A new canonical induction formula for <i>p</i>-permutation modules

Laurence Barker; Hatice Mutlu

doi:10.1016/j.crma.2019.04.004

Group theory

A new canonical induction formula for p-permutation modules

Presented by: the Editorial Board

Laurence Barker ¹; Hatice Mutlu ¹

¹Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey

Comptes Rendus. Mathématique, Volume 357 (2019) no. 4, pp. 327-332

Abstract (Original language)
Résumé

Applying Robert Boltje's theory of canonical induction, we give a restriction-preserving formula expressing any p-permutation module as a $Z [1 / p]$ -linear combination of modules induced and inflated from projective modules associated with subquotient groups. The underlying constructions include, for any given finite group, a ring with a $Z$ -basis indexed by conjugacy classes of triples $(U, K, E)$ where U is a subgroup, K is a $p^{'}$ -residue-free normal subgroup of U, and E is an indecomposable projective module of the group algebra of $U / K$ .

En application de la théorie de l'induction canonique de Robert Boltje, nous présentons une formule stable par restriction au moyen de laquelle tout module de p-permutation est exprimé sous forme de combinaison $Z [1 / p]$ -linéaire des inductions des inflations des modules projectifs associés à des groupes de sous-quotients. Les constructions concernées comprennent, pour tout groupe fini, un anneau qui a une $Z$ -base indexée par les classes de conjugaison des triplets $(U, K, E)$ avec U un sous-groupe, $O^{p^{'}} (K) = K ⊴ U$ et E un module projectif indécomposable de l'algèbre de groupe de $U / K$ .

Received: 2018-10-01
Accepted: 2019-04-09
Published online: 2019-04-01

DOI: 10.1016/j.crma.2019.04.004

Author's affiliations:

Laurence Barker ¹ ; Hatice Mutlu ¹

¹ Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey

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     title = {A new canonical induction formula for \protect\emph{p}-permutation modules},
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     pages = {327--332},
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     doi = {10.1016/j.crma.2019.04.004},
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Laurence Barker; Hatice Mutlu. A new canonical induction formula for p-permutation modules. Comptes Rendus. Mathématique, Volume 357 (2019) no. 4, pp. 327-332. doi: 10.1016/j.crma.2019.04.004

References
Cited by

[1] L. Barker An inversion formula for the primitive idempotents of the trivial source algebra, J. Pure Appl. Math. (2019) (in press) | DOI

[2] R. Boltje A general theory of canonical induction formulae, J. Algebra, Volume 206 (1998), pp. 293-343

[3] R. Boltje Linear source modules and trivial source modules, Proc. Symp. Pure Math., Volume 63 (1998), pp. 7-30

[4] R. Boltje, Representation rings of finite groups, their species and idempotent formulae, preprint.

[5] R. Boltje; G. Raggi-Cárdenas; L. Valero-Elizondo The $-_{+}$ and $-^{+}$ constructions for biset functors, J. Algebra, Volume 523 (2019), pp. 241-273

[6] S. Bouc; J. Thévenaz The primitive idempotents of the p-permutation ring, J. Algebra, Volume 323 (2010), pp. 2905-2915

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