Around evaluations of biset functors
[Évaluations de foncteurs à bi-ensembles]
Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 805-843.

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.

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.

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DOI : https://doi.org/10.5802/aif.3259
Classification : 19A22,  20C99,  16G10,  18E10
Mots clés : Bi-ensemble, Anneau de Burnside, foncteur à bi-ensemble, algèbre quasi-héréditaire
@article{AIF_2019__69_2_805_0,
     author = {Rognerud, Baptiste},
     title = {Around evaluations of biset functors},
     journal = {Annales de l'Institut Fourier},
     pages = {805--843},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {2},
     year = {2019},
     doi = {10.5802/aif.3259},
     zbl = {07067420},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3259/}
}
Rognerud, Baptiste. Around evaluations of biset functors. Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 805-843. doi : 10.5802/aif.3259. http://archive.numdam.org/articles/10.5802/aif.3259/

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