# Jacques Thévenaz

#### Professor Emeritus

jacques.thevenaz@epfl.ch +41 21 693 03 74 http://ctg.epfl.ch/thevenaz

## Publications

### Infoscience publications

#### LIST OF PUBLICATIONS

### On the lifting of the Dade group

*Journal Of Group Theory*. 2019-05-01.

DOI : 10.1515/jgth-2018-0145.

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.

### Correspondence functors and lattices

*Journal of Algebra*. 2019.

DOI : 10.1016/j.jalgebra.2018.10.019.

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.

### The algebra of Boolean matrices, correspondence functors, and simplicity

*ArXiv*. 2018-11-19.

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.

### Tensor product of correspondence functors

*Journal of Algebra*. 2018-11-19.

DOI : 10.1016/j.jalgebra.2020.02.005.

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.

### Correspondence functors and finiteness conditions

*Journal of Algebra*. 2018.

DOI : 10.1016/j.jalgebra.2017.11.010.

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.

### Lifting endo-p-permutation modules

*Archiv der Mathematik*. 2018.

DOI : 10.1007/s00013-017-1115-3.

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

### Universal p'-central extensions

*Expositiones Mathematicae*. 2017.

DOI : 10.1016/j.exmath.2016.12.004.

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.

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

*Pacific Journal of Mathematics*. 2017.

DOI : 10.2140/pjm.2017.287.423.

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.

### The algebra of essential relations on a finite set

*Journal für die Reine und Angewandte Mathematik*. 2016.

DOI : 10.1515/crelle-2014-0019.

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.

### The monoid algebra of all relations on a finite set

*preprint, will never be published in this form*. 2015.

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

*, preprint, will never be published in this form*. 2015.

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

*Algebra & Number Theory*. 2015.

DOI : 10.2140/ant.2015.9.749.

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.

### Mathématiques et immortalité : une équation impossible ?

*L'immortalité, un sujet d'avenir*; Lausanne: Editions Favre, 2014.

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

### Vanishing evaluations of simple functors

*Journal of Pure and Applied Algebra*. 2014.

DOI : 10.1016/j.jpaa.2013.05.007.

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.

### Torsion-free endotrivial modules

*Journal of Algebra*. 2014.

DOI : 10.1016/j.jalgebra.2013.01.020.

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.

### Simple biset functors and double Burnside ring

*Journal of Pure and Applied Algebra*. 2013.

DOI : 10.1016/j.jpaa.2012.08.005.

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.

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

*Journal of the European Mathematical Society*. 2013.

DOI : 10.4171/Jems/358.

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.

### Stabilizing bisets

*Advances in Mathematics*. 2012.

DOI : 10.1016/j.aim.2011.12.008.

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.

### Endotrivial modules for p-solvable groups

*Transactions of the American Mathematical Society*. 2011.

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.

### The primitive idempotents of the p-permutation ring

*Journal of Algebra*. 2010.

DOI : 10.1016/j.jalgebra.2009.11.036.

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.

### The poset of elementary abelian subgroups of rank at least 2

*Guido's Book of Conjectures*; Genève: L'Enseignement Mathématique, 2008. p. 41-45.

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

*Journal of the London Mathematical Society, Second Series***. 2008.

DOI : 10.1112/jlms/jdn039.

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.

### A sectional characterization of the Dade group

*Journal of Group Theory*. 2008.

DOI : 10.1515/JGT.2008.010.

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.

### Endo-permutation modules, a guided tour

*Group representation theory*; Lausanne: EPFL Press, 2007. p. 115-147.

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

*Archiv der Mathematik*. 2007.

DOI : 10.1007/s00013-007-2365-2.

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.

### The classification of torsion endo-trivial modules

*Annals of Mathematics, series 2*. 2005.

DOI : 10.4007/annals.2005.162.823.

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).

### The classification of endo-trivial modules

*Inventiones Mathematicae*. 2004.

DOI : 10.1007/s00222-004-0385-0.

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.

### Finite simple groups and localization

*Israel Journal of Mathematics*. 2002.

DOI : 10.1007/BF02785857.

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.

### Torsion endo-trivial modules

*Algebras and Representation Theory*. 2000.

DOI : 10.1023/A:1009988424910.

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.

### The group of endo-permutation modules

*Inventiones Mathematicae*. 2000.

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.

### Maximal subgroups of direct products

*Journal of Algebra*. 1997.

DOI : 10.1006/jabr.1997.7095.

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.

### 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.

### 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.

### Research

#### Collaborations

Regular collaborations with :

- Serge Bouc, Université de Picardie Jules Verne, Amiens

- Jon F. Carlson, University of Georgia, USA

- Nadia Mazza, University of Lancaster, GB

- Radu Stancu, Université de Picardie Jules Verne, Amiens

#### Research interests

- Representations of finite groups :

Modular representations, endo-permutation modules, endo-trivial modules, block theory, G-algebras, p-local theory, Alperin's conjecture, almost split sequences, Clifford theory, representations in characteristic p^r.

- Mackey functors and biset functors :

Structure theory, simple functors, G-algebras, cohomology, representation rings.

- Posets of subgroups of finite groups :

Moebius function, Euler characteristic, Burnside rings.

- Finite simplicial complexes :

Brown's and Quillen's complexes, Tits buildings, subgroup complexes, topological K-theory.