Representation theory of symmetric groups and related Hecke algebras

Kleshchev, Alexander

doi:10.1090/S0273-0979-09-01277-4

Representation theory of symmetric groups and related Hecke algebras

Author: Alexander Kleshchev
Journal: Bull. Amer. Math. Soc. 47 (2010), 419-481
MSC (2000): Primary 20C30, 20C08, 17B37, 20C20, 17B67
DOI: https://doi.org/10.1090/S0273-0979-09-01277-4
Published electronically: October 27, 2009
MathSciNet review: 2651085
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Abstract: We survey some fundamental trends in representation theory of symmetric groups and related objects which became apparent in the last fifteen years. The emphasis is on connections with Lie theory via categorification. We present results on branching rules and crystal graphs, decomposition numbers and canonical bases, graded representation theory, connections with cyclotomic and affine Hecke algebras, Khovanov-Lauda-Rouquier algebras, category ${\mathcal O}$ , -algebras, etc.

1. H. H. Andersen, J. C. Jantzen, and W. Soergel, Representations of quantum groups at a 𝑝th root of unity and of semisimple groups in characteristic 𝑝: independence of 𝑝, Astérisque 220 (1994), 321 (English, with English and French summaries). MR 1272539
2. Tomoyuki Arakawa and Takeshi Suzuki, Duality between 𝔰𝔩_{𝔫}(ℭ) and the degenerate affine Hecke algebra, J. Algebra 209 (1998), no. 1, 288–304. MR 1652134, https://doi.org/10.1006/jabr.1998.7530
3. 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. MR 1443748, https://doi.org/10.1215/kjm/1250518452
4. Susumu Ariki, On the classification of simple modules for cyclotomic Hecke algebras of type 𝐺(𝑚,1,𝑛) and Kleshchev multipartitions, Osaka J. Math. 38 (2001), no. 4, 827–837. MR 1864465
5. Susumu Ariki, Representations of quantum algebras and combinatorics of Young tableaux, University Lecture Series, vol. 26, American Mathematical Society, Providence, RI, 2002. Translated from the 2000 Japanese edition and revised by the author. MR 1911030
6. Susumu Ariki, Proof of the modular branching rule for cyclotomic Hecke algebras, J. Algebra 306 (2006), no. 1, 290–300. MR 2271584, https://doi.org/10.1016/j.jalgebra.2006.04.033
7. S. Ariki and N. Jacon, Dipper-James-Murphy's conjecture for Hecke algebras of type , arXiv:math/0703447.
8. S. Ariki, N. Jacon and C. Lecouvey, The modular branching rule for affine Hecke algebras of type A, arXiv:0808.3915
9. Susumu Ariki and Kazuhiko Koike, A Hecke algebra of (𝑍/𝑟𝑍)≀𝔖_{𝔫} and construction of its irreducible representations, Adv. Math. 106 (1994), no. 2, 216–243. MR 1279219, https://doi.org/10.1006/aima.1994.1057
10. Susumu Ariki, Victor Kreiman, and Shunsuke Tsuchioka, On the tensor product of two basic representations of 𝑈ᵥ(̂𝔰𝔩ₑ), Adv. Math. 218 (2008), no. 1, 28–86. MR 2409408, https://doi.org/10.1016/j.aim.2007.11.018
11. Susumu Ariki and Andrew Mathas, The number of simple modules of the Hecke algebras of type 𝐺(𝑟,1,𝑛), Math. Z. 233 (2000), no. 3, 601–623. MR 1750939, https://doi.org/10.1007/s002090050489
12. Susumu Ariki, Andrew Mathas, and Hebing Rui, Cyclotomic Nazarov-Wenzl algebras, Nagoya Math. J. 182 (2006), 47–134. MR 2235339, https://doi.org/10.1017/S0027763000026842
13. M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), no. 3, 469–514. MR 746539, https://doi.org/10.1007/BF01388470
14. 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. MR 1443188, https://doi.org/10.1090/S0002-9947-99-02003-6
15. 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. MR 1706817, https://doi.org/10.1006/jabr.1999.7923
16. C. Bessenrodt and J. B. Olsson, Residue symbols and Jantzen-Seitz partitions, J. Combin. Theory Ser. A 81 (1998), no. 2, 201–230. MR 1603889, https://doi.org/10.1006/jcta.1997.2838
17. Ben Ford and Alexander S. Kleshchev, A proof of the Mullineux conjecture, Math. Z. 226 (1997), no. 2, 267–308. MR 1477629, https://doi.org/10.1007/PL00004340
18. C. Bessenrodt and J. B. Olsson, Branching of modular representations of the alternating groups, J. Algebra 209 (1998), no. 1, 143–174. MR 1652118, https://doi.org/10.1006/jabr.1998.7505
19. Christine Bessenrodt, Jørn B. Olsson, and Maozhi Xu, On properties of the Mullineux map with an application to Schur modules, Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 3, 443–459. MR 1684242, https://doi.org/10.1017/S0305004199003424
20. 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. MR 1722888, https://doi.org/10.2140/pjm.1999.190.201
21. 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. MR 1750169, https://doi.org/10.1112/S0024609300007098
22. Christine Bessenrodt and Alexander S. Kleshchev, Irreducible tensor products over alternating groups, J. Algebra 228 (2000), no. 2, 536–550. MR 1764578, https://doi.org/10.1006/jabr.2000.8284
23. Christine Bessenrodt and Alexander S. Kleshchev, On Kronecker products of spin characters of the double covers of the symmetric groups, Pacific J. Math. 198 (2001), no. 2, 295–305. MR 1835510, https://doi.org/10.2140/pjm.2001.198.295
24. Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527. MR 1322847, https://doi.org/10.1090/S0894-0347-96-00192-0
25. Michel Broué, Equivalences of blocks of group algebras, Finite-dimensional algebras and related topics (Ottawa, ON, 1992) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 424, Kluwer Acad. Publ., Dordrecht, 1994, pp. 1–26. MR 1308978, https://doi.org/10.1007/978-94-017-1556-0_1
26. Michel Broué and Gunter Malle, Zyklotomische Heckealgebren, Astérisque 212 (1993), 119–189 (German). Représentations unipotentes génériques et blocs des groupes réductifs finis. MR 1235834
27. 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. MR 1643413, https://doi.org/10.1112/S0024611598000562
28. Jonathan Brundan, Centers of degenerate cyclotomic Hecke algebras and parabolic category 𝒪, Represent. Theory 12 (2008), 236–259. MR 2424964, https://doi.org/10.1090/S1088-4165-08-00333-6
29. 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. MR 1719176, https://doi.org/10.1112/S0024611500012132
30. Jonathan Brundan and Alexander S. Kleshchev, Representations of the symmetric group which are irreducible over subgroups, J. Reine Angew. Math. 530 (2001), 145–190. MR 1807270, https://doi.org/10.1515/crll.2001.002
31. Jonathan Brundan and Alexander Kleshchev, Hecke-Clifford superalgebras, crystals of type 𝐴_{2𝑙}⁽²⁾ and modular branching rules for 𝑆_{𝑛}, Represent. Theory 5 (2001), 317–403. MR 1870595, https://doi.org/10.1090/S1088-4165-01-00123-6
32. Jonathan Brundan and Alexander Kleshchev, Projective representations of symmetric groups via Sergeev duality, Math. Z. 239 (2002), no. 1, 27–68. MR 1879328, https://doi.org/10.1007/s002090100282
33. Jonathan Brundan and Alexander Kleshchev, Cartan determinants and Shapovalov forms, Math. Ann. 324 (2002), no. 3, 431–449. MR 1938453, https://doi.org/10.1007/s00208-002-0346-0
34. 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. MR 1994959, https://doi.org/10.1142/9789812564481_0003
35. Jonathan Brundan and Alexander Kleshchev, James’ regularization theorem for double covers of symmetric groups, J. Algebra 306 (2006), no. 1, 128–137. MR 2271575, https://doi.org/10.1016/j.jalgebra.2006.01.055
36. Jonathan Brundan and Alexander Kleshchev, Representations of shifted Yangians and finite 𝑊-algebras, Mem. Amer. Math. Soc. 196 (2008), no. 918, viii+107. MR 2456464, https://doi.org/10.1090/memo/0918
37. Jonathan Brundan and Alexander Kleshchev, Schur-Weyl duality for higher levels, Selecta Math. (N.S.) 14 (2008), no. 1, 1–57. MR 2480709, https://doi.org/10.1007/s00029-008-0059-7
38. J. Brundan and A. Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, Invent. Math., to appear; arXiv:0808.2032.
39. J. Brundan and A. Kleshchev, The degenerate analogue of Ariki's categorification theorem, Math. Z., to appear; arXiv:0901.0057.
40. J. Brundan and A. Kleshchev, Graded decomposition numbers for cyclotomic Hecke algebras, Adv. in Math., to appear; arXiv:0901.4450.
41. Jonathan Brundan, Alexander Kleshchev, and Irina Suprunenko, Semisimple restrictions from 𝐺𝐿(𝑛) to 𝐺𝐿(𝑛-1), J. Reine Angew. Math. 500 (1998), 83–112. MR 1637489
42. J. Brundan, A. Kleshchev and W. Wang, Graded Specht modules, arXiv:0901.0218.
43. Jonathan Brundan and Jonathan Kujawa, A new proof of the Mullineux conjecture, J. Algebraic Combin. 18 (2003), no. 1, 13–39. MR 2002217, https://doi.org/10.1023/A:1025113308552
44. I. V. Cherednik, A new interpretation of Gel′fand-Tzetlin bases, Duke Math. J. 54 (1987), no. 2, 563–577. MR 899405, https://doi.org/10.1215/S0012-7094-87-05423-8
45. Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132
46. Joseph Chuang, The derived categories of some blocks of symmetric groups and a conjecture of Broué, J. Algebra 217 (1999), no. 1, 114–155. MR 1700479, https://doi.org/10.1006/jabr.1998.7780
47. Joseph Chuang and Radha Kessar, Symmetric groups, wreath products, Morita equivalences, and Broué’s abelian defect group conjecture, Bull. London Math. Soc. 34 (2002), no. 2, 174–184. MR 1874244, https://doi.org/10.1112/S0024609301008839
48. Joseph Chuang and Raphaël Rouquier, Derived equivalences for symmetric groups and 𝔰𝔩₂-categorification, Ann. of Math. (2) 167 (2008), no. 1, 245–298. MR 2373155, https://doi.org/10.4007/annals.2008.167.245
49. Joseph Chuang and Kai Meng Tan, Filtrations in Rouquier blocks of symmetric groups and Schur algebras, Proc. London Math. Soc. (3) 86 (2003), no. 3, 685–706. MR 1974395, https://doi.org/10.1112/S0024611502013953
50. Joseph Chuang and Kai Meng Tan, Some canonical basis vectors in the basic 𝑈_{𝑞}(̂𝔰𝔩_{𝔫})-module, J. Algebra 248 (2002), no. 2, 765–779. MR 1882121, https://doi.org/10.1006/jabr.2001.9030
51. Edward Cline, Brian Parshall, and Leonard Scott, Stratifying endomorphism algebras, Mem. Amer. Math. Soc. 124 (1996), no. 591, viii+119. MR 1350891, https://doi.org/10.1090/memo/0591
52. P. Diaconis and C. Green, Applications of Murphy's elements, Stanford University Technical Report, (1989), no. 335.
53. Richard Dipper and Gordon James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 52 (1986), no. 1, 20–52. MR 812444, https://doi.org/10.1112/plms/s3-52.1.20
54. Richard Dipper and Gordon James, Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 54 (1987), no. 1, 57–82. MR 872250, https://doi.org/10.1112/plms/s3-54.1.57
55. Richard Dipper, Gordon James, and Andrew Mathas, Cyclotomic 𝑞-Schur algebras, Math. Z. 229 (1998), no. 3, 385–416. MR 1658581, https://doi.org/10.1007/PL00004665
56. Jie Du and Hebing Rui, Specht modules for Ariki-Koike algebras, Comm. Algebra 29 (2001), no. 10, 4701–4719. MR 1855120, https://doi.org/10.1081/AGB-100106782
57. Harald Ellers, Searching for more general weight conjectures, using the symmetric group as an example, J. Algebra 225 (2000), no. 2, 602–629. MR 1741554, https://doi.org/10.1006/jabr.1999.8127
58. Harald Ellers and John Murray, Branching rules for Specht modules, J. Algebra 307 (2007), no. 1, 278–286. MR 2278054, https://doi.org/10.1016/j.jalgebra.2006.07.032
59. Karin Erdmann and Gerhard O. Michler, Blocks for symmetric groups and their covering groups and quadratic forms, Beiträge Algebra Geom. 37 (1996), no. 1, 103–118. MR 1407809
60. Matthew Fayers, Weights of multipartitions and representations of Ariki-Koike algebras, Adv. Math. 206 (2006), no. 1, 112–144. MR 2261752, https://doi.org/10.1016/j.aim.2005.07.017
61. Matthew Fayers, An extension of James’s conjecture, Int. Math. Res. Not. IMRN 10 (2007), Art. ID rnm032, 24. MR 2344574, https://doi.org/10.1093/imrn/rnm032
62. Matthew Fayers, James’s Conjecture holds for weight four blocks of Iwahori-Hecke algebras, J. Algebra 317 (2007), no. 2, 593–633. MR 2362933, https://doi.org/10.1016/j.jalgebra.2007.08.006
63. Matthew Fayers, Decomposition numbers for weight three blocks of symmetric groups and Iwahori-Hecke algebras, Trans. Amer. Math. Soc. 360 (2008), no. 3, 1341–1376. MR 2357698, https://doi.org/10.1090/S0002-9947-07-04156-6
64. Matthew Fayers, Regularisation and the Mullineux map, Electron. J. Combin. 15 (2008), no. 1, Research Paper 142, 15. MR 2465766
65. M. Fayers, General runner removal and the Mullineux map, arXiv:0712.2390.
66. Matthew Fayers and Kai Meng Tan, The ordinary quiver of a weight three block of the symmetric group is bipartite, Adv. Math. 209 (2007), no. 1, 69–98. MR 2294218, https://doi.org/10.1016/j.aim.2006.04.007
67. Paul Fong and Morton E. Harris, On perfect isometries and isotypies in alternating groups, Trans. Amer. Math. Soc. 349 (1997), no. 9, 3469–3516. MR 1390981, https://doi.org/10.1090/S0002-9947-97-01793-5
68. Ben Ford and Alexander S. Kleshchev, A proof of the Mullineux conjecture, Math. Z. 226 (1997), no. 2, 267–308. MR 1477629, https://doi.org/10.1007/PL00004340
69. Andrew R. Francis and John J. Graham, Centres of Hecke algebras: the Dipper-James conjecture, J. Algebra 306 (2006), no. 1, 244–267. MR 2271582, https://doi.org/10.1016/j.jalgebra.2006.05.010
70. Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 232821
71. Wee Liang Gan and Victor Ginzburg, Quantization of Slodowy slices, Int. Math. Res. Not. 5 (2002), 243–255. MR 1876934, https://doi.org/10.1155/S107379280210609X
72. Meinolf Geck, Representations of Hecke algebras at roots of unity, Astérisque 252 (1998), Exp. No. 836, 3, 33–55 (English, with French summary). Séminaire Bourbaki. Vol. 1997/98. MR 1685620
73. Meinolf Geck, Kazhdan-Lusztig cells, 𝑞-Schur algebras and James’ conjecture, J. London Math. Soc. (2) 63 (2001), no. 2, 336–352. MR 1810133, https://doi.org/10.1017/S0024610700001873
74. Meinolf Geck and Jürgen Müller, James’ conjecture for Hecke algebras of exceptional type. I, J. Algebra 321 (2009), no. 11, 3274–3298. MR 2510049, https://doi.org/10.1016/j.jalgebra.2008.10.024
75. 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. MR 1722904, https://doi.org/10.1006/jabr.1999.7943
76. John Graham and Gordon James, On a conjecture of Gow and Kleshchev concerning tensor products, J. Algebra 227 (2000), no. 2, 767–782. MR 1759843, https://doi.org/10.1006/jabr.1999.8247
77. J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), no. 1, 1–34. MR 1376244, https://doi.org/10.1007/BF01232365
78. Jean-Baptiste Gramain, On defect groups for generalized blocks of the symmetric group, J. Lond. Math. Soc. (2) 78 (2008), no. 1, 155–171. MR 2427057, https://doi.org/10.1112/jlms/jdn023
79. Andrew Granville and Ken Ono, Defect zero 𝑝-blocks for finite simple groups, Trans. Amer. Math. Soc. 348 (1996), no. 1, 331–347. MR 1321575, https://doi.org/10.1090/S0002-9947-96-01481-X
80. I. Grojnowski, Representations of affine Hecke algebras (and affine quantum 𝐺𝐿_{𝑛}) at roots of unity, Internat. Math. Res. Notices 5 (1994), 215 ff., approx. 3 pp.}, issn=1073-7928, review=\MR{1270135}, doi=10.1155/S1073792894000243,.
81. I. Grojnowski, Affine $\mathfrak{sl}_p$ controls the representation theory of the symmetric group and related Hecke algebras, arXiv:math.RT/9907129.
82. I. Grojnowski and M. Vazirani, Strong multiplicity one theorems for affine Hecke algebras of type A, Transform. Groups 6 (2001), no. 2, 143–155. MR 1835669, https://doi.org/10.1007/BF01597133
83. Takahiro Hayashi, 𝑞-analogues of Clifford and Weyl algebras—spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys. 127 (1990), no. 1, 129–144. MR 1036118
84. David Hill, Elementary divisors of the Shapovalov form on the basic representation of Kac-Moody algebras, J. Algebra 319 (2008), no. 12, 5208–5246. MR 2423824, https://doi.org/10.1016/j.jalgebra.2007.11.009
85. P. Hoefsmit, Representations of Hecke Algebras of Finite Groups with $\operatorname{BN}$ -Pairs of Classical Type, Ph.D. thesis, University of British Columbia, 1974.
86. Jun Hu, Branching rules for Hecke algebras of type 𝐷_{𝑛}, Math. Nachr. 280 (2007), no. 1-2, 93–104. MR 2290385, https://doi.org/10.1002/mana.200410467
87. Jun Hu, Mullineux involution and twisted affine Lie algebras, J. Algebra 304 (2006), no. 1, 557–576. MR 2256406, https://doi.org/10.1016/j.jalgebra.2006.03.025
88. J. Hu and A. Mathas, Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A; arXiv:0907.2985.
89. V. G. Drinfel′d, Degenerate affine Hecke algebras and Yangians, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 69–70 (Russian). MR 831053
90. Nicolas Jacon and Cédric Lecouvey, On the Mullineux involution for Ariki-Koike algebras, J. Algebra 321 (2009), no. 8, 2156–2170. MR 2501515, https://doi.org/10.1016/j.jalgebra.2008.09.033
91. G. D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR 513828
92. Gordon James, The decomposition matrices of 𝐺𝐿_{𝑛}(𝑞) for 𝑛\le10, Proc. London Math. Soc. (3) 60 (1990), no. 2, 225–265. MR 1031453, https://doi.org/10.1112/plms/s3-60.2.225
93. Gordon James, Sinéad Lyle, and Andrew Mathas, Rouquier blocks, Math. Z. 252 (2006), no. 3, 511–531. MR 2207757, https://doi.org/10.1007/s00209-005-0863-0
94. Gordon James and Andrew Mathas, Symmetric group blocks of small defect, J. Algebra 279 (2004), no. 2, 566–612. MR 2078931, https://doi.org/10.1016/j.jalgebra.2004.04.006
95. Gordon James and Adrian Williams, Decomposition numbers of symmetric groups by induction, J. Algebra 228 (2000), no. 1, 119–142. MR 1760959, https://doi.org/10.1006/jabr.1999.8248
96. Michio Jimbo, Kailash C. Misra, Tetsuji Miwa, and Masato Okado, Combinatorics of representations of 𝑈_{𝑞}(̂𝔰𝔩(𝔫)) at 𝔮=0, Comm. Math. Phys. 136 (1991), no. 3, 543–566. MR 1099695
97. Naihuan Jing and Weiqiang Wang, Twisted vertex representations and spin characters, Math. Z. 239 (2002), no. 4, 715–746. MR 1902059, https://doi.org/10.1007/s002090100340
98. Andrew R. Jones, The structure of the Young symmetrizers for spin representations of the symmetric group. I, J. Algebra 205 (1998), no. 2, 626–660. MR 1632785, https://doi.org/10.1006/jabr.1997.7400
99. Andrew R. Jones, The structure of the Young symmetrizers for spin representations of the symmetric group. II, J. Algebra 213 (1999), no. 2, 381–404. MR 1673462, https://doi.org/10.1006/jabr.1998.7667
100. A. R. Jones and M. L. Nazarov, Affine Sergeev algebra and 𝑞-analogues of the Young symmetrizers for projective representations of the symmetric group, Proc. London Math. Soc. (3) 78 (1999), no. 3, 481–512. MR 1674836, https://doi.org/10.1112/S002461159900177X
101. Tadeusz Józefiak, Relating spin representations of symmetric and hyperoctahedral groups, J. Pure Appl. Algebra 152 (2000), no. 1-3, 187–193. Commutative algebra, homological algebra and representation theory (Catania/Genoa/Rome, 1998). MR 1783994, https://doi.org/10.1016/S0022-4049(99)00143-7
102. A. A. Jucis, On the Young operators of symmetric groups, Litovsk. Fiz. Sb. 6 (1966), 163–180 (Russian, with English and Lithuanian summaries). MR 202866
103. A. Jucis, Factorization of Young’s projection operators for symmetric groups, Litovsk. Fiz. Sb. 11 (1971), 1–10 (Russian, with English and Lithuanian summaries). MR 290671
104. A.-A. A. Jucys, Symmetric polynomials and the center of the symmetric group ring, Rep. Mathematical Phys. 5 (1974), no. 1, 107–112. MR 419576, https://doi.org/10.1016/0034-4877(74)90019-6
105. Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219
106. Masaki Kashiwara, Crystalizing the 𝑞-analogue of universal enveloping algebras, Comm. Math. Phys. 133 (1990), no. 2, 249–260. MR 1090425
107. M. Kashiwara, On crystal bases of the 𝑄-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. MR 1115118, https://doi.org/10.1215/S0012-7094-91-06321-0
108. Masaki Kashiwara, Global crystal bases of quantum groups, Duke Math. J. 69 (1993), no. 2, 455–485. MR 1203234, https://doi.org/10.1215/S0012-7094-93-06920-7
109. Masaki Kashiwara, On crystal bases, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 155–197. MR 1357199
110. David Kazhdan and George Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), no. 1, 153–215. MR 862716, https://doi.org/10.1007/BF01389157
111. Radha Kessar, Blocks and source algebras for the double covers of the symmetric and alternating groups, J. Algebra 186 (1996), no. 3, 872–933. MR 1424598, https://doi.org/10.1006/jabr.1996.0400
112. Radha Kessar and Mary Schaps, Crossover Morita equivalences for blocks of the covering groups of the symmetric and alternating groups, J. Group Theory 9 (2006), no. 6, 715–730. MR 2272713, https://doi.org/10.1515/JGT.2006.046
113. M. Khovanov and A. Lauda, A diagrammatic approach to categorification of quantum groups I, Represent. Theory 13 (2009), 309-347.
114. M. Khovanov and A. Lauda, A diagrammatic approach to categorification of quantum groups II; arXiv:0804.2080
115. M. Khovanov and A. Lauda, A diagrammatic approach to categorification of quantum groups III; arXiv:0807.3250.
116. 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. MR 1057341
117. 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. MR 1102819, https://doi.org/10.1016/0021-8693(91)90181-7
118. 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. MR 1289862, https://doi.org/10.1112/plms/s3-69.3.515
119. A. S. Kleshchev, Branching rules for modular representations of symmetric groups. I, J. Algebra 178 (1995), no. 2, 493–511. MR 1359899, https://doi.org/10.1006/jabr.1995.1362
120. A. S. Kleshchev, Branching rules for modular representations of symmetric groups. I, J. Algebra 178 (1995), no. 2, 493–511. MR 1359899, https://doi.org/10.1006/jabr.1995.1362
121. A. S. Kleshchev, Branching rules for modular representations of symmetric groups. I, J. Algebra 178 (1995), no. 2, 493–511. MR 1359899, https://doi.org/10.1006/jabr.1995.1362
122. Alexander Kleshchev, Completely splittable representations of symmetric groups, J. Algebra 181 (1996), no. 2, 584–592. MR 1383482, https://doi.org/10.1006/jabr.1996.0135
123. 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. MR 1466660, https://doi.org/10.1112/S0024611597000427
124. Alexander Kleshchev, Branching rules for modular representations of symmetric groups. IV, J. Algebra 201 (1998), no. 2, 547–572. MR 1612335, https://doi.org/10.1006/jabr.1997.7302
125. Alexander Kleshchev, Linear and projective representations of symmetric groups, Cambridge Tracts in Mathematics, vol. 163, Cambridge University Press, Cambridge, 2005. MR 2165457
126. A. Kleshchev and D. Nash, An interpretation of the LLT algorithm, preprint, University of Oregon, 2009.
127. A. Kleshchev and A. Ram, Homogeneous representations of Khovanov-Lauda algebras, J. European Math. Soc., to appear.
128. 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. MR 1785073, https://doi.org/10.1007/s002090000125
129. 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. MR 1863400, https://doi.org/10.1112/S002461150101320X
130. 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. MR 2031049, https://doi.org/10.1090/S0002-9947-03-03364-6
131. A. S. Kleshchev and A. E. Zalesski, Minimal polynomials of elements of order 𝑝 in 𝑝-modular projective representations of alternating groups, Proc. Amer. Math. Soc. 132 (2004), no. 6, 1605–1612. MR 2051120, https://doi.org/10.1090/S0002-9939-03-07242-3
132. Jonathan Kujawa, Crystal structures arising from representations of 𝐺𝐿(𝑚\vert𝑛), Represent. Theory 10 (2006), 49–85. MR 2209849, https://doi.org/10.1090/S1088-4165-06-00219-6
133. Burkhard Külshammer, Jørn B. Olsson, and Geoffrey R. Robinson, Generalized blocks for symmetric groups, Invent. Math. 151 (2003), no. 3, 513–552. MR 1961337, https://doi.org/10.1007/s00222-002-0258-3
134. M. Künzer and G. Nebe, Elementary divisors of Gram matrices of certain Specht modules, Comm. Algebra 31 (2003), no. 7, 3377–3427. MR 1990280, https://doi.org/10.1081/AGB-120022231
135. Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), no. 1, 205–263. MR 1410572
136. Bernard Leclerc, Dual canonical bases, quantum shuffles and 𝑞-characters, Math. Z. 246 (2004), no. 4, 691–732. MR 2045836, https://doi.org/10.1007/s00209-003-0609-9
137. Bernard Leclerc and Jean-Yves Thibon, Canonical bases of 𝑞-deformed Fock spaces, Internat. Math. Res. Notices 9 (1996), 447–456. MR 1399410, https://doi.org/10.1155/S1073792896000293
138. Bernard Leclerc and Jean-Yves Thibon, Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials, Combinatorial methods in representation theory (Kyoto, 1998) Adv. Stud. Pure Math., vol. 28, Kinokuniya, Tokyo, 2000, pp. 155–220. MR 1864481, https://doi.org/10.2969/aspm/02810155
139. Bernard Leclerc, Jean-Yves Thibon, and Eric Vasserot, Zelevinsky’s involution at roots of unity, J. Reine Angew. Math. 513 (1999), 33–51. MR 1713318, https://doi.org/10.1515/crll.1999.062
140. George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599–635. MR 991016, https://doi.org/10.1090/S0894-0347-1989-0991016-9
141. George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
142. Sinéad Lyle, Some reducible Specht modules, J. Algebra 269 (2003), no. 2, 536–543. MR 2015852, https://doi.org/10.1016/S0021-8693(03)00537-4
143. Sinéad Lyle and Andrew Mathas, Blocks of cyclotomic Hecke algebras, Adv. Math. 216 (2007), no. 2, 854–878. MR 2351381, https://doi.org/10.1016/j.aim.2007.06.008
144. T. E. Lynch, Generalized Whittaker vectors and representation theory, PhD thesis, M.I.T., 1979.
145. Gunter Malle and Andrew Mathas, Symmetric cyclotomic Hecke algebras, J. Algebra 205 (1998), no. 1, 275–293. MR 1631350, https://doi.org/10.1006/jabr.1997.7339
146. Andrei Marcus, On equivalences between blocks of group algebras: reduction to the simple components, J. Algebra 184 (1996), no. 2, 372–396. MR 1409219, https://doi.org/10.1006/jabr.1996.0265
147. Andrei Marcus, Broué’s abelian defect group conjecture for alternating groups, Proc. Amer. Math. Soc. 132 (2004), no. 1, 7–14. MR 2021243, https://doi.org/10.1090/S0002-9939-03-07214-9
148. Stuart Martin and Kai Meng Tan, Defect 3 blocks of symmetric group algebras. I, J. Algebra 237 (2001), no. 1, 95–120. MR 1813902, https://doi.org/10.1006/jabr.2000.8564
149. Stuart Martin and Kai Meng Tan, [3:2]-pairs of symmetric group algebras and their intermediate defect 4 blocks, J. Algebra 288 (2005), no. 2, 505–526. MR 2146142, https://doi.org/10.1016/j.jalgebra.2005.01.036
150. Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, vol. 15, American Mathematical Society, Providence, RI, 1999. MR 1711316
151. Andrew Mathas, Seminormal forms and Gram determinants for cellular algebras, J. Reine Angew. Math. 619 (2008), 141–173. With an appendix by Marcos Soriano. MR 2414949, https://doi.org/10.1515/CRELLE.2008.042
152. Kailash Misra and Tetsuji Miwa, Crystal base for the basic representation of 𝑈_{𝑞}(𝔰𝔩(𝔫)), Comm. Math. Phys. 134 (1990), no. 1, 79–88. MR 1079801
153. Jürgen Müller, Brauer trees for the Schur cover of the symmetric group, J. Algebra 266 (2003), no. 2, 427–445. MR 1995123, https://doi.org/10.1016/S0021-8693(03)00342-9
154. G. Mullineux, Bijections of 𝑝-regular partitions and 𝑝-modular irreducibles of the symmetric groups, J. London Math. Soc. (2) 20 (1979), no. 1, 60–66. MR 545202, https://doi.org/10.1112/jlms/s2-20.1.60
155. G. E. Murphy, A new construction of Young’s seminormal representation of the symmetric groups, J. Algebra 69 (1981), no. 2, 287–297. MR 617079, https://doi.org/10.1016/0021-8693(81)90205-2
156. G. E. Murphy, The idempotents of the symmetric group and Nakayama’s conjecture, J. Algebra 81 (1983), no. 1, 258–265. MR 696137, https://doi.org/10.1016/0021-8693(83)90219-3
157. G. E. Murphy, The representations of Hecke algebras of type 𝐴_{𝑛}, J. Algebra 173 (1995), no. 1, 97–121. MR 1327362, https://doi.org/10.1006/jabr.1995.1079
158. Constantin Năstăsescu and Freddy Van Oystaeyen, Methods of graded rings, Lecture Notes in Mathematics, vol. 1836, Springer-Verlag, Berlin, 2004. MR 2046303
159. M. L. Nazarov, Young’s orthogonal form of irreducible projective representations of the symmetric group, J. London Math. Soc. (2) 42 (1990), no. 3, 437–451. MR 1087219, https://doi.org/10.1112/jlms/s2-42.3.437
160. Maxim Nazarov, Young’s symmetrizers for projective representations of the symmetric group, Adv. Math. 127 (1997), no. 2, 190–257. MR 1448714, https://doi.org/10.1006/aima.1997.1621
161. Andrei Okounkov and Anatoly Vershik, A new approach to representation theory of symmetric groups, Selecta Math. (N.S.) 2 (1996), no. 4, 581–605. MR 1443185, https://doi.org/10.1007/PL00001384
162. Rowena Paget, Induction and decomposition numbers for RoCK blocks, Q. J. Math. 56 (2005), no. 2, 251–262. MR 2143501, https://doi.org/10.1093/qmath/hah028
163. Rowena Paget, The Mullineux map for RoCK blocks, Comm. Algebra 34 (2006), no. 9, 3245–3253. MR 2252669, https://doi.org/10.1080/00927870600778498
164. 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. MR 2107009, https://doi.org/10.1112/S0024611504014893
165. Alexander Premet, Special transverse slices and their enveloping algebras, Adv. Math. 170 (2002), no. 1, 1–55. With an appendix by Serge Skryabin. MR 1929302, https://doi.org/10.1006/aima.2001.2063
166. Lluís Puig, The Nakayama conjecture and the Brauer pairs, Séminaire sur les groupes finis, Tome III, Publ. Math. Univ. Paris VII, vol. 25, Univ. Paris VII, Paris, 1986, pp. ii, 171–189 (English, with French summary). MR 907478
167. Lluís Puig, On Joanna Scopes’ criterion of equivalence for blocks of symmetric groups, Algebra Colloq. 1 (1994), no. 1, 25–55. MR 1262662
168. Arun Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Proc. London Math. Soc. (3) 75 (1997), no. 1, 99–133. MR 1444315, https://doi.org/10.1112/S0024611597000282
169. Jeremy Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436–456. MR 1002456, https://doi.org/10.1112/jlms/s2-39.3.436
170. Jeremy Rickard, Derived equivalences as derived functors, J. London Math. Soc. (2) 43 (1991), no. 1, 37–48. MR 1099084, https://doi.org/10.1112/jlms/s2-43.1.37
171. Jeremy Rickard, Equivalences of derived categories for symmetric algebras, J. Algebra 257 (2002), no. 2, 460–481. MR 1947972, https://doi.org/10.1016/S0021-8693(02)00520-3
172. Raphaël Rouquier, Isométries parfaites dans les blocs à défaut abélien des groupes symétriques et sporadiques, J. Algebra 168 (1994), no. 2, 648–694 (French). MR 1292784, https://doi.org/10.1006/jabr.1994.1248
173. Raphaël Rouquier, Derived equivalences and finite dimensional algebras, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 191–221. MR 2275594
174. R. Rouquier, -Kac-Moody algebras, arXiv:0812.5023.
175. Jan Saxl, The complex characters of the symmetric groups that remain irreducible in subgroups, J. Algebra 111 (1987), no. 1, 210–219. MR 913206, https://doi.org/10.1016/0021-8693(87)90251-1
176. Vladimir Shchigolev, Generalization of modular lowering operators for 𝐺𝐿_{𝑛}, Comm. Algebra 36 (2008), no. 4, 1250–1288. MR 2406583, https://doi.org/10.1080/00927870701863389
177. V. V. Shchigolev, On extensions and branching rules of modules that are close to completely splittable, Mat. Sb. 196 (2005), no. 8, 119–160 (Russian, with Russian summary); English transl., Sb. Math. 196 (2005), no. 7-8, 1209–1249. MR 2188366, https://doi.org/10.1070/SM2005v196n08ABEH002338
178. Issai Schur, Gesammelte Abhandlungen. Band I, Springer-Verlag, Berlin-New York, 1973 (German). Herausgegeben von Alfred Brauer und Hans Rohrbach. MR 0462891
179. Joanna Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric groups, J. Algebra 142 (1991), no. 2, 441–455. MR 1127075, https://doi.org/10.1016/0021-8693(91)90319-4
180. Leonard L. Scott, Representations in characteristic 𝑝, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979) Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, pp. 319–331. MR 604599
181. A. N. Sergeev, Tensor algebra of the identity representation as a module over the Lie superalgebras 𝐺𝑙(𝑛,𝑚) and 𝑄(𝑛), Mat. Sb. (N.S.) 123(165) (1984), no. 3, 422–430 (Russian). MR 735715
182. Alexander Sergeev, The Howe duality and the projective representations of symmetric groups, Represent. Theory 3 (1999), 416–434. MR 1722115, https://doi.org/10.1090/S1088-4165-99-00085-0
183. Bin Shu and Weiqiang Wang, Modular representations of the ortho-symplectic supergroups, Proc. Lond. Math. Soc. (3) 96 (2008), no. 1, 251–271. MR 2392322, https://doi.org/10.1112/plms/pdm040
184. Eugene Stern, Semi-infinite wedges and vertex operators, Internat. Math. Res. Notices 4 (1995), 201–219. MR 1326065, https://doi.org/10.1155/S107379289500016X
185. Wolfgang Soergel, Kategorie 𝒪, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), no. 2, 421–445 (German, with English summary). MR 1029692, https://doi.org/10.1090/S0894-0347-1990-1029692-5
186. Kai Meng Tan, Martin’s conjecture holds for weight 3 blocks of symmetric groups, J. Algebra 320 (2008), no. 3, 1115–1132. MR 2427632, https://doi.org/10.1016/j.jalgebra.2008.04.024
187. Shunsuke Tsuchioka, A modular branching rule for the generalized symmetric groups, J. Algebra 316 (2007), no. 1, 459–470. MR 2354872, https://doi.org/10.1016/j.jalgebra.2006.12.007
188. W. Turner, Rock blocks, to appear in Mem. Amer. Math. Soc.; arXiv:0710.5462.
189. Denis Uglov, Canonical bases of higher-level 𝑞-deformed Fock spaces and Kazhdan-Lusztig polynomials, Physical combinatorics (Kyoto, 1999) Progr. Math., vol. 191, Birkhäuser Boston, Boston, MA, 2000, pp. 249–299. MR 1768086
190. Michela Varagnolo and Eric Vasserot, On the decomposition matrices of the quantized Schur algebra, Duke Math. J. 100 (1999), no. 2, 267–297. MR 1722955, https://doi.org/10.1215/S0012-7094-99-10010-X
191. M. Varagnolo and E. Vasserot, Canonical bases and Khovanov-Lauda algebras, arXiv:0901.3992.
192. M. Vazirani, Parameterizing Hecke algebra modules: Bernstein-Zelevinsky multisegments, Kleshchev multipartitions, and crystal graphs, Transform. Groups 7 (2002), no. 3, 267–303. MR 1923974, https://doi.org/10.1007/s00031-002-0014-1
193. Jinkui Wan and Weiqiang Wang, Modular representations and branching rules for wreath Hecke algebras, Int. Math. Res. Not. IMRN , posted on (2008), Art. ID rnn128, 31. MR 2449051, https://doi.org/10.1093/imrn/rnn128
194. Weiqiang Wang, Spin Hecke algebras of finite and affine types, Adv. Math. 212 (2007), no. 2, 723–748. MR 2329318, https://doi.org/10.1016/j.aim.2006.11.007
195. Hans Wenzl, Hecke algebras of type 𝐴_{𝑛} and subfactors, Invent. Math. 92 (1988), no. 2, 349–383. MR 936086, https://doi.org/10.1007/BF01404457
196. Adrian Williams, Symmetric group decomposition numbers for some three-part partitions, Comm. Algebra 34 (2006), no. 5, 1599–1613. MR 2229479, https://doi.org/10.1080/00927870500542523
197. Maozhi Xu, On 𝑝-series and the Mullineux conjecture, Comm. Algebra 27 (1999), no. 11, 5255–5265. MR 1713034, https://doi.org/10.1080/00927879908826755
198. Manabu Yamaguchi, A duality of a twisted group algebra of the hyperoctahedral group and the queer Lie superalgebra, Combinatorial methods in representation theory (Kyoto, 1998) Adv. Stud. Pure Math., vol. 28, Kinokuniya, Tokyo, 2000, pp. 401–422. MR 1864491, https://doi.org/10.2969/aspm/02810401
199. Manabu Yamaguchi, A duality of the twisted group algebra of the symmetric group and a Lie superalgebra, J. Algebra 222 (1999), no. 1, 301–327. MR 1728157, https://doi.org/10.1006/jabr.1999.8049
200. Xavier Yvonne, Canonical bases of higher-level 𝑞-deformed Fock spaces, J. Algebraic Combin. 26 (2007), no. 3, 383–414. MR 2348103, https://doi.org/10.1007/s10801-007-0062-7