Abstract
In an attempt to get some information on the multiplicative structure of the Green ring we study algebraic modules for simple groups, and associated groups such as quasisimple and almost-simple groups. We prove that, for almost all groups of Lie type in defining characteristic, the natural module is non-algebraic. For alternating and symmetric groups, we prove that the simple modules in p-blocks with defect groups of order p 2 are algebraic, for p ≤ 5. Finally, we analyze nine sporadic groups, finding that all simple modules are algebraic for various primes and sporadic groups.
Similar content being viewed by others
References
Alperin, J.: On modules for the linear fractional groups. In: Iwahori, N. (ed.) Finite Groups (Sapporo and Kyoto, l974), pp. 157–164. Japan Society for the Promotion of Science, Tokyo (1976)
Alperin, J.: Projective modules for SL(2, 2n). J. Pure Appl. Algebra 15, 219–234 (1979)
Archer, L.: On certain quotients of the Green rings of dihedral 2-groups. J. Pure Appl. Algebra 212, 1888–1897 (2008)
Auslander, M., Carlson, J.: Almost-split sequences and group rings. J. Algebra 103, 122–140 (1986)
Benson, D., Carlson, J.: Nilpotent elements in the Green ring. J. Algebra 104, 329–350 (1986)
Conlon, S.: The modular representation algebra of groups with Sylow 2-subgroup Z 2 × Z 2. J. Aust. Math. Soc. 6, 76–88 (1966)
Craven, D.: Algebraic Modules for Finite Groups. Ph.D. thesis, University of Oxford, Oxford (2008)
Craven, D.: Simple modules for groups with abelian Sylow 2-subgroups are algebraic. J. Algebra 321, 1473–1479 (2009) (A list of corrections is available on the author’s homepage.)
Craven, D.: Algebraic modules and the Auslander–Reiten quiver. J. Pure Appl. Algebra 215, 221–231 (2011)
Craven, D.: Sources of simple modules for weight 2 blocks of symmetric groups. Q. J. Math. Oxford. doi:10.1093/qmath/har016
Craven, D., Eaton, C., Kessar, R., Linckelmann, M.: The structure of blocks with Klein four defect group. Math. Z. 268, 441–476 (2011)
Donkin, S.: On tilting modules for algebraic groups. Math. Z. 212, 39–60 (1993)
Doty, S., Henke, A.: Decomposition of tensor products of modular irreducibles for SL2. Q. J. Math. 56, 189–207 (2005)
Erdmann, K.: Principal blocks of groups with dihedral Sylow 2-subgroups. Commun. Algebra 5, 665–694 (1977)
Erdmann, K.: On 2-blocks with semidihedral defect groups. Trans. Am. Math. Soc. 256, 267–287 (1979)
Erdmann, K., Henke, A.: On Ringel duality for Schur algebras. Math. Proc. Camb. Philos. Soc. 132, 97–116 (2002)
Feit, W.: Irreducible modules of p-solvable groups. In: Cooperstein, B., Mason., G. (eds.) The Santa Cruz Conference on Finite Groups, pp. 405–412. American Mathematical Society, Providence (1980)
Feit, W.: The Representation Theory of Finite Groups. North-Holland, Amsterdam (1982)
Fong, P., Harris, M.: On perfect isometries and isotypies in finite groups. Invent. Math. 114, 139–191 (1993)
Jansen, C., Lux, K., Parker, R., Wilson, R.: An Atlas of Brauer Characters. Oxford University Press, New York (1995)
Jennings, S.: The structure of the group ring of a p-group over a modular field. Trans. Am. Math. Soc. 50, 175–185 (1941)
Kawata, S., Michler, G., Uno, K.: On Auslander-Reiten components and simple modules for finite groups of Lie type. Osaka J. Math. 38, 21–26 (2001)
Kleidman, P.: The maximal subgroups of the Chevalley groups G 2(q) with q odd, the Ree groups 2 G 2(q), and their automorphism groups. J. Algebra 117, 30–71 (1988)
Koshitani, S., Kunugi, N.: Broué’s conjecture holds for principal 3-blocks with elementary abelian defect group of order 9. J. Algebra 248, 575–604 (2002)
Koshitani, S., Kunugi, N., Waki, K.: Broué’s abelian defect group conjecture holds for the Held group and the sporadic Suzuki group. J. Algebra 279, 638–666 (2004)
Koshitani, S., Miyachi, H.: The principal 3-blocks of four- and five-dimensional projective special linear groups in non-defining characteristic. J. Algebra 226, 788–806 (2000)
Kovács, L.: Some Indecomposables for SL2. Research Report no. 11, Mathematics Research Report Series of the Australian National University (1981)
Liebeck, M., Seitz, G.: Reductive subgroups of exceptional algebraic groups. Mem. Amer. Math. Soc. 121(580), vi+111 (1996)
Puig, L.: Algèbres de source de certains blocs des groupes de Chevalley. Astérisque 181–182, 221–236 (1990)
Puig, L.: On the Local Structure of Morita and Rickard Equivalences Between Brauer Blocks. Birkhäuser, Basel (1999)
Thévenaz, J.: G-Algebras and Modular Representation Theory. Oxford University Press, New York (1995)
Wilson, R.: The Finite Simple Groups. Springer, London (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Craven, D.A. On Tensor Products of Simple Modules for Simple Groups. Algebr Represent Theor 16, 377–404 (2013). https://doi.org/10.1007/s10468-011-9311-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-011-9311-5