Jacques Thévenaz
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Professor Emeritus
jacques.thevenaz@epfl.ch +41 21 693 03 74 http://ctg.epfl.ch/thevenaz
Publications
Infoscience publications
LIST OF PUBLICATIONS
Correspondence functors and duality
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/).
Journal Of Algebra
2023-03-01
DOI : 10.1016/j.jalgebra.2022.10.028
Large evanescently-induced Brillouin scattering at the surrounding of a nanofibre
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.
Nature Communications
2022-03-17
DOI : 10.1038/s41467-022-29051-8
Idempotents in the endomorphism algebra of a finite lattice
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.
Order-A Journal On The Theory Of Ordered Sets And Its Applications
2022
DOI : 10.1007/s11083-021-09588-x
Simple and projective correspondence functors
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.
Representation Theory
2021
DOI : 10.1090/ert/564
The algebra of Boolean matrices, correspondence functors, and simplicity
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.
Journal of Combinatorial Algebra
2020-01-01
DOI : 10.4171/JCA/44
Tensor product of correspondence functors
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.
Journal of Algebra
2020
DOI : 10.1016/j.jalgebra.2020.02.005
On the lifting of the Dade group
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.
Journal Of Group Theory
2019-05-01
DOI : 10.1515/jgth-2018-0145
Correspondence functors and lattices
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.
Journal of Algebra
2019
DOI : 10.1016/j.jalgebra.2018.10.019
Correspondence functors and finiteness conditions
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.
Journal of Algebra
2018
DOI : 10.1016/j.jalgebra.2017.11.010
Lifting endo-p-permutation modules
We prove that all endo-p-permutation modules for a finite group are liftable from characteristic p to characteristic 0.
Archiv der Mathematik
2018
DOI : 10.1007/s00013-017-1115-3
Universal p'-central extensions
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.
Expositiones Mathematicae
2017
DOI : 10.1016/j.exmath.2016.12.004
Endo-trivial modules: a reduction to p'-central extensions
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.
Pacific Journal of Mathematics
2017
DOI : 10.2140/pjm.2017.287.423
The algebra of essential relations on a finite set
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.
Journal für die Reine und Angewandte Mathematik
2016
DOI : 10.1515/crelle-2014-0019
The monoid algebra of all relations on a finite set
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.
preprint, will never be published in this form
2015
The representation theory of finite sets and correspondences
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.
preprint, will never be published in this form
2015
The torsion group of endotrivial modules
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.
Algebra & Number Theory
2015
DOI : 10.2140/ant.2015.9.749
Mathématiques et immortalité : une équation impossible ?
Réflexions historiques et philosophiques sur le lien entre mathématiques, infini, et immortalité.
L'immortalité, un sujet d'avenir; Lausanne: Editions Favre,2014.
Vanishing evaluations of simple functors
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.
Journal of Pure and Applied Algebra
2014
DOI : 10.1016/j.jpaa.2013.05.007
Torsion-free endotrivial modules
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.
Journal of Algebra
2014
DOI : 10.1016/j.jalgebra.2013.01.020
Simple biset functors and double Burnside ring
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.
Journal of Pure and Applied Algebra
2013
DOI : 10.1016/j.jpaa.2012.08.005
Endotrivial modules over groups with quaternion or semi-dihedral Sylow 2-subgroup
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.
Journal of the European Mathematical Society
2013
DOI : 10.4171/Jems/358
Stabilizing bisets
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.
Advances in Mathematics
2012
DOI : 10.1016/j.aim.2011.12.008
Endotrivial modules for p-solvable groups
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.
Transactions of the American Mathematical Society
2011
DOI : 10.1090/S0002-9947-2011-05307-9
The primitive idempotents of the p-permutation ring
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.
Journal of Algebra
2010
DOI : 10.1016/j.jalgebra.2009.11.036
The poset of elementary abelian subgroups of rank at least 2
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).
Guido's Book of Conjectures; Genève: L'Enseignement Mathématique,2008.
p. 41-45.Gluing torsion endo-permutation modules
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.
Journal of the London Mathematical Society, Second Series**
2008
DOI : 10.1112/jlms/jdn039
A sectional characterization of the Dade group
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.
Journal of Group Theory
2008
DOI : 10.1515/JGT.2008.010
Endo-permutation modules, a guided tour
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.
Group representation theory; Lausanne: EPFL Press,2007.
p. 115-147.Endotrivial modules in the cyclic case
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.
Archiv der Mathematik
2007
DOI : 10.1007/s00013-007-2365-2
The classification of torsion endo-trivial modules
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).
Annals of Mathematics, series 2
2005
DOI : 10.4007/annals.2005.162.823
The classification of endo-trivial modules
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.
Inventiones Mathematicae
2004
DOI : 10.1007/s00222-004-0385-0
Finite simple groups and localization
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.
Israel Journal of Mathematics
2002
DOI : 10.1007/BF02785857
Torsion endo-trivial modules
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.
Algebras and Representation Theory
2000
DOI : 10.1023/A:1009988424910
The group of endo-permutation modules
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.
Inventiones Mathematicae
2000
DOI : 10.1007/s002229900026
Maximal subgroups of direct products
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.
Journal of Algebra
1997
DOI : 10.1006/jabr.1997.7095
The structure of Mackey functors
Transactions of the American Mathematical Society
1995
DOI : 10.2307/2154915
The parametrization of interior algebras
Mathematische Zeitschrift
1993
DOI : 10.1007/BF02571667
Most finite groups are p-nilpotent
Expositiones Mathematicae
1993
Equivariant K-theory and Alperin's conjecture
Journal of Pure and Applied Algebra
1993
DOI : 10.1016/0022-4049(93)90052-U
Locally determined functions and Alperin's conjecture
Journal of the London Mathematical Society, Second Series**
1992
DOI : 10.1112/jlms/s2-45.3.446
Polynomial identities for partitions
European Journal of Combinatorics
1992
DOI : 10.1016/0195-6698(92)90044-Z
On a conjecture of Webb
Archiv der Mathematik
1992
DOI : 10.1007/BF01191872
On Alperin's conjecture for finite reductive groups (Sur la conjecture d'Alperin pour les groupes réductifs finis)
Comptes Rendus de l'Académie des Sciences, série I
1992
Homotopy equivalence of posets with a group action
Journal of Combinatorial Theory, Series A
1991
DOI : 10.1016/0097-3165(91)90030-K
Defect theory for maximal ideals and simple functors
Journal of Algebra
1991
DOI : 10.1016/0021-8693(91)90168-8
The Z*-theorem for compact Lie groups
Mathematische Annalen
1991
DOI : 10.1007/BF01445193
A Mackey functor version of a conjecture of Alperin
Astérisque
1990
Simple Mackey functors
Rendiconti del Circolo Matematico di Palermo
1990
A visit to the kingdom of the Mackey functors
Bayreuther Mathematische Schriften
1990
A characterization of cyclic groups
Archiv der Mathematik
1989
DOI : 10.1007/BF01194381
G-algebras, Jacobson radical and almost split sequences
Inventiones Mathematicae
1988
DOI : 10.1007/BF01393690
Duality in G-algebras
Mathematische Zeitschrift
1988
DOI : 10.1007/BF01161746
Isomorphic Burnside rings
Communications in Algebra
1988
DOI : 10.1080/00927878808823668
Some remarks on G-functors and the Brauer morphism
Journal für die Reine und Angewandte Mathematik
1988
DOI : 10.1515/crll.1988.384.24
A generalization of Sylow's third theorem
Journal of Algebra
1988
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)
Séminaire sur les groupes finis; Paris: Univ. Paris VII,1987.
p. iii, 207-217.Permutation representations arising from simplicial complexes
Journal of Combinatorial Theory, Series A
1987
DOI : 10.1016/0097-3165(87)90078-1
The top homology of the lattice of subgroups of a soluble group
Discrete Mathematics
1985
DOI : 10.1016/S0012-365X(85)80005-4
Relative projective covers and almost split sequences
Communications in Algebra
1985
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)
Commentarii Mathematici Helvetici
1985
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)
Commentarii Mathematici Helvetici
1984
DOI : 10.1007/BF02566359
Lifting idempotents and Clifford theory
Commentarii Mathematici Helvetici
1983
DOI : 10.1007/BF02564626
Extensions of group representations from a normal subgroup
Communications in Algebra
1983
DOI : 10.1080/00927878308822855
Representations of groups of order ph in characteristic p^r
Journal of Algebra
1982
DOI : 10.1016/0021-8693(82)90325-8
Representations of finite groups in characteristic p^r
Journal of Algebra
1981
DOI : 10.1016/0021-8693(81)90305-7