In this short note, we investigate some consequences of the vanishing of simple biset functors. As a corollary, if there is no non-trivial vanishing of simple biset functors (e.g., if the group G is commutative), then we show that is a quasi-hereditary algebra in characteristic zero. In general, this is not true without the non-vanishing condition, as over a field of characteristic zero, the double Burnside algebra of the alternating group of degree 5 has infinite global dimension.
Dans cette note, on s'intéresse à quelques conséquences du phénomène dit de disparition des foncteurs à bi-ensembles simples. On démontre que, dans le cas où il n'y a pas de disparitions non triviales de foncteurs simples (par exemple, si le groupe est commutatif), alors l'algèbre de Burnside double en caractéristique zéro est quasi-héréditaire. Sans l'hypothèse de non-disparitions triviales, ce résultat est en général faux. En effet, l'algèbre de Burnside double du groupe alterné de degré 5 en caractéristique zéro est de dimension globale infinie.
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Baptiste Rognerud 1
@article{CRMATH_2015__353_8_689_0, author = {Baptiste Rognerud}, title = {Quasi-hereditary property of double {Burnside} algebras}, journal = {Comptes Rendus. Math\'ematique}, pages = {689--693}, publisher = {Elsevier}, volume = {353}, number = {8}, year = {2015}, doi = {10.1016/j.crma.2015.05.008}, language = {en}, }
Baptiste Rognerud. Quasi-hereditary property of double Burnside algebras. Comptes Rendus. Mathématique, Volume 353 (2015) no. 8, pp. 689-693. doi : 10.1016/j.crma.2015.05.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.05.008/
[1] Rhetorical biset functors, rational p-biset functors and their semisimplicity in characteristic zero, J. Algebra, Volume 319 (2008) no. 9, pp. 3810-3853
[2] Biset Functors for Finite Groups, Lecture Notes in Mathematics, vol. 1990, Springer, 2010
[3] Simple biset functors and double Burnside ring, J. Pure Appl. Algebra, Volume 217 (2013) no. 3, pp. 546-566
[4] Vanishing evaluations of simple functors, J. Pure Appl. Algebra, Volume 218 (2014) no. 2, pp. 218-227
[5] Krull–Remak–Schmidt categories and projective covers http://www.math.uni-bielefeld.de/~hkrause/krs.pdf
[6] Stratifications and Mackey functors II: globally defined Mackey functors, J. K-Theory, Volume 6 (2010) no. 1, pp. 99-170
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