Maximal ideals in modular group algebras of the finitary symmetric and alternating groups
HTML articles powered by AMS MathViewer
- by Alexander Baranov and Alexander Kleshchev PDF
- Trans. Amer. Math. Soc. 351 (1999), 595-617 Request permission
Abstract:
The main result of the paper is a description of the maximal ideals in the modular group algebras of the finitary symmetric and alternating groups (provided the characteristic $p$ of the ground field is greater than 2). For the symmetric group there are exactly $p-1$ such ideals and for the alternating group there are $(p-1)/2$ of them. The description is obtained in terms of the annihilators of certain systems of the ‘completely splittable’ irreducible modular representations of the finite symmetric and alternating groups. The main tools used in the proofs are the modular branching rules (obtained earlier by the second author) and the ‘Mullineux conjecture’ proved recently by Ford-Kleshchev and Bessenrodt-Olsson. The results obtained are relevant to the theory of PI-algebras. They are used in a later paper by the authors and A. E. Zalesskii on almost simple group algebras and asymptotic properties of modular representations of symmetric groups.References
- S. A. Amitsur, The polynomial identities of associative rings, in “Noetherian Rings and Rings with Polynomial Identities”, Proc. Conf. Univ. Durham, 1979, pp. 1–38.
- C. Bessenrodt and J.B. Olsson. On residue symbols and the Mullineux conjecture, J. Algebraic Combin. 7 (1998), 227–251.
- Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0144979
- Ben Ford, Irreducible representations of the alternating group in odd characteristic, Proc. Amer. Math. Soc. 125 (1997), no. 2, 375–380. MR 1353385, DOI 10.1090/S0002-9939-97-03621-6
- B. Ford and A. Kleshchev, A proof of the Mullineux conjecture, Math. Z. 226 (1997), 257–308.
- Edward Formanek and John Lawrence, The group algebra of the infinite symmetric group, Israel J. Math. 23 (1976), no. 3-4, 325–331. MR 417267, DOI 10.1007/BF02761809
- Edward Formanek and Claudio Procesi, Mumford’s conjecture for the general linear group, Advances in Math. 19 (1976), no. 3, 292–305. MR 404279, DOI 10.1016/0001-8708(76)90026-8
- G. D. James, On the decomposition matrices of the symmetric groups. I, J. Algebra 43 (1976), no. 1, 42–44. MR 430049, DOI 10.1016/0021-8693(76)90142-3
- G. D. James, On a conjecture of Carter concerning irreducible Specht modules, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 1, 11–17. MR 463281, DOI 10.1017/S0305004100054232
- G. D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR 513828
- Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
- G. D. James and G. E. Murphy, The determinant of the Gram matrix for a Specht module, J. Algebra 59 (1979), no. 1, 222–235. MR 541676, DOI 10.1016/0021-8693(79)90158-3
- A. S. Kleshchev, Branching rules for modular representations of symmetric groups. I, J. Algebra 178 (1995), no. 2, 493–511. MR 1359899, DOI 10.1006/jabr.1995.1362
- A. S. Kleshchev, Branching rules for modular representations of symmetric groups. I, J. Algebra 178 (1995), no. 2, 493–511. MR 1359899, DOI 10.1006/jabr.1995.1362
- Alexander Kleshchev, Completely splittable representations of symmetric groups, J. Algebra 181 (1996), no. 2, 584–592. MR 1383482, DOI 10.1006/jabr.1996.0135
- A. S. Kleshchev, On decomposition numbers and branching coefficients for symmetric and special linear groups , Proc. London Math. Soc. (3) 75 (1997), 497–558.
- G. Mullineux, On the $p$-cores of $p$-regular diagrams, J. London Math. Soc. (2) 20 (1979), no. 2, 222–226. MR 551448, DOI 10.1112/jlms/s2-20.2.222
- Ju. P. Razmyslov, Identities with trace in full matrix algebras over a field of characteristic zero, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 723–756 (Russian). MR 0506414
- Yu. P. Razmyslov, Tozhdestva algebr i ikh predstavleniĭ , Sovremennaya Algebra. [Modern Algebra], “Nauka”, Moscow, 1989 (Russian). With an English summary. MR 1007304
- Louis Halle Rowen, Polynomial identities in ring theory, Pure and Applied Mathematics, vol. 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 576061
- S. K. Sehgal and A. E. Zalesskiĭ, Induced modules and some arithmetic invariants of the finitary symmetric groups, Nova J. Algebra Geom. 2 (1993), no. 1, 89–105. MR 1254154
- A. E. Zalesski, Modular group rings of the finitary symmetric group. part B, Israel J. Math. 96 (1996), no. part B, 609–621. MR 1433709, DOI 10.1007/BF02937325
- A. E. Zalesskiĭ, Group rings of simple locally finite groups, Finite and locally finite groups (Istanbul, 1994) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 471, Kluwer Acad. Publ., Dordrecht, 1995, pp. 219–246. MR 1362812, DOI 10.1007/978-94-011-0329-9_{9}
Additional Information
- Alexander Baranov
- Affiliation: Institute of Mathematics, Academy of Sciences of Belarus, Surganova 11, Minsk, 220072, Belarus
- Email: baranov@im.bas-net.by
- Alexander Kleshchev
- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
- MR Author ID: 268538
- Email: klesh@math.uoregon.edu
- Received by editor(s): November 25, 1996
- Additional Notes: Supported by the Fundamental Research Foundation of Belarus and the National Science Foundation
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 595-617
- MSC (1991): Primary 20C05, 16S34
- DOI: https://doi.org/10.1090/S0002-9947-99-02003-6
- MathSciNet review: 1443188