On restrictions of modular spin representations of symmetric and alternating groups. (English) Zbl 1065.20013
A major problem both in representation theory and the classification problem for maximal subgroups of (almost) simple finite groups is to try to classify triples \((H,G,V)\) where \(V\) is an absolutely irreducible modular representation of the finite group \(H\) and \(V\) remains irreducible upon restriction to \(G\). In particular, the case where \(H\) is almost simple (modulo its center) is the critical case. In this paper, the authors consider the case where \(H\) is a double cover of \(A_n\) or \(S_n\) (and \(V\) is a faithful \(H\)-module – the case where \(H=A_n\) or \(S_n\) has been studied earlier).
The authors prove quite definitive results in Theorems B, C, D and E describing when such a \(G\) exists. In Theorem A, they also extend Wagner’s results about the faithful representations of \(H\) of smallest dimension. In Theorem F, they show that tensor products of irreducibles are reducible in many instances (in characteristic larger than 3).
This is an important contribution to the field. We refer the reader to the paper for the more detailed statements of the results.
The authors prove quite definitive results in Theorems B, C, D and E describing when such a \(G\) exists. In Theorem A, they also extend Wagner’s results about the faithful representations of \(H\) of smallest dimension. In Theorem F, they show that tensor products of irreducibles are reducible in many instances (in characteristic larger than 3).
This is an important contribution to the field. We refer the reader to the paper for the more detailed statements of the results.
Reviewer: Robert Guralnick (Los Angeles)
MSC:
| 20C20 | Modular representations and characters |
| 20C30 | Representations of finite symmetric groups |
| 20C25 | Projective representations and multipliers |
| 20B35 | Subgroups of symmetric groups |
| 20B20 | Multiply transitive finite groups |
Keywords:
representation theory; finite groups; almost simple groups; maximal subgroups; irreducible restrictions; double covers; faithful modules; basic spin modules; alternating groups; tensor productsReferences:
| [1] | M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), no. 3, 469 – 514. · Zbl 0537.20023 · doi:10.1007/BF01388470 |
| [2] | Antal Balog, Christine Bessenrodt, Jørn B. Olsson, and Ken Ono, Prime power degree representations of the symmetric and alternating groups, J. London Math. Soc. (2) 64 (2001), no. 2, 344 – 356. · Zbl 1018.20008 · doi:10.1112/S0024610701002356 |
| [3] | Christine Bessenrodt, On mixed products of complex characters of the double covers of the symmetric groups, Pacific J. Math. 199 (2001), no. 2, 257 – 268. · Zbl 1056.20009 · doi:10.2140/pjm.2001.199.257 |
| [4] | C. Bessenrodt and A. Kleshchev, On Kronecker products of complex representations of the symmetric and alternating groups, Pacific J. Math. 190 (1999), no. 2, 201 – 223. · Zbl 1009.20013 · doi:10.2140/pjm.1999.190.201 |
| [5] | Christine Bessenrodt and Alexander S. Kleshchev, On tensor products of modular representations of symmetric groups, Bull. London Math. Soc. 32 (2000), no. 3, 292 – 296. · Zbl 1021.20010 · doi:10.1112/S0024609300007098 |
| [6] | Christine Bessenrodt and Alexander S. Kleshchev, Irreducible tensor products over alternating groups, J. Algebra 228 (2000), no. 2, 536 – 550. · Zbl 0998.20013 · doi:10.1006/jabr.2000.8284 |
| [7] | Jonathan Brundan and Alexander S. Kleshchev, Representations of the symmetric group which are irreducible over subgroups, J. Reine Angew. Math. 530 (2001), 145 – 190. · Zbl 1059.20016 · doi:10.1515/crll.2001.002 |
| [8] | Jonathan Brundan and Alexander Kleshchev, Projective representations of symmetric groups via Sergeev duality, Math. Z. 239 (2002), no. 1, 27 – 68. · Zbl 1029.20008 · doi:10.1007/s002090100282 |
| [9] | Jonathan Brundan and Alexander Kleshchev, Hecke-Clifford superalgebras, crystals of type \?_{2\?}\?²\? and modular branching rules for \?_{\?}, Represent. Theory 5 (2001), 317 – 403. · Zbl 1005.17010 |
| [10] | Peter J. Cameron, Finite permutation groups and finite simple groups, Bull. London Math. Soc. 13 (1981), no. 1, 1 – 22. · Zbl 0463.20003 · doi:10.1112/blms/13.1.1 |
| [11] | Peter J. Cameron, Peter M. Neumann, and Jan Saxl, An interchange property in finite permutation groups, Bull. London Math. Soc. 11 (1979), no. 2, 161 – 169. · Zbl 0433.20001 · doi:10.1112/blms/11.2.161 |
| [12] | J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. · Zbl 0568.20001 |
| [13] | Roderick Gow and Alexander Kleshchev, Connections between the representations of the symmetric group and the symplectic group in characteristic 2, J. Algebra 221 (1999), no. 1, 60 – 89. · Zbl 0943.20013 · doi:10.1006/jabr.1999.7943 |
| [14] | Robert M. Guralnick and Pham Huu Tiep, Low-dimensional representations of special linear groups in cross characteristics, Proc. London Math. Soc. (3) 78 (1999), no. 1, 116 – 138. · Zbl 0974.20014 · doi:10.1112/S0024611599001720 |
| [15] | P. N. Hoffman and J. F. Humphreys, Projective representations of the symmetric groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1992. \?-functions and shifted tableaux; Oxford Science Publications. · Zbl 0777.20005 |
| [16] | I. Martin Isaacs, Character theory of finite groups, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Pure and Applied Mathematics, No. 69. · Zbl 0337.20005 |
| [17] | G. D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. · Zbl 0393.20009 |
| [18] | Jens C. Jantzen and Gary M. Seitz, On the representation theory of the symmetric groups, Proc. London Math. Soc. (3) 65 (1992), no. 3, 475 – 504. · Zbl 0779.20004 · doi:10.1112/plms/s3-65.3.475 |
| [19] | Christoph Jansen, Klaus Lux, Richard Parker, and Robert Wilson, An atlas of Brauer characters, London Mathematical Society Monographs. New Series, vol. 11, The Clarendon Press, Oxford University Press, New York, 1995. Appendix 2 by T. Breuer and S. Norton; Oxford Science Publications. · Zbl 0831.20001 |
| [20] | William M. Kantor, \?-homogeneous groups, Math. Z. 124 (1972), 261 – 265. · Zbl 0232.20003 · doi:10.1007/BF01113919 |
| [21] | William M. Kantor, Homogeneous designs and geometric lattices, J. Combin. Theory Ser. A 38 (1985), no. 1, 66 – 74. · Zbl 0559.05015 · doi:10.1016/0097-3165(85)90022-6 |
| [22] | Peter B. Kleidman and David B. Wales, The projective characters of the symmetric groups that remain irreducible on subgroups, J. Algebra 138 (1991), no. 2, 440 – 478. · Zbl 0792.20012 · doi:10.1016/0021-8693(91)90181-7 |
| [23] | Peter Kleidman and Martin Liebeck, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990. · Zbl 0697.20004 |
| [24] | Alexander S. Kleshchev, On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups. I, Proc. London Math. Soc. (3) 69 (1994), no. 3, 515 – 540. , https://doi.org/10.1112/plms/s3-69.3.515 Alexander S. Kleshchev, On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups. II, Comm. Algebra 22 (1994), no. 15, 6175 – 6208. · Zbl 0829.20062 · doi:10.1080/00927879408825183 |
| [25] | A. S. Kleshchev and J. K. Sheth, Representations of the symmetric group are reducible over simply transitive subgroups, Math. Z. 235 (2000), no. 1, 99 – 109. · Zbl 0986.20011 · doi:10.1007/s002090000125 |
| [26] | Alexander S. Kleshchev and Jagat Sheth, Representations of the alternating group which are irreducible over subgroups, Proc. London Math. Soc. (3) 84 (2002), no. 1, 194 – 212. · Zbl 1018.20011 · doi:10.1112/S002461150101320X |
| [27] | Martin W. Liebeck and Gary M. Seitz, On the subgroup structure of classical groups, Invent. Math. 134 (1998), no. 2, 427 – 453. · Zbl 0920.20039 · doi:10.1007/s002220050270 |
| [28] | Donald Livingstone and Ascher Wagner, Transitivity of finite permutation groups on unordered sets, Math. Z. 90 (1965), 393 – 403. · Zbl 0136.28101 · doi:10.1007/BF01112361 |
| [29] | A. Phillips, Branching problems for projective representations of the symmetric and alternating groups, preprint, 2003. |
| [30] | Jan Saxl, The complex characters of the symmetric groups that remain irreducible in subgroups, J. Algebra 111 (1987), no. 1, 210 – 219. · Zbl 0633.20008 · doi:10.1016/0021-8693(87)90251-1 |
| [31] | Ascher Wagner, An observation on the degrees of projective representations of the symmetric and alternating group over an arbitrary field, Arch. Math. (Basel) 29 (1977), no. 6, 583 – 589. · Zbl 0383.20009 · doi:10.1007/BF01220457 |
| [32] | David B. Wales, Some projective representations of \?_{\?}, J. Algebra 61 (1979), no. 1, 37 – 57. · Zbl 0433.20010 · doi:10.1016/0021-8693(79)90304-1 |
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