×

zbMATH — the first resource for mathematics

Modular branching rules for projective representations of symmetric groups and lowering operators for the supergroup \(Q(n)\). (English) Zbl 1326.20011
Mem. Am. Math. Soc. 1034, xviii, 123 p. (2012).
Summary: There are two approaches to projective representation theory of symmetric and alternating groups, which are powerful enough to work for modular representations. One is based on Sergeev duality, which connects projective representation theory of the symmetric group and representation theory of the algebraic supergroup \(Q(n)\) via appropriate Schur (super)algebras and Schur functors. The second approach follows the work of Grojnowski for classical affine and cyclotomic Hecke algebras and connects projective representation theory of symmetric groups in characteristic \(p\) to the crystal graph of the basic module of the twisted affine Kac-Moody algebra of type \(A_{p-1}^{(2)}\).
The goal of this work is to connect the two approaches mentioned above and to obtain new branching results for projective representations of symmetric groups. This is achieved by developing the theory of lowering operators for the supergroup \(Q(n)\) which is parallel to (although much more intricate than) the similar theory for \(GL(n)\) developed by the first author. The theory of lowering operators for \(GL(n)\) is a non-trivial generalization of Carter’s work in characteristic zero, and it has received a lot of attention. So this part of our work might be of independent interest.
One of the applications of lowering operators is to tensor products of irreducible \(Q(n)\)-modules with natural and dual natural modules, which leads to important special translation functors. We describe the socles and primitive vectors in such tensor products.
MSC:
20C30 Representations of finite symmetric groups
20C25 Projective representations and multipliers
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
20G05 Representation theory for linear algebraic groups
05E10 Combinatorial aspects of representation theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Susumu Ariki, On the decomposition numbers of the Hecke algebra of \(G(m,1,n)\), J. Math. Kyoto Univ. 36 (1996), no. 4, 789-808. · Zbl 0888.20011
[2] Susumu Ariki, Proof of the modular branching rule for cyclotomic Hecke algebras, J. Algebra 306 (2006), no. 1, 290-300. · Zbl 1130.20005 · doi:10.1016/j.jalgebra.2006.04.033
[3] Hussam Arisha and Mary Schaps, Maximal strings in the crystal graph of spin representations of the symmetric and alternating groups, Comm. Algebra 37 (2009), no. 11, 3779-3795. · Zbl 1185.20011 · doi:10.1080/00927870802502654
[4] Alexander Baranov and Alexander Kleshchev, Maximal ideals in modular group algebras of the finitary symmetric and alternating groups, Trans. Amer. Math. Soc. 351 (1999), no. 2, 595-617. · Zbl 0915.16024 · doi:10.1090/S0002-9947-99-02003-6
[5] A. A. Baranov, A. S. Kleshchev, and A. E. Zalesskii, Asymptotic results on modular representations of symmetric groups and almost simple modular group algebras, J. Algebra 219 (1999), no. 2, 506-530. · Zbl 0946.20004 · doi:10.1006/jabr.1999.7923
[6] Jonathan Brundan, Modular branching rules and the Mullineux map for Hecke algebras of type \(A\), Proc. London Math. Soc. (3) 77 (1998), no. 3, 551-581. · Zbl 0904.20007 · doi:10.1112/S0024611598000562
[7] Jonathan Brundan, Lowering operators for \({\mathrm GL}(n)\) and quantum \({\mathrm GL}(n)\), Group representations: cohomology, group actions and topology (Seattle, WA, 1996) Proc. Sympos. Pure Math., vol. 63, Amer. Math. Soc., Providence, RI, 1998, pp. 95-114. · Zbl 0895.20034 · doi:10.1090/pspum/063/1603139
[8] Jonathan Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra \({\mathfrak q}(n)\), Adv. Math. 182 (2004), no. 1, 28-77. · Zbl 1048.17003 · doi:10.1016/S0001-8708(03)00073-2
[9] Jonathan Brundan, Modular representations of the supergroup \(Q(n)\). II, Pacific J. Math. 224 (2006), no. 1, 65-90. · Zbl 1122.20022 · doi:10.2140/pjm.2006.224.65
[10] Jonathan Brundan and Alexander Kleshchev, Modular Littlewood-Richardson coefficients, Math. Z. 232 (1999), no. 2, 287-320. · Zbl 0945.20027 · doi:10.1007/s002090050516
[11] Jonathan Brundan and Alexander Kleshchev, On translation functors for general linear and symmetric groups, Proc. London Math. Soc. (3) 80 (2000), no. 1, 75-106. · Zbl 1011.20042 · doi:10.1112/S0024611500012132
[12] 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
[13] Jonathan Brundan and Alexander Kleshchev, Hecke-Clifford superalgebras, crystals of type \(A_{2l}^{(2)}\) and modular branching rules for \(\hat S_n\), Represent. Theory 5 (2001), 317-403. · Zbl 1005.17010 · doi:10.1090/S1088-4165-01-00123-6
[14] 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
[15] Jonathan Brundan and Alexander Kleshchev, Representation theory of symmetric groups and their double covers, Groups, combinatorics & geometry (Durham, 2001) World Sci. Publ., River Edge, NJ, 2003, pp. 31-53. · Zbl 1043.20005 · doi:10.1142/9789812564481_0003
[16] Jonathan Brundan and Alexander Kleshchev, Cartan determinants and Shapovalov forms, Math. Ann. 324 (2002), no. 3, 431-449. · Zbl 1047.17012 · doi:10.1007/s00208-002-0346-0
[17] Jonathan Brundan and Alexander Kleshchev, Modular representations of the supergroup \(Q(n)\). I, J. Algebra 260 (2003), no. 1, 64-98. Special issue celebrating the 80th birthday of Robert Steinberg. · Zbl 1027.17004 · doi:10.1016/S0021-8693(02)00620-8
[18] Jonathan Brundan and Alexander Kleshchev, James’ regularization theorem for double covers of symmetric groups, J. Algebra 306 (2006), no. 1, 128-137. · Zbl 1112.20009 · doi:10.1016/j.jalgebra.2006.01.055
[19] Jonathan Brundan and Alexander Kleshchev, Graded decomposition numbers for cyclotomic Hecke algebras, Adv. Math. 222 (2009), no. 6, 1883-1942. · Zbl 1241.20003 · doi:10.1016/j.aim.2009.06.018
[20] R. W. Carter, Raising and lowering operators for \({\mathfrak s}{\mathfrak l}_n\), with applications to orthogonal bases of \({\mathfrak s}{\mathfrak l}_n\)-modules, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 351-366.
[21] Roger W. Carter and George Lusztig, On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193-242. · Zbl 0298.20009
[22] I. Grojnowski, Affine \(\mathfrak{sl}_p\) controls the representation theory of the symmetric group and related Hecke algebras, arXiv:math.RT/9907129.
[23] Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. · Zbl 1034.20041
[24] Seok-Jin Kang, Crystal bases for quantum affine algebras and combinatorics of Young walls, Proc. London Math. Soc. (3) 86 (2003), no. 1, 29-69. · Zbl 1030.17013 · doi:10.1112/S0024611502013734
[25] S.-J. Kang, M. Kashiwara, and S. Tsuchioka, Quiver Hecke superalgebras
[26] Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups. I, Represent. Theory 13 (2009), 309-347. · Zbl 1188.81117 · doi:10.1090/S1088-4165-09-00346-X
[27] Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups II, Trans. Amer. Math. Soc. 363 (2011), no. 5, 2685-2700. · Zbl 1214.81113 · doi:10.1090/S0002-9947-2010-05210-9
[28] M. Khovanov and A.D. Lauda, A diagrammatic approach to categorification of quantum groups. III, arXiv:0807.3250. · Zbl 1188.81117
[29] 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. · Zbl 0808.20039 · doi:10.1112/plms/s3-69.3.515
[30] A. S. Kleshchev, Branching rules for modular representations of symmetric groups. I, J. Algebra 178 (1995), no. 2, 493-511. · Zbl 0854.20013 · doi:10.1006/jabr.1995.1362
[31] Alexander S. Kleshchev, Branching rules for modular representations of symmetric groups. II, J. Reine Angew. Math. 459 (1995), 163-212. · Zbl 0817.20009 · doi:10.1515/crll.1995.459.163
[32] A. S. Kleshchev, Branching rules for modular representations of symmetric groups. III. Some corollaries and a problem of Mullineux, J. London Math. Soc. (2) 54 (1996), no. 1, 25-38. · Zbl 0854.20014 · doi:10.1112/jlms/54.1.25
[33] Alexander Kleshchev, On decomposition numbers and branching coefficients for symmetric and special linear groups, Proc. London Math. Soc. (3) 75 (1997), no. 3, 497-558. · Zbl 0907.20023 · doi:10.1112/S0024611597000427
[34] Alexander Kleshchev, Branching rules for modular representations of symmetric groups. IV, J. Algebra 201 (1998), no. 2, 547-572. · Zbl 0931.20014 · doi:10.1006/jabr.1997.7302
[35] Alexander Kleshchev, Linear and projective representations of symmetric groups, Cambridge Tracts in Mathematics, vol. 163, Cambridge University Press, Cambridge, 2005. · Zbl 1080.20011
[36] Alexander S. Kleshchev and Pham Huu Tiep, On restrictions of modular spin representations of symmetric and alternating groups, Trans. Amer. Math. Soc. 356 (2004), no. 5, 1971-1999 (electronic). · Zbl 1065.20013 · doi:10.1090/S0002-9947-03-03364-6
[37] A. Kleshchev and P.-H. Tiep, Small dimension projective representations of symmetric and alternating groups, arXiv:1106.3123. To appear in Algebra & Number Theory.
[38] Jonathan Kujawa, Crystal structures arising from representations of \({\mathrm GL}(m| n)\), Represent. Theory 10 (2006), 49-85. · Zbl 1196.17011 · doi:10.1090/S1088-4165-06-00219-6
[39] Bernard Leclerc and Jean-Yves Thibon, \(q\)-deformed Fock spaces and modular representations of spin symmetric groups, J. Phys. A 30 (1997), no. 17, 6163-6176. · Zbl 1039.17509 · doi:10.1088/0305-4470/30/17/023
[40] Maxim Nazarov, Young’s symmetrizers for projective representations of the symmetric group, Adv. Math. 127 (1997), no. 2, 190-257. · Zbl 0930.20011 · doi:10.1006/aima.1997.1621
[41] Aaron M. Phillips, Restricting modular spin representations of symmetric and alternating groups to Young-type subgroups, Proc. London Math. Soc. (3) 89 (2004), no. 3, 623-654. · Zbl 1085.20003 · doi:10.1112/S0024611504014893
[42] Arun Ram and Peter Tingley, Universal Verma modules and the Misra-Miwa Fock space, Int. J. Math. Math. Sci. , posted on (2010), Art. ID 326247, 19. · Zbl 1207.81042 · doi:10.1155/2010/326247
[43] R. Rouquier, \(2\)-Kac-Moody algebras, arXiv:0812.5023.
[44] A. N. Sergeev, The centre of enveloping algebra for Lie superalgebra \(Q(n,\,{\mathbf C})\), Lett. Math. Phys. 7 (1983), no. 3, 177-179. · Zbl 0539.17003 · doi:10.1007/BF00400431
[45] A. N. Sergeev, Tensor algebra of the identity representation as a module over the Lie superalgebras \({\mathrm Gl}(n,\,m)\) and \(Q(n)\), Mat. Sb. (N.S.) 123(165) (1984), no. 3, 422-430 (Russian).
[46] Alexander Sergeev, The Howe duality and the projective representations of symmetric groups, Represent. Theory 3 (1999), 416-434 (electronic). · Zbl 0999.17014 · doi:10.1090/S1088-4165-99-00085-0
[47] V. V. Shchigolev, Iterating lowering operators, J. Pure Appl. Algebra 206 (2006), no. 1-2, 111-122. · Zbl 1109.20042 · doi:10.1016/j.jpaa.2005.04.015
[48] Vladimir Shchigolev, Generalization of modular lowering operators for \({\mathrm GL}_n\), Comm. Algebra 36 (2008), no. 4, 1250-1288. · Zbl 1151.20036 · doi:10.1080/00927870701863389
[49] Vladimir Shchigolev, Rectangular low level case of modular branching problem for \({\mathrm GL}_n(K)\), J. Algebra 321 (2009), no. 1, 28-85. · Zbl 1208.20042 · doi:10.1016/j.jalgebra.2008.08.034
[50] Vladimir Shchigolev, Weyl submodules in restrictions of simple modules, J. Algebra 321 (2009), no. 5, 1453-1462. · Zbl 1177.20052 · doi:10.1016/j.jalgebra.2008.11.034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.