Skip to main content Accessibility help
×
Home

Stratifications and Mackey Functors II: Globally Defined Mackey Functors

  • Peter Webb (a1)

Abstract

We describe structural properties of globally defined Mackey functors related to the stratification theory of algebras. We show that over a field of characteristic zero they form a highest weight category and we also determine precisely when this category is semisimple. This approach is used to show that the Cartan matrix is often symmetric and non-singular, and we are able to compute finite parts of it in some instances. We also develop a theory of vertices of globally defined Mackey functors in the spirit of group representation theory, as well as giving information about extensions between simple functors.

Copyright

References

Hide All
1.Adams, J.F., Gunawardena, J.H.C. and Miller, H., The Segal conjecture for elementary abelian p-groups, Topology 24 (1985), 435460.
2.Alperin, J.L., Local representation theory, Cambridge University Press 1986.
3.Auslander, M., Reiten, I. and Smalø, S.O., Representation theory of Artin algebras, Cambridge studies in advanced mathematics 36, Cambridge University Press 1995.
4.Barker, L., Rhetorical biset functors, rational p-biset functors and their semisimplicity in characteristic zero, preprint.
5.Benson, D.J., Representations and cohomology I: basic representation theory of finite groups and associative algebras, Cambridge studies in advanced mathematics 30, Cambridge University Press 1991.
6.Benson, D.J. and Feshbach, M., Stable splittings of classifying spaces of finite groups, Topology 31 (1992), 157176.
7.Bouc, S., Foncteurs d'ensembles munis d'une double action, J. Algebra 183 (1996), 664736.
8.Bouc, S., The functor of rational representations for p-groups, Adv. Math. 186 (2004), 267306.
9.Bouc, S., The Dade group of a p-group, Invent. Math. 164 (2006), 189231.
10.Bouc, S., The functor of units of Burnside rings for p-groups, Comment. Math. Helv. 82 (2007), 583615.
11.Bouc, S., Rational p-biset functors, J. Algebra 319 (2008), 17761800.
12.Bouc, S. and Thévenaz, J., The group of endo-permutation modules, Invent. Math. 139 (2000), 275349.
13.Bourbaki, N., Algebra I, Springer-Verlag (1989).
14.Bourizk, I., Sur des foncteurs simples, Pacific J. Math. 215 (2004), 201221.
15.Bourizk, I., A remark on a functor of rational representations, Bull. Belg. Math. Soc. Simon Stevin 13 (2006), 149157.
16.Cline, E., Parshall, B. and Scott, L., Finite dimensional algebras and highest weight categories, J. reine angew. Math. 391 (1988), 8599.
17.Cline, E., Parshall, B. and Scott, L., Stratifying endomorphism algebras, Mem. Amer. Math. Soc. 124 (1996), no. 591.
18.tom Dieck, T., Transformation Groups, Walter de Gruyter, Berlin – New York 1987.
19.Dlab, V. and Ringel, C.M., The module theoretical approach to quasi-hereditary algebras, in ‘Representations of Algebras and Related Topics,’ Tachikawa, H. and Brenner, S., eds.), pp. 200224. London Mathematical Society Lecture Note Series, vol 168 (1992).
20.Erdmann, K., Symmetric groups and quasi-hereditary algebras, pp. 123161 in Dlab, V. and Scott, L.L. (eds.), Finite Dimensional Algberas and Related Topics, Kluwer 1994.
21.Gabriel, P., Des catégories abéliennes, Bull. Soc. Math. Franc. 90 (1962), 323448.
22.Geck, M. and Rouquier, R., Centers and simple modules for Iwahori-Hecke algebras, pp. 251272 in Finite reductive groups (Luminy, 1994), Progr. Math. 141, BirkhŁuser Boston, Boston, MA, 1997.
23.Green, J.A., Polynmial representations of GLn, Lecture Notes in Math. 830, Springer-Verlag 1980.
24.Lewis, L.G. Jr., The theory of Green functors, unpublished notes, 1981.
25.Lewis, L.G. Jr., May, J.P. and McClure, J.E., Classifying G-spaces and the Segal conjecture, pp. 165179 in: Kane, R.M. et al. (eds.), Current Trends in Algebraic Topology, CMS Conference Proc. 2 (1982).
26.Martino, J.R., Classifying spaces and their maps, pp. 161198 in ‘Homotopy theory and its applications’ (Cocoyoc 1993), Contemp. Math. 188, American Math. Soc. 1995.
27.Miller, H., Letter to J.F. Adams, 1981.
28.Symonds, P., A splitting principle for group representations, Comment. Math. Helv. 66 (1991), 169184.
29.Thévenaz, J., G-Algebras and modular representation theory, Oxford University Press 1995.
30.Thévenaz, J. and Webb, P.J., Simple Mackey Functors, Proc. of 2nd international group theory conference, Bressanone (1989), Supplement to Rendiconti del Circolo Matematico di Palermo 23 (1990), 299319.
31.Thévenaz, J. and Webb, P.J., The structure of Mackey functors, Trans. Amer. Math. Soc. 347 (1995), 18651961.
32.Webb, P.J., Two classifications of simple Mackey functors with applications to group cohomology and the decomposition of classifying spaces, J. Pure Appl. Algebra 88 (1993), 265304.
33.Webb, P.J., A guide to Mackey functors, Hazewinkel, M. (ed.), Handbook of Algebra vol. 2, Elsevier 2000, pp. 805836.
34.Webb, P.J., Stratifications and Mackey functors I: functors for a single group, Proc. London Math. Soc. (3) 82 (2001), 299336.
35.Webb, P.J., Weight Theory in the Context of Arbitrary Finite Groups, Fields Institute Communications 40 (2004), 277289.
36.Webb, P.J., Standard stratifications of EI categories and Alperin's weight conjecture, to appear in Journal of Algebra.
37.Witten, C.M., Self-maps of classifying spaces of finite groups and classification of low-dimensional Poincaré duality spaces, Ph.D. thesis, Stanford University 1978.
38.Xu, F., Homological Properties of Category Algebras, Ph.D. thesis, University of Minnesota 2006.

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed