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On primordial groups for the Green ring

Part of: Grothendieck groups and $K_0$ Representation theory of groups

Published online by Cambridge University Press:  24 October 2012

Alberto G. Raggi-Cárdenas  and
Nadia Romero
Alberto G. Raggi-Cárdenas
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, AP 61-3, CP 58089, Morelia, Michoacán, Mexico (graggi@shi.matmor.unam.mx; nadia@matmor.unam.mx)
Nadia Romero
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, AP 61-3, CP 58089, Morelia, Michoacán, Mexico (graggi@shi.matmor.unam.mx; nadia@matmor.unam.mx)
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Abstract

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Consider the Mackey functor that assigns to each finite group G the Green ring of finitely generated kG-modules, where k is a field of characteristic p > 0. Thévenaz foresaw in 1988 that the class of primordial groups for this functor is the family of k-Dress groups. In this paper we prove that this is true for the subfunctor defined by the Green ring of finitely generated kG-modules of trivial source.

Keywords

Green ringprimordial grouptrivial source module

MSC classification

Secondary: 20C05: Group rings of finite groups and their modules 19A22: Frobenius induction, Burnside and representation rings
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 1 , February 2013 , pp. 337 - 347
Copyright
Copyright © Edinburgh Mathematical Society 2012

References

1.Benson, D. J., Representations and cohomology, Volume I, Basic representation theory of finite groups and associative algebras (Cambridge University Press, 1995).Google Scholar
2.Boltje, R., Explicit and canonical Dress induction, Alg. Representat. Theory 8 (2005), 731–746.CrossRefGoogle Scholar
3.Bouc, S., Foncteurs d'ensembles munis d'une double action, J. Alg. 183 (1996), 664–736.CrossRefGoogle Scholar
4.Coşkun, O., Mackey functors, induction from restriction functors and coinduction from transfer functors, J. Alg. 315 (2007), 224–248.CrossRefGoogle Scholar
5.Curtis, C. W. and Reiner, I., Methods of representation theory: with applications to finite groups and orders, Volume I (Wiley, 1981).Google Scholar
6.Dress, A., Contributions to the theory of induced representations, in Classical algebraic K-theory and connections with arithmetic, Springer Lecture Notes in Mathematics, Volume 342, pp. 181–240 (Springer, 1973).CrossRefGoogle Scholar
7.Raggi, A., Primordial subgroups for mod(kG), J. Alg. 139 (1991), 155–158.CrossRefGoogle Scholar
8.Thévenaz, J., Some remarks on G-functors and the Brauer morphism, J. Reine Angew. Math. 384 (1988), 24–56.Google Scholar
9.Webb, P., Stratifications and Mackey functors, II, Globally defined Mackey functors, J. K-Theory 6 (2010), 99–170.CrossRefGoogle Scholar
10.Yoshida, T., Idempotents of Burnside rings and Dress Induction Theorem, J. Alg. 80 (1983), 90–105.CrossRefGoogle Scholar