Around evaluations of biset functors
Annales de l'Institut Fourier, to appear, 39 p.

Our purpose here, is to study double Burnside algebras via evaluations of biset functors. In order to avoid the difficult problem of vanishing of simple functors, we look at finite groups for which there is no non-trivial vanishing and we call them non-vanishing groups. This family contains all the abelian groups, but also infinitely many others. We show that for a non-vanishing group, there is an equivalence between the category of modules over the double Burnside algebra and a specific category of biset functors. Then, we deduce results about the highest-weight structure, and the self-injective property of the double Burnside algebra. We also revisit Barker’s Theorem on the semi-simplicity of the category of biset functors.

On se propose dans cet article d’étudier l’algèbre de Burnside double via des évaluations de foncteurs à bi-ensembles. Afin d’éviter le problème notoirement difficile de la disparition des foncteurs simples, on s’intéresse aux groupes finis pour lesquels il n’y a pas de disparitions non triviales qu’on appelle groupes sans disparitions. Cette famille de groupes contient les groupes abéliens, mais aussi une infinité d’autres. On démontre que la catégorie de modules sur l’algèbre de Burnside double d’un tel groupe est équivalente à une catégorie de foncteurs à bi-ensembles. On en déduit des résultats sur la structure de plus haut poids ainsi que sur l’auto-injectivité de l’algèbre de Burnside double. Finalement, on revisite un théorème de Barker sur la semi-simplicité de la catégorie des foncteurs à bi-ensembles.

Received : 2017-03-23
Revised : 2018-03-15
Accepted : 2018-04-26
Classification:  19A22,  20C99,  16G10,  18E10
Keywords: Biset, Burnside ring, biset functor, quasi-hereditary algebra
@unpublished{AIF_0__0_0_A23_0,
     author = {Rognerud, Baptiste},
     title = {Around evaluations of biset functors},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
Rognerud, Baptiste. Around evaluations of biset functors. Annales de l'Institut Fourier, to appear, 39 p.

[1] An, Lijian; Ding, Jianfang; Zhang, Qinhai Finite self dual groups, J. Algebra, Tome 341 (2011) no. 1, pp. 35-44 | Article

[2] Barker, Laurence Rhetorical biset functors, rational p-biset functors and their semisimplicity in characteristic zero, J. Algebra, Tome 319 (2008) no. 9, pp. 3810-3853 | Article

[3] Boltje, Robert; Danz, Susanne A ghost ring for the left-free double Burnside ring and an application to fusion systems, Adv. Math., Tome 229 (2012) no. 3, pp. 1688-1733 | Article

[4] Boltje, Robert; Külshammer, Burkhard Central idempotents of the bifree and left-free double Burnside ring, Isr. J. Math., Tome 202 (2014) no. 1, pp. 161-193 | Article | Zbl 1321.19002

[5] Bouc, Serge Some simple bisets functors (in preparation)

[6] Bouc, Serge Biset functors for finite groups, Springer, Lecture Notes in Mathematics, Tome 1990 (2010), ix+399 pages | Zbl 1205.19002

[7] Bouc, Serge; Stancu, Radu; Thévenaz, Jacques Simple biset functors and double Burnside ring, J. Pure Appl. Algebra, Tome 217 (2013) no. 3, pp. 546-566 | Article

[8] Bouc, Serge; Stancu, Radu; Thévenaz, Jacques Vanishing evaluations of simple functors, J. Pure Appl. Algebra, Tome 218 (2014) no. 2, pp. 218-227 | Article

[9] Bouc, Serge; Thévenaz, Jacques The monoid algebra of all relations on a finite set (2015) (https://arxiv.org/abs/1511.01741 )

[10] Bouc, Serge; Thévenaz, Jacques The representation theory of finite sets and correspondences (2015) (https://arxiv.org/abs/1510.03034 )

[11] Chevalley, Rosalie Sur quelques foncteurs de bi-ensembles, École Polytechnique Fédérale de Lausanne (Switzerland) (2015) (Ph. D. Thesis)

[12] Franjou, Vincent; Pirashvili, Teimuraz Comparison of abelian categories recollements, Doc. Math., Tome 9 (2004), pp. 41-56 | Zbl 1060.18008

[13] Humphreys, James E. Representations of Semisimple Lie Algebras in the BGG Category O, American Mathematical Society, Graduate Studies in Mathematics, Tome 94 (2008), xvi+289 pages | Zbl 1177.17001

[14] Koenig, Steffen; Külshammer, Julian; Ovsienko, Sergiy Quasi-hereditary algebras, exact Borel subalgebras, -categories and boxes, Adv. Math., Tome 262 (2014), pp. 546-592 | Article

[15] König, Steffen Exact borel subalgebras of quasi-hereditary algebras, I, Math. Z., Tome 220 (195) no. 1, pp. 399-426

[16] Lam, Tsit-Yuen A first course in noncommutative rings, Springer, Graduate Texts in Mathematics, Tome 131 (2001), xx+385 pages | Article | MR 1838439 | Zbl 0728.16001

[17] Lindner, Harald A remark on Mackey-functors, Manuscr. Math., Tome 18 (1976) no. 3, pp. 273-278

[18] May, John P. Picard Groups, Grothendieck Rings, and Burnside Rings of Categories, Adv. Math., Tome 163 (2001) no. 1, pp. 1-16 | Article | Zbl 0994.18004

[19] Rognerud, Baptiste Equivalences between blocks of cohomological Mackey algebras, Math. Z., Tome 280 (2015) no. 1-2, pp. 421-449 | Article | MR 3343914

[20] Rognerud, Baptiste Equivalences between blocks of p-local Mackey algebras, J. Algebra, Tome 428 (2015), pp. 205-229 | Article | MR 3314292

[21] Rognerud, Baptiste Quasi-hereditary property of double Burnside algebras, C. R. Math. Acad. Sci. Paris, Tome 353 (2015) no. 8, pp. 689-693 | Article

[22] Rognerud, Baptiste Trace maps for Mackey algebras, J. Algebra, Tome 426 (2015), pp. 288-312 | Article

[23] Thévenaz, Jacques; Webb, Peter The structure of Mackey functors, Trans. Am. Math. Soc., Tome 347 (1995) no. 6, pp. 1865-1961 | Article | MR 1261590

[24] Webb, Peter Stratifications and Mackey Functors II: Globally Defined Mackey Functors, J. K-Theory, Tome 6 (2010) no. 1, pp. 99-170 | Article | Zbl 1243.20014

[25] Zimmermann, Alexander Representation Theory: A Homological Algebra Point of View, Springer, Algebra and Applications, Tome 19 (2014), xx+707 pages | Zbl 1306.20001