correspondences to the category of k-modules, where k is a commutative ring. By means of a suitably defined duality, new correspondence functors are constructed, having remarkable properties. In particular, their evaluation at any finite set is always a free k-module and an explicit formula is obtained for its rank. The results use some subtle new ingredients from the theory of finite lattices. (c) 2022 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

DOI : 10.1016/j.jalgebra.2022.10.028

Large evanescently-induced Brillouin scattering at the surrounding of a nanofibre

F. Yang; F. Gyger; A. Godet; J. Chrétien; L. Zhang  et al.

Brillouin scattering has been widely exploited for advanced photonics functionalities such as microwave photonics, signal processing, sensing, lasing, and more recently in micro- and nano-photonic waveguides. Most of the works have focused on the opto-acoustic interaction driven from the core region of micro- and nano-waveguides. Here we observe, for the first time, an efficient Brillouin scattering generated by an evanescent field nearby a single-pass sub-wavelength waveguide embedded in a pressurised gas cell, with a maximum gain coefficient of 18.90 ± 0.17 m−1W−1. This gain is 11 times larger than the highest Brillouin gain obtained in a hollow-core fibre and 79 times larger than in a standard single-mode fibre. The realisation of strong free-space Brillouin scattering from a waveguide benefits from the flexibility of confined light while providing a direct access to the opto-acoustic interaction, as required in free-space optoacoustics such as Brillouin spectroscopy and microscopy. Therefore, our work creates an important bridge between Brillouin scattering in waveguides, Brillouin spectroscopy and microscopy, and opens new avenues in light-sound interactions, optomechanics, sensing, lasing and imaging.

DOI : 10.1038/s41467-022-29051-8

Idempotents in the endomorphism algebra of a finite lattice

S. Bouc; J. Thévenaz

We give a direct construction of a specific central idempotent in the endomorphism algebra of a finite lattice T. This idempotent is associated with all possible sublattices of T which are totally ordered. A generalization is considered in a conjectural fashion and proved experimentally to hold in many cases.

DOI : 10.1007/s11083-021-09588-x

Simple and projective correspondence functors

S. Bouc; J. Thévenaz

A correspondence functor is a functor from the category of finite sets and correspondences to the category of k-modules, where k is a commutative ring. We determine exactly which simple correspondence functors are projective. We also determine which simple modules are projective for the algebra of all relations on a finite set. Moreover, we analyze the occurrence of such simple projective functors inside the correspondence functor F associated with a finite lattice and we deduce a direct sum decomposition of F.

DOI : 10.1090/ert/564

The algebra of Boolean matrices, correspondence functors, and simplicity

S. Bouc; J. Thévenaz

We determine the dimension of every simple module for the algebra of the monoid of all relations on a finite set (i.e. Boolean matrices). This is in fact the same question as the determination of the dimension of every evaluation of a simple correspondence functor. The method uses the theory of such functors developed in previous papers, as well as some new ingredients in the theory of finite lattices.

DOI : 10.4171/JCA/44

Tensor product of correspondence functors

S. Bouc; J. Thévenaz

As part of the study of correspondence functors, the present paper investigates their tensor product and proves some of its main properties. In particular, the correspondence functor associated to a finite lattice has the structure of a commutative algebra in the tensor category of all correspondence functors.

DOI : 10.1016/j.jalgebra.2020.02.005

On the lifting of the Dade group

C. Lassueur; J. Thevenaz

For the group of endo-permutation modules of a finite p-group, there is a surjective reduction homomorphism from a complete discrete valuation ring of characteristic 0 to its residue field of characteristic p. We prove that this reduction map always has a section which is a group homomorphism.

DOI : 10.1515/jgth-2018-0145

Correspondence functors and lattices

S. Bouc; J. Thévenaz

A correspondence functor is a functor from the category of finite sets and correspondences to the category of k-modules, where k is a commutative ring. A main tool for this study is the construction of a correspondence functor associated to any finite lattice T. We prove for instance that this functor is projective if and only if the lattice T is distributive. Moreover, it has quotients which play a crucial role in the analysis of simple functors. The special case of total orders yields some more specific and complete results.

DOI : 10.1016/j.jalgebra.2018.10.019

Correspondence functors and finiteness conditions

S. Bouc; J. Thévenaz

We investigate the representation theory of finite sets. The correspondence functors are the functors from the category of finite sets and correspondences to the category of k-modules, where k is a commutative ring. They have various specific properties which do not hold for other types of functors. In particular, if k is a field and if F is a correspondence functor, then F is finitely generated if and only if the dimension of F(X) grows exponentially in terms of the cardinality of the finite set X. Moreover, in such a case, F has actually finite length. Also, if k is noetherian, then any subfunctor of a finitely generated functor is finitely generated.

DOI : 10.1016/j.jalgebra.2017.11.010

Lifting endo-p-permutation modules

C. Lassueur; J. Thévenaz

We prove that all endo-p-permutation modules for a finite group are liftable from characteristic p to characteristic 0.

DOI : 10.1007/s00013-017-1115-3

Universal p'-central extensions

C. Lassueur; J. Thévenaz

It is well-known that a finite group possesses a universal central extension if and only if it is a perfect group. Similarly, given a prime number p, we show that a finite group possesses a universal p′-central extension if and only if the p′-part of its abelianization is trivial. This question arises naturally when working with group representations over a field of characteristic p.

DOI : 10.1016/j.exmath.2016.12.004

Endo-trivial modules: a reduction to p'-central extensions

C. Lassueur; J. Thévenaz

We examine how, in prime characteristic p, the group of endotrivial modules of a finite group G and the group of endotrivial modules of a quotient of G modulo a normal subgroup of order prime to p are related. There is always an inflation map, but examples show that this map is in general not surjective. We prove that the situation is controlled by a single central extension, namely, the central extension given by a p′-representation group of the quotient of G by its largest normal p′-subgroup.

DOI : 10.2140/pjm.2017.287.423

The algebra of essential relations on a finite set

S. Bouc; J. Thévenaz

Let X be a finite set and let k be a commutative ring. We consider the k-algebra of the monoid of all relations on X, modulo the ideal generated by the relations factorizing through a set of cardinality strictly smaller than Card(X), called inessential relations. This quotient is called the essential algebra associated to X. We then define a suitable nilpotent ideal of the essential algebra and describe completely the structure of the corresponding quotient, a product of matrix algebras over suitable group algebras. In particular, we obtain a description of all the simple modules for the essential algebra.

DOI : 10.1515/crelle-2014-0019

The monoid algebra of all relations on a finite set

S. Bouc; J. Thévenaz

We classify all the simple modules for the algebra of relations on a finite set, give their dimension, and find the dimension of the Jacobson radical of the algebra.

The representation theory of finite sets and correspondences

S. Bouc; J. Thévenaz

We investigate correspondence functors, namely the functors from the category of finite sets and correspondences to the category of k-modules, where k is a commutative ring. They have various specific properties which do not hold for other types of functors. In particular, if k is a field and if F is a correspondence functor, then F is finitely generated if and only if the dimension of F(X) grows exponentially in terms of the cardinality of the finite set X. In such a case, F has finite length. Also, if k is noetherian, then any subfunctor of a finitely generated functor is finitely generated. When k is a field, we give a description of all the simple functors and we determine the dimension of their evaluations at any finite set. A main tool is the construction of a functor associated to any finite lattice T. We prove for instance that this functor is projective if and only if the lattice T is distributive. Moreover, it has quotients which play a crucial role in the analysis of simple functors. The special case of total orders yields some more specific results. Several other properties are also discussed, such as projectivity, duality, and symmetry. In an appendix, all the lattices associated to a given poset are described.

The torsion group of endotrivial modules

J. F. Carlson; J. Thévenaz

Let G be a finite group and let T(G) be the abelian group of equivalence classes of endotrivial kG-modules, where k is an algebraically closed field of characteristic p. We determine, in terms of the structure of G, the kernel of the restriction map from T(G) to T(S), where S is a Sylow p-subgroup of G, in the case when S is abelian. This provides a classification of all torsion endotrivial kG-modules in that case.

DOI : 10.2140/ant.2015.9.749

Mathématiques et immortalité : une équation impossible ?

J. Thévenaz

Réflexions historiques et philosophiques sur le lien entre mathématiques, infini, et immortalité.

Vanishing evaluations of simple functors

S. Bouc; R. Stancu; J. Thévenaz

The classification of simple biset functors is known, but the evaluation of a simple biset functor at a finite group G may be zero. We investigate various situations where this happens, as well as cases where this does not occur. We also prove a closed formula for such an evaluation under some restrictive conditions on G.

DOI : 10.1016/j.jpaa.2013.05.007

Torsion-free endotrivial modules

J. F. Carlson; N. Mazza; J. Thévenaz

Let G be a finite group and let T(G) be the abelian group of equivalence classes of endotrivial kG-modules, where k is an algebraically closed field of characteristic p. We investigate the torsion-free part TF(G) of the group T(G) and look for generators of TF(G). We describe three methods for obtaining generators. Each of them only gives partial answers to the question but we obtain more precise results in some specific cases. We also conjecture that TF(G) can be generated by modules belonging to the principal block and we prove the conjecture in some cases.

DOI : 10.1016/j.jalgebra.2013.01.020

Simple biset functors and double Burnside ring

S. Bouc; R. Stancu; J. Thévenaz

Let G be a finite group and let k be a field. Our purpose is to investigate the simple modules for the double Burnside ring kB(G,G). It turns out that they are evaluations at G of simple biset functors. For a fixed finite group H, we introduce a suitable bilinear form on kB(G,H) and we prove that the quotient of kB(-,H) by the radical of the bilinear form is a semi-simple functor. This allows for a description of the evaluation of simple functors, hence of simple modules for the double Burnside ring.

DOI : 10.1016/j.jpaa.2012.08.005

Endotrivial modules over groups with quaternion or semi-dihedral Sylow 2-subgroup

J. F. Carlson; N. Mazza; J. Thévenaz

Let G be a finite group with a Sylow 2-subgroup P which is either quaternion or semi-dihedral. Let k be an algebraically closed field of characteristic 2. We prove the existence of exotic endotrivial kG-modules, whose restrictions to P are isomorphic to the direct sum of the known exotic endotrivial kP-modules and some projective modules. This provides a description of the group T(G) of endotrivial kG-modules.

DOI : 10.4171/Jems/358

Stabilizing bisets

S. Bouc; J. Thévenaz

Let G be a finite group and let R be a commutative ring. We analyse the (G,G)-bisets which stabilize an indecomposable RG-module. We prove that the minimal ones are unique up to equivalence. This result has the same flavor as the uniqueness of vertices and sources up to conjugation and resembles also the theory of cuspidal characters in the context of Harish-Chandra induction for reductive groups, but it is different and very general. We show that stabilizing bisets have rather strong properties and we explore two situations where they occur. Moreover, we prove some specific results for simple modules and also for p-groups.

DOI : 10.1016/j.aim.2011.12.008

Endotrivial modules for p-solvable groups

J. F. Carlson; N. Mazza; J. Thévenaz

We determine the torsion subgroup of the group of endotrivial modules for a finite solvable group in characteristic p. We also prove that our result would hold for p-solvable groups, provided a conjecture can be proved about the case of p-nilpotent groups.

DOI : 10.1090/S0002-9947-2011-05307-9

The primitive idempotents of the p-permutation ring

S. Bouc; J. Thévenaz

Let G be a finite group, let p be a prime number, and let K be a field of characteristic 0 and k be a field of characteristic p, both large enough. In this note, we state explicit formulae for the primitive idempotents of the ring of p-permutation kG-modules, tensored with K.

DOI : 10.1016/j.jalgebra.2009.11.036

The poset of elementary abelian subgroups of rank at least 2

S. Bouc; J. Thévenaz

Let P be a finite p-group. In this note, we prove that the poset of elementary abelian subgroups of P of rank at least 2 has the homotopy type of a wedge of spheres (of possibly different dimensions).

Gluing torsion endo-permutation modules

S. Bouc; J. Thévenaz

Given a finite p-group P, the main result gives necessary and sufficient conditions for obtaining a torsion endo-permutation module for P by gluing a compatible family of torsion endo-permutation modules for all sections N(Q)/Q, where Q runs among non-trivial subgroups of P. This problem has applications in block theory. The paper also contains an appendix about biset functors (with a generalized Mackey formula) and an appendix about the Dade functor.

DOI : 10.1112/jlms/jdn039

A sectional characterization of the Dade group

S. Bouc; J. Thévenaz

Let k be a field of characteristic p, let P be a finite p-group, where p is an odd prime, and let D(P) be the Dade group of endo-permutation kP-modules. It is known that D(P) is detected via deflation-restriction by the family of all sections of P which are elementary abelian of rank at most 2. In this paper, we improve this result by characterizing D(P) as the limit (with respect to deflation-restriction maps and conjugation maps) of all groups D(T/S) where T/S runs through all sections of P which are either elementary abelian of rank at most 3 or extraspecial of order p^3 and exponent p.

DOI : 10.1515/JGT.2008.010

Endo-permutation modules, a guided tour

J. Thévenaz

Shortly after the final classification of all endo-trivial modules for a finite p- group P by Jon Carlson and Jacques Thévenaz, the complete classification of all endo-permutation modules was obtained by Serge Bouc (The Dade group of a p-group, Inventiones Mathematicae 64 (2006) 189-231). The purpose of this paper is to give an overview of this subject, as well as a survey of the steps of the proof and of the various techniques involved in the classification.

Endotrivial modules in the cyclic case

N. Mazza; J. Thévenaz

We determine all endotrivial modules in prime characteristic p, for a finite group having a cyclic Sylow p-subgroup. In other words, we describe completely the group of endotrivial modules in that case.

DOI : 10.1007/s00013-007-2365-2

The classification of torsion endo-trivial modules

J. F. Carlson; J. Thévenaz

The main purpose of this paper is to show that, if G is a finite p-group which is not cyclic, quaternion or semi-dihedral, then the group of endo-trivial G-modules is detected on restriction to elementary abelian p-subgroups of rank 2. Consequently, the torsion subgroup of the group of endo-trivial modules is trivial for any finite p-group, except in the known cases of cyclic, quaternion and semi-dihedral groups. This requires a large amount of group cohomology, recent bounds for the number of Bocksteins in Serre's theorem, Carlson's recent theorem expressing Serre's theorem in terms of modules, and finally the theory of varieties attached to modules. A detection theorem for the torsion subgroup of the Dade group of all endo-permutation modules is deduced from the main theorem. The structure of this torsion subgroup can then be fully described when p is odd (using the results of Bouc-Thévenaz).

DOI : 10.4007/annals.2005.162.823

The classification of endo-trivial modules

J. F. Carlson; J. Thévenaz

This paper settles the classification of all endo-trivial modules for a finite p-group, by treating the remaining case of a p-group having maximal elementary abelian subgroups of rank 2. It is shown that the modules constructed by Alperin using relative syzygies provide the complete list of endo-trivial modules in this case.

DOI : 10.1007/s00222-004-0385-0

Finite simple groups and localization

J. L. Rodríguez; J. Scherer; J. Thévenaz

The purpose of this paper is to explore the concept of localization, which comes from homotopy theory, in the context of finite simple groups. We give an easy criterion for a finite simple group to be a localization of some simple subgroup and we apply it in various cases. Iterating this process allows us to connect many simple groups by a sequence of localizations. We prove that all sporadic simple groups (except possibly the Monster) and several groups of Lie type are connected to alternating groups. The question remains open whether or not there are several connected components within the family of finite simple groups. In some cases, we also consider automorphism groups and universal covering groups and we show that a localization of a finite simple group may not be simple.

DOI : 10.1007/BF02785857

Torsion endo-trivial modules

J. F. Carlson; J. Thévenaz

We prove that the group T(G) of endo-trivial modules for a non-cyclic finite p-group G is detected on restriction to the family of subgroups which are either elementary abelian of rank 2 or (almost) extraspecial. This result is closely related to the problem of finding the torsion subgroup of T(G). We give the complete structure of T(G) when G is dihedral, semi-dihedral, or quaternion.

DOI : 10.1023/A:1009988424910

The group of endo-permutation modules

S. Bouc; J. Thévenaz

The group D(P) of all endo-permutation modules for a finite p-group P is a finitely generated abelian group. We prove that its torsion-free rank is equal to the number of conjugacy classes of non-cyclic subgroups of P. We also obtain partial results on its torsion subgroup. We determine next the structure of Q\otimes D(-) viewed as a functor, which turns out to be a simple functor S_{E,Q}, indexed by the elementary group E of order p^2 and the trivial Out(E)-module Q. Finally we describe a rather strange exact sequence relating Q\otimes D(P), Q\otimes B(P), and Q\otimes R(P), where B(P) is the Burnside ring and R(P) is the Grothendieck ring of QP-modules.

DOI : 10.1007/s002229900026

Maximal subgroups of direct products

J. Thévenaz

We determine all maximal subgroups of the direct product G^n of n copies of a group G. If G is finite, we show that the number of maximal subgroups of G^n is a quadratic function of n if G is perfect, but grows exponentially otherwise. We deduce a theorem of Wiegold about the growth behaviour of the number of generators of G^n.

DOI : 10.1006/jabr.1997.7095

The structure of Mackey functors

J. Thévenaz; P. J. Webb

DOI : 10.2307/2154915

The parametrization of interior algebras

J. Thévenaz

DOI : 10.1007/BF02571667

Most finite groups are p-nilpotent

J. Thévenaz

Equivariant K-theory and Alperin's conjecture

J. Thévenaz

DOI : 10.1016/0022-4049(93)90052-U

Locally determined functions and Alperin's conjecture

J. Thévenaz

DOI : 10.1112/jlms/s2-45.3.446

Polynomial identities for partitions

J. Thévenaz

DOI : 10.1016/0195-6698(92)90044-Z

On a conjecture of Webb

J. Thévenaz

DOI : 10.1007/BF01191872

On Alperin's conjecture for finite reductive groups (Sur la conjecture d'Alperin pour les groupes réductifs finis)

G. I. Lehrer; J. Thévenaz

Homotopy equivalence of posets with a group action

J. Thévenaz; P. J. Webb

DOI : 10.1016/0097-3165(91)90030-K

Defect theory for maximal ideals and simple functors

J. Thévenaz

DOI : 10.1016/0021-8693(91)90168-8

The Z*-theorem for compact Lie groups

G. Mislin; J. Thévenaz

DOI : 10.1007/BF01445193

A Mackey functor version of a conjecture of Alperin

J. Thévenaz; P. J. Webb

Simple Mackey functors

J. Thévenaz; P. J. Webb

A visit to the kingdom of the Mackey functors

J. Thévenaz

A characterization of cyclic groups

J. Thévenaz

DOI : 10.1007/BF01194381

G-algebras, Jacobson radical and almost split sequences

J. Thévenaz

DOI : 10.1007/BF01393690

Duality in G-algebras

J. Thévenaz

DOI : 10.1007/BF01161746

Isomorphic Burnside rings

J. Thévenaz

DOI : 10.1080/00927878808823668

Some remarks on G-functors and the Brauer morphism

J. Thévenaz

DOI : 10.1515/crll.1988.384.24

A generalization of Sylow's third theorem

K. S. Brown; J. Thévenaz

DOI : 10.1016/0021-8693(88)90268-2

Idempotents de l'anneau de Burnside et caractéristique d'Euler (Idempotents of the Burnside ring and Euler characteristic)

J. Thévenaz

Permutation representations arising from simplicial complexes

J. Thévenaz

DOI : 10.1016/0097-3165(87)90078-1

The top homology of the lattice of subgroups of a soluble group

J. Thévenaz

DOI : 10.1016/S0012-365X(85)80005-4

Relative projective covers and almost split sequences

J. Thévenaz

DOI : 10.1080/00927878508823237

Type d'homotopie des treillis et treillis des sous-groupes d'un groupe fini (Homotopy type of lattices and lattices of subgroups of a finite groups)

C. Kratzer; J. Thévenaz

DOI : 10.1007/BF02567401

Fonction de Möbius d'un groupe fini et anneau de Burnside (Möbius function of a finite group and the Burnside ring)

C. Kratzer; J. Thévenaz

DOI : 10.1007/BF02566359

Lifting idempotents and Clifford theory

J. Thévenaz

DOI : 10.1007/BF02564626

Extensions of group representations from a normal subgroup

J. Thévenaz

DOI : 10.1080/00927878308822855

Representations of groups of order ph in characteristic p^r

J. Thévenaz

DOI : 10.1016/0021-8693(82)90325-8

Representations of finite groups in characteristic p^r

J. Thévenaz

DOI : 10.1016/0021-8693(81)90305-7