Abstract
Let F be an arbitrary field of characteristic p ≥ 0, and Σ n be the symmetric group on n letters. We shall discuss finite-dimensional representations of Σ n over F or, equivalently, finite-dimensional FΣ n -modules. Basic facts important for our topic can be found in [18, 21, 20]. We recall here some of them.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Amitsur S.A. (1979) The polynomial identities of associative rings, in A. Goldie and T. Willmore (eds.), Noetherian Rings and Rings with Polynomial Identities, Proc. Conf. Univ. Durham, pp. 1–38.
Baranov, A. and Kleshchev, A. (1997) Maximal ideals in modular group algebras of the finitary symmetric and alternating groups, Trans. Amer. Math. Soc., to appear.
Benson, D. (1987) Some remarks on the decomposition numbers for the symmetric groups, in P. Fong (ed.), The Arcata Conference on Representations of Finite Groups, Proc. Symp. Pure Math. 47 (1), Amer. Math. Soc., Providence, pp. 381–394.
Bessenrodt, C. and Olsson, J.B. (1996) On residue symbols and the Mullineux conjecture, preprint, University of Copenhagen (to appear in J. Alg. Comb.).
Brundan, J. (1997) Modular branching rules and the Mullineux map for Hecke algebras of type A, Proc. London Math. Soc., to appear.
Brundan, J., Kleshchev, A. and Suprunenko, I. (1997) Semisimple restrictions from GL(n) to GL(n - 1), preprint.
Dipper, R. (1990) On quotients of Hom-functors and representations of finite general linear groups, I, J. Algebra 130, 235–259.
Erdmann, K. (1995) Tensor products and dimensions of simple modules for symmetric groups, Manuscripta Math. 88, 357–386.
Erdmann, K. (1996) Decomposition numbers for symmetric groups and composition factors of Weyl modules, J. Algebra 180, 316–320.
Ford, B. (1995) Irreducible restrictions of representations of the symmetric groups, Bull. London Math. Soc. 27, 453–459.
Ford, B. (1997) Irreducible representations of the alternating groups in odd characteristic, Proc. Amer. Math. Soc. 125, 375–380.
Ford, B. and Kleshchev, A. (1997) A proof of the Mullineux conjecture, Math. Z. 226, 267–308.
Formanek, E. and Lawrence, D. (1976) The group algebra of the infinite symmetric group, Israel J. Math. 23, 325–331.
Formanek, E. and. Procesi, C. (1976) Mumford’s conjecture for the general linear group, Adv. in Math. 19, 292–305.
Green, J.A. (1980) Polynomial Representations of GL n , Lecture Notes in Mathematics 830, Springer-Verlag, Berlin, Heidelberg, New York.
James, G.D. (1976) On the decomposition matrices of the symmetric groups, II, J. Algebra 43, 45–54.
James, G.D. (1978) On a conjecture of Carter concerning irreducible Specht modules, Math. Proc. Carob. Phil. Soc. 83, 11–17.
James, G.D. (1978) The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682, Springer-Verlag, Berlin, Heidelberg, New York.
James, G.D. (1983) On the minimal dimensions of irreducible representations of symmetric groups. Math. Proc. Cambridge Philos. Soc. 94, 417–424.
James, G.D. (1987) The representation theory of the symmetric groups, in P. Fong (ed.), The Arcata Conference on Representations of Finite Groups, Proc. Symp. Pure Math. 47 (1), Amer. Math. Soc., Providence, pp. 111–126.
James, G.D. and Kerber, A. (1981) The Representation Theory of the Symmetric Group, Addison-Wesley, London.
James, G.D. and Murphy, G.E. (1979) The determinant of the Gram matrix for a Specht module. J. Algebra 59, 222–235.
Jantzen, J.C. and Seitz, G.M. (1992) On the representation theory of the symmetric groups, Proc. London Math. Soc. 65, 475–504.
Kemer, A. (1996) Remarks on the prime varieties, Israel J. Math. 96, 341–356.
Kleshchev, A.S. (1994) On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups, I, Proc. London Math. Soc. 69, 515–540.
Kleshchev, A.S. (1994) On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups, II, Comm. Alg. 22, 6175–6208.
Kleshchev, A.S. (1995) Branching rules for modular representations of symmetric groups, I, J. Algebra 178, 493–511.
Kleshchev, A.S. (1995) Branching rules for modular representations of symmetric groups, II, J. Reine Angew. Math. 459, 163–212.
Kleshchev, A.S. (1996) Branching rules for modular representations of symmetric groups, III, J. London Math. Soc. 54, 25–38.
Kleshchev, A.S. (1998) Branching rules for modular representations of symmetric groups, IV, J. Algebra 201, 547–572.
Kleshchev, A.S. (1996) Completely splittable representations of symmetric groups, J. Algebra 181, 584–592.
Kleshchev, A.S. (1997) On decomposition numbers and branching coefficients for symmetric and special linear groups, Proc. Lond. Math. Soc. 75, 497–558.
Kleshchev, A. and Premet, A. (1997) The globally irreducible representations of symmetric groups, Math. Proc. Camb. Phil. Soc., to appear.
Lascoux, A., Leclerc, B. and Thibon, J.-Y. (1996) Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181, 205–263.
Mathieu, O. (1996) On the dimension of some modular irreducible representations of the symmetric group, Lett. Math. Phys. 38, 23–32.
Mathieu, O. and Papadopoulo, G. (1997) A character formula for a family of simple modular representations of GL n , preprint.
Mullineux, G. (1979) Bijections of p-regular partitions and p-modular irreducibles of the symmetric groups, J. London Math. Soc. 20, 60–66.
Mullineux, G. (1979) On the p-cores of p-regular diagrams, J. London Math. Soc. 20, 222–226.
Razmyslov, Yu.P. (1974) Trace identities of full matrix algebras over a field of characteristic zero, Math. USSR. Izvestiya 8, 727–760.
Razmyslov, Yu.P. (1989) Identities of Algebras and Their Representations, Nauka, Moscow (in Russian).
Ryba, A. (1994) Fibonacci representations of the symmetric groups, J. Algebra 170, 678–686.
Seitz, G.M. (1987) The maximal subgroups of classical algebraic groups, Mem. Amer. Math. Soc. 365, 1–286.
Seitz, G.M. (1992) Subgroups of finite and algebraic groups, in Groups, Combinatorics, and Geometry, London Math. Soc. Lecture Notes Series 165, Cambridge University Press, pp. 316–326.
Sheth, J. (1997) Branching rules for two-part partitions and inductive systems, preprint, University of Oregon.
Smith, S.D. (1982) Irreducible modules and parabolic subgroups. J. Algebra 74, 286–289.
Xu, M. (1997) On Mullineux’ conjecture in the representation theory of symmetric groups, Comm. Alg. 25, 1797–1803.
Zalesskiĭ, A.E. (1995) Group rings of simple locally finite groups, in B. Hartley et al. (eds.) Finite and Locally Finite Groups, Kluwer Academic Publishers, Dordrecht, pp. 219–246.
Zalesskiĭ, A.E. (1996) Modular group rings of the finitary symmetric groups, Israel J. Math. 96, 609–621.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Kleshchev, A.S. (1998). Branching Rules for Symmetric Groups and Applications. In: Carter, R.W., Saxl, J. (eds) Algebraic Groups and their Representations. NATO ASI Series, vol 517. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5308-9_7
Download citation
DOI: https://doi.org/10.1007/978-94-011-5308-9_7
Publisher Name: Springer, Dordrecht
Print ISBN: 978-0-7923-5292-1
Online ISBN: 978-94-011-5308-9
eBook Packages: Springer Book Archive