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
HTML articles powered by AMS MathViewer

by Alexander Kleshchev PDF

Bull. Amer. Math. Soc. 47 (2010), 419-481 Request permission

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 ${\mathcal O}$, $W$-algebras, etc.

References

H. H. Andersen, J. C. Jantzen, and W. Soergel, Representations of quantum groups at a $p$th 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
Tomoyuki Arakawa and Takeshi Suzuki, Duality between $\mathfrak {s}\mathfrak {l}_n(\textbf {C})$ and the degenerate affine Hecke algebra, J. Algebra 209 (1998), no. 1, 288–304. MR 1652134, DOI 10.1006/jabr.1998.7530
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, DOI 10.1215/kjm/1250518452
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
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, DOI 10.1090/ulect/026
Susumu Ariki, Proof of the modular branching rule for cyclotomic Hecke algebras, J. Algebra 306 (2006), no. 1, 290–300. MR 2271584, DOI 10.1016/j.jalgebra.2006.04.033
S. Ariki and N. Jacon, Dipper-James-Murphy’s conjecture for Hecke algebras of type $B$, arXiv:math/0703447.
S. Ariki, N. Jacon and C. Lecouvey, The modular branching rule for affine Hecke algebras of type A, arXiv:0808.3915
Susumu Ariki and Kazuhiko Koike, A Hecke algebra of $(\textbf {Z}/r\textbf {Z})\wr {\mathfrak {S}}_n$ and construction of its irreducible representations, Adv. Math. 106 (1994), no. 2, 216–243. MR 1279219, DOI 10.1006/aima.1994.1057
Susumu Ariki, Victor Kreiman, and Shunsuke Tsuchioka, On the tensor product of two basic representations of $U_v(\widehat {\mathfrak {sl}}_e)$, Adv. Math. 218 (2008), no. 1, 28–86. MR 2409408, DOI 10.1016/j.aim.2007.11.018
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, DOI 10.1007/s002090050489
Susumu Ariki, Andrew Mathas, and Hebing Rui, Cyclotomic Nazarov-Wenzl algebras, Nagoya Math. J. 182 (2006), 47–134. MR 2235339, DOI 10.1017/S0027763000026842
M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), no. 3, 469–514. MR 746539, DOI 10.1007/BF01388470
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, DOI 10.1090/S0002-9947-99-02003-6
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, DOI 10.1006/jabr.1999.7923
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, DOI 10.1006/jcta.1997.2838
C. Bessenrodt and J. B. Olsson, On residue symbols and the Mullineux conjecture, J. Algebraic Combin. 7 (1998), no. 3, 227–251. MR 1616083, DOI 10.1023/A:1008618621557
C. Bessenrodt and J. B. Olsson, Branching of modular representations of the alternating groups, J. Algebra 209 (1998), no. 1, 143–174. MR 1652118, DOI 10.1006/jabr.1998.7505
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, DOI 10.1017/S0305004199003424
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, DOI 10.2140/pjm.1999.190.201
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, DOI 10.1112/S0024609300007098
Christine Bessenrodt and Alexander S. Kleshchev, Irreducible tensor products over alternating groups, J. Algebra 228 (2000), no. 2, 536–550. MR 1764578, DOI 10.1006/jabr.2000.8284
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, DOI 10.2140/pjm.2001.198.295
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, DOI 10.1090/S0894-0347-96-00192-0
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, DOI 10.1007/978-94-017-1556-0_{1}
Michel Broué and Gunter Malle, Zyklotomische Heckealgebren, Astérisque 212 (1993), 119–189 (German). Représentations unipotentes génériques et blocs des groupes réductifs finis. MR 1235834
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, DOI 10.1112/S0024611598000562
Jonathan Brundan, Centers of degenerate cyclotomic Hecke algebras and parabolic category $\scr O$, Represent. Theory 12 (2008), 236–259. MR 2424964, DOI 10.1090/S1088-4165-08-00333-6
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, DOI 10.1112/S0024611500012132
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, DOI 10.1515/crll.2001.002
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. MR 1870595, DOI 10.1090/S1088-4165-01-00123-6
Jonathan Brundan and Alexander Kleshchev, Projective representations of symmetric groups via Sergeev duality, Math. Z. 239 (2002), no. 1, 27–68. MR 1879328, DOI 10.1007/s002090100282
Jonathan Brundan and Alexander Kleshchev, Cartan determinants and Shapovalov forms, Math. Ann. 324 (2002), no. 3, 431–449. MR 1938453, DOI 10.1007/s00208-002-0346-0
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, DOI 10.1142/9789812564481_{0}003
Jonathan Brundan and Alexander Kleshchev, James’ regularization theorem for double covers of symmetric groups, J. Algebra 306 (2006), no. 1, 128–137. MR 2271575, DOI 10.1016/j.jalgebra.2006.01.055
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, DOI 10.1090/memo/0918
Jonathan Brundan and Alexander Kleshchev, Schur-Weyl duality for higher levels, Selecta Math. (N.S.) 14 (2008), no. 1, 1–57. MR 2480709, DOI 10.1007/s00029-008-0059-7
J. Brundan and A. Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, Invent. Math., to appear; arXiv:0808.2032.
J. Brundan and A. Kleshchev, The degenerate analogue of Ariki’s categorification theorem, Math. Z., to appear; arXiv:0901.0057.
J. Brundan and A. Kleshchev, Graded decomposition numbers for cyclotomic Hecke algebras, Adv. in Math., to appear; arXiv:0901.4450.
Jonathan Brundan, Alexander Kleshchev, and Irina Suprunenko, Semisimple restrictions from $\textrm {GL}(n)$ to $\textrm {GL}(n-1)$, J. Reine Angew. Math. 500 (1998), 83–112. MR 1637489
J. Brundan, A. Kleshchev and W. Wang, Graded Specht modules, arXiv:0901.0218.
Jonathan Brundan and Jonathan Kujawa, A new proof of the Mullineux conjecture, J. Algebraic Combin. 18 (2003), no. 1, 13–39. MR 2002217, DOI 10.1023/A:1025113308552
I. V. Cherednik, A new interpretation of Gel′fand-Tzetlin bases, Duke Math. J. 54 (1987), no. 2, 563–577. MR 899405, DOI 10.1215/S0012-7094-87-05423-8
Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132
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, DOI 10.1006/jabr.1998.7780
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, DOI 10.1112/S0024609301008839
Joseph Chuang and Raphaël Rouquier, Derived equivalences for symmetric groups and $\mathfrak {sl}_2$-categorification, Ann. of Math. (2) 167 (2008), no. 1, 245–298. MR 2373155, DOI 10.4007/annals.2008.167.245
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, DOI 10.1112/S0024611502013953
Joseph Chuang and Kai Meng Tan, Some canonical basis vectors in the basic $U_q(\widehat {\mathfrak {s}\mathfrak {l}}_n)$-module, J. Algebra 248 (2002), no. 2, 765–779. MR 1882121, DOI 10.1006/jabr.2001.9030
Edward Cline, Brian Parshall, and Leonard Scott, Stratifying endomorphism algebras, Mem. Amer. Math. Soc. 124 (1996), no. 591, viii+119. MR 1350891, DOI 10.1090/memo/0591
P. Diaconis and C. Green, Applications of Murphy’s elements, Stanford University Technical Report, (1989), no. 335.
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, DOI 10.1112/plms/s3-52.1.20
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, DOI 10.1112/plms/s3-54.1.57
Richard Dipper, Gordon James, and Andrew Mathas, Cyclotomic $q$-Schur algebras, Math. Z. 229 (1998), no. 3, 385–416. MR 1658581, DOI 10.1007/PL00004665
Jie Du and Hebing Rui, Specht modules for Ariki-Koike algebras, Comm. Algebra 29 (2001), no. 10, 4701–4719. MR 1855120, DOI 10.1081/AGB-100106782
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, DOI 10.1006/jabr.1999.8127
Harald Ellers and John Murray, Branching rules for Specht modules, J. Algebra 307 (2007), no. 1, 278–286. MR 2278054, DOI 10.1016/j.jalgebra.2006.07.032
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
Matthew Fayers, Weights of multipartitions and representations of Ariki-Koike algebras, Adv. Math. 206 (2006), no. 1, 112–144. MR 2261752, DOI 10.1016/j.aim.2005.07.017
Matthew Fayers, An extension of James’s conjecture, Int. Math. Res. Not. IMRN 10 (2007), Art. ID rnm032, 24. MR 2344574, DOI 10.1093/imrn/rnm032
Matthew Fayers, James’s Conjecture holds for weight four blocks of Iwahori-Hecke algebras, J. Algebra 317 (2007), no. 2, 593–633. MR 2362933, DOI 10.1016/j.jalgebra.2007.08.006
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, DOI 10.1090/S0002-9947-07-04156-6
Matthew Fayers, Regularisation and the Mullineux map, Electron. J. Combin. 15 (2008), no. 1, Research Paper 142, 15. MR 2465766
M. Fayers, General runner removal and the Mullineux map, arXiv:0712.2390.
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, DOI 10.1016/j.aim.2006.04.007
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, DOI 10.1090/S0002-9947-97-01793-5
Ben Ford and Alexander S. Kleshchev, A proof of the Mullineux conjecture, Math. Z. 226 (1997), no. 2, 267–308. MR 1477629, DOI 10.1007/PL00004340
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, DOI 10.1016/j.jalgebra.2006.05.010
Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 232821
Wee Liang Gan and Victor Ginzburg, Quantization of Slodowy slices, Int. Math. Res. Not. 5 (2002), 243–255. MR 1876934, DOI 10.1155/S107379280210609X
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
Meinolf Geck, Kazhdan-Lusztig cells, $q$-Schur algebras and James’ conjecture, J. London Math. Soc. (2) 63 (2001), no. 2, 336–352. MR 1810133, DOI 10.1017/S0024610700001873
Meinolf Geck and Jürgen Müller, James’ conjecture for Hecke algebras of exceptional type. I, J. Algebra 321 (2009), no. 11, 3274–3298. MR 2510049, DOI 10.1016/j.jalgebra.2008.10.024
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, DOI 10.1006/jabr.1999.7943
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, DOI 10.1006/jabr.1999.8247
J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), no. 1, 1–34. MR 1376244, DOI 10.1007/BF01232365
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, DOI 10.1112/jlms/jdn023
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, DOI 10.1090/S0002-9947-96-01481-X
I. Grojnowski, Representations of affine Hecke algebras (and affine quantum $\textrm {GL}_n$) 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,
I. Grojnowski, Affine $\mathfrak {sl}_p$ controls the representation theory of the symmetric group and related Hecke algebras, arXiv:math.RT/9907129.
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, DOI 10.1007/BF01597133
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
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, DOI 10.1016/j.jalgebra.2007.11.009
P. Hoefsmit, Representations of Hecke Algebras of Finite Groups with $\operatorname {BN}$-Pairs of Classical Type, Ph.D. thesis, University of British Columbia, 1974.
Jun Hu, Branching rules for Hecke algebras of type $D_n$, Math. Nachr. 280 (2007), no. 1-2, 93–104. MR 2290385, DOI 10.1002/mana.200410467
Jun Hu, Mullineux involution and twisted affine Lie algebras, J. Algebra 304 (2006), no. 1, 557–576. MR 2256406, DOI 10.1016/j.jalgebra.2006.03.025
J. Hu and A. Mathas, Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A; arXiv:0907.2985.
V. G. Drinfel′d, Degenerate affine Hecke algebras and Yangians, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 69–70 (Russian). MR 831053
Nicolas Jacon and Cédric Lecouvey, On the Mullineux involution for Ariki-Koike algebras, J. Algebra 321 (2009), no. 8, 2156–2170. MR 2501515, DOI 10.1016/j.jalgebra.2008.09.033
G. D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR 513828
Gordon James, The decomposition matrices of $\textrm {GL}_n(q)$ for $n\le 10$, Proc. London Math. Soc. (3) 60 (1990), no. 2, 225–265. MR 1031453, DOI 10.1112/plms/s3-60.2.225
Gordon James, Sinéad Lyle, and Andrew Mathas, Rouquier blocks, Math. Z. 252 (2006), no. 3, 511–531. MR 2207757, DOI 10.1007/s00209-005-0863-0
Gordon James and Andrew Mathas, Symmetric group blocks of small defect, J. Algebra 279 (2004), no. 2, 566–612. MR 2078931, DOI 10.1016/j.jalgebra.2004.04.006
Gordon James and Adrian Williams, Decomposition numbers of symmetric groups by induction, J. Algebra 228 (2000), no. 1, 119–142. MR 1760959, DOI 10.1006/jabr.1999.8248
Michio Jimbo, Kailash C. Misra, Tetsuji Miwa, and Masato Okado, Combinatorics of representations of $U_q(\widehat {{\mathfrak {s}}{\mathfrak {l}}}(n))$ at $q=0$, Comm. Math. Phys. 136 (1991), no. 3, 543–566. MR 1099695
Naihuan Jing and Weiqiang Wang, Twisted vertex representations and spin characters, Math. Z. 239 (2002), no. 4, 715–746. MR 1902059, DOI 10.1007/s002090100340
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, DOI 10.1006/jabr.1997.7400
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, DOI 10.1006/jabr.1998.7667
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, DOI 10.1112/S002461159900177X
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, DOI 10.1016/S0022-4049(99)00143-7
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
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
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, DOI 10.1016/0034-4877(74)90019-6
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
Masaki Kashiwara, Crystalizing the $q$-analogue of universal enveloping algebras, Comm. Math. Phys. 133 (1990), no. 2, 249–260. MR 1090425
M. Kashiwara, On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. MR 1115118, DOI 10.1215/S0012-7094-91-06321-0
Masaki Kashiwara, Global crystal bases of quantum groups, Duke Math. J. 69 (1993), no. 2, 455–485. MR 1203234, DOI 10.1215/S0012-7094-93-06920-7
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
David Kazhdan and George Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), no. 1, 153–215. MR 862716, DOI 10.1007/BF01389157
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, DOI 10.1006/jabr.1996.0400
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, DOI 10.1515/JGT.2006.046
M. Khovanov and A. Lauda, A diagrammatic approach to categorification of quantum groups I, Represent. Theory 13 (2009), 309–347.
M. Khovanov and A. Lauda, A diagrammatic approach to categorification of quantum groups II; arXiv:0804.2080
M. Khovanov and A. Lauda, A diagrammatic approach to categorification of quantum groups III; arXiv:0807.3250.
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, DOI 10.1017/CBO9780511629235
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, DOI 10.1016/0021-8693(91)90181-7
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, DOI 10.1112/plms/s3-69.3.515
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 S. Kleshchev, Branching rules for modular representations of symmetric groups. II, J. Reine Angew. Math. 459 (1995), 163–212. MR 1319521, DOI 10.1515/crll.1995.459.163
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. MR 1395065, DOI 10.1112/jlms/54.1.25
Alexander Kleshchev, Completely splittable representations of symmetric groups, J. Algebra 181 (1996), no. 2, 584–592. MR 1383482, DOI 10.1006/jabr.1996.0135
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, DOI 10.1112/S0024611597000427
Alexander Kleshchev, Branching rules for modular representations of symmetric groups. IV, J. Algebra 201 (1998), no. 2, 547–572. MR 1612335, DOI 10.1006/jabr.1997.7302
Alexander Kleshchev, Linear and projective representations of symmetric groups, Cambridge Tracts in Mathematics, vol. 163, Cambridge University Press, Cambridge, 2005. MR 2165457, DOI 10.1017/CBO9780511542800
A. Kleshchev and D. Nash, An interpretation of the LLT algorithm, preprint, University of Oregon, 2009.
A. Kleshchev and A. Ram, Homogeneous representations of Khovanov-Lauda algebras, J. European Math. Soc., to appear.
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, DOI 10.1007/s002090000125
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, DOI 10.1112/S002461150101320X
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, DOI 10.1090/S0002-9947-03-03364-6
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, DOI 10.1090/S0002-9939-03-07242-3
Jonathan Kujawa, Crystal structures arising from representations of $\textrm {GL}(m\vert n)$, Represent. Theory 10 (2006), 49–85. MR 2209849, DOI 10.1090/S1088-4165-06-00219-6
Burkhard Külshammer, Jørn B. Olsson, and Geoffrey R. Robinson, Generalized blocks for symmetric groups, Invent. Math. 151 (2003), no. 3, 513–552. MR 1961337, DOI 10.1007/s00222-002-0258-3
M. Künzer and G. Nebe, Elementary divisors of Gram matrices of certain Specht modules, Comm. Algebra 31 (2003), no. 7, 3377–3427. MR 1990280, DOI 10.1081/AGB-120022231
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
Bernard Leclerc, Dual canonical bases, quantum shuffles and $q$-characters, Math. Z. 246 (2004), no. 4, 691–732. MR 2045836, DOI 10.1007/s00209-003-0609-9
Bernard Leclerc and Jean-Yves Thibon, Canonical bases of $q$-deformed Fock spaces, Internat. Math. Res. Notices 9 (1996), 447–456. MR 1399410, DOI 10.1155/S1073792896000293
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, DOI 10.2969/aspm/02810155
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, DOI 10.1515/crll.1999.062
George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599–635. MR 991016, DOI 10.1090/S0894-0347-1989-0991016-9
George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
Sinéad Lyle, Some reducible Specht modules, J. Algebra 269 (2003), no. 2, 536–543. MR 2015852, DOI 10.1016/S0021-8693(03)00537-4
Sinéad Lyle and Andrew Mathas, Blocks of cyclotomic Hecke algebras, Adv. Math. 216 (2007), no. 2, 854–878. MR 2351381, DOI 10.1016/j.aim.2007.06.008
T. E. Lynch, Generalized Whittaker vectors and representation theory, PhD thesis, M.I.T., 1979.
Gunter Malle and Andrew Mathas, Symmetric cyclotomic Hecke algebras, J. Algebra 205 (1998), no. 1, 275–293. MR 1631350, DOI 10.1006/jabr.1997.7339
Andrei Marcus, On equivalences between blocks of group algebras: reduction to the simple components, J. Algebra 184 (1996), no. 2, 372–396. MR 1409219, DOI 10.1006/jabr.1996.0265
Andrei Marcus, Broué’s abelian defect group conjecture for alternating groups, Proc. Amer. Math. Soc. 132 (2004), no. 1, 7–14. MR 2021243, DOI 10.1090/S0002-9939-03-07214-9
Stuart Martin and Kai Meng Tan, Defect 3 blocks of symmetric group algebras. I, J. Algebra 237 (2001), no. 1, 95–120. MR 1813902, DOI 10.1006/jabr.2000.8564
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, DOI 10.1016/j.jalgebra.2005.01.036
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, DOI 10.1090/ulect/015
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, DOI 10.1515/CRELLE.2008.042
Kailash Misra and Tetsuji Miwa, Crystal base for the basic representation of $U_q(\mathfrak {s}\mathfrak {l}(n))$, Comm. Math. Phys. 134 (1990), no. 1, 79–88. MR 1079801
Jürgen Müller, Brauer trees for the Schur cover of the symmetric group, J. Algebra 266 (2003), no. 2, 427–445. MR 1995123, DOI 10.1016/S0021-8693(03)00342-9
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, DOI 10.1112/jlms/s2-20.1.60
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, DOI 10.1016/0021-8693(81)90205-2
G. E. Murphy, The idempotents of the symmetric group and Nakayama’s conjecture, J. Algebra 81 (1983), no. 1, 258–265. MR 696137, DOI 10.1016/0021-8693(83)90219-3
G. E. Murphy, The representations of Hecke algebras of type $A_n$, J. Algebra 173 (1995), no. 1, 97–121. MR 1327362, DOI 10.1006/jabr.1995.1079
Constantin Năstăsescu and Freddy Van Oystaeyen, Methods of graded rings, Lecture Notes in Mathematics, vol. 1836, Springer-Verlag, Berlin, 2004. MR 2046303, DOI 10.1007/b94904
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, DOI 10.1112/jlms/s2-42.3.437
Maxim Nazarov, Young’s symmetrizers for projective representations of the symmetric group, Adv. Math. 127 (1997), no. 2, 190–257. MR 1448714, DOI 10.1006/aima.1997.1621
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, DOI 10.1007/PL00001384
Rowena Paget, Induction and decomposition numbers for RoCK blocks, Q. J. Math. 56 (2005), no. 2, 251–262. MR 2143501, DOI 10.1093/qmath/hah028
Rowena Paget, The Mullineux map for RoCK blocks, Comm. Algebra 34 (2006), no. 9, 3245–3253. MR 2252669, DOI 10.1080/00927870600778498
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, DOI 10.1112/S0024611504014893
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, DOI 10.1006/aima.2001.2063
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
Lluís Puig, On Joanna Scopes’ criterion of equivalence for blocks of symmetric groups, Algebra Colloq. 1 (1994), no. 1, 25–55. MR 1262662
Arun Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Proc. London Math. Soc. (3) 75 (1997), no. 1, 99–133. MR 1444315, DOI 10.1112/S0024611597000282
Jeremy Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436–456. MR 1002456, DOI 10.1112/jlms/s2-39.3.436
Jeremy Rickard, Derived equivalences as derived functors, J. London Math. Soc. (2) 43 (1991), no. 1, 37–48. MR 1099084, DOI 10.1112/jlms/s2-43.1.37
Jeremy Rickard, Equivalences of derived categories for symmetric algebras, J. Algebra 257 (2002), no. 2, 460–481. MR 1947972, DOI 10.1016/S0021-8693(02)00520-3
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, DOI 10.1006/jabr.1994.1248
Raphaël Rouquier, Derived equivalences and finite dimensional algebras, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 191–221. MR 2275594
R. Rouquier, $2$-Kac-Moody algebras, arXiv:0812.5023.
Jan Saxl, The complex characters of the symmetric groups that remain irreducible in subgroups, J. Algebra 111 (1987), no. 1, 210–219. MR 913206, DOI 10.1016/0021-8693(87)90251-1
Vladimir Shchigolev, Generalization of modular lowering operators for $\textrm {GL}_n$, Comm. Algebra 36 (2008), no. 4, 1250–1288. MR 2406583, DOI 10.1080/00927870701863389
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, DOI 10.1070/SM2005v196n08ABEH002338
Issai Schur, Gesammelte Abhandlungen. Band I, Springer-Verlag, Berlin-New York, 1973 (German). Herausgegeben von Alfred Brauer und Hans Rohrbach. MR 0462891
Joanna Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric groups, J. Algebra 142 (1991), no. 2, 441–455. MR 1127075, DOI 10.1016/0021-8693(91)90319-4
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
A. N. Sergeev, Tensor algebra of the identity representation as a module over the Lie superalgebras $\textrm {Gl}(n,\,m)$ and $Q(n)$, Mat. Sb. (N.S.) 123(165) (1984), no. 3, 422–430 (Russian). MR 735715
Alexander Sergeev, The Howe duality and the projective representations of symmetric groups, Represent. Theory 3 (1999), 416–434. MR 1722115, DOI 10.1090/S1088-4165-99-00085-0
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, DOI 10.1112/plms/pdm040
Eugene Stern, Semi-infinite wedges and vertex operators, Internat. Math. Res. Notices 4 (1995), 201–219. MR 1326065, DOI 10.1155/S107379289500016X
Wolfgang Soergel, Kategorie $\scr O$, 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, DOI 10.1090/S0894-0347-1990-1029692-5
Kai Meng Tan, Martin’s conjecture holds for weight 3 blocks of symmetric groups, J. Algebra 320 (2008), no. 3, 1115–1132. MR 2427632, DOI 10.1016/j.jalgebra.2008.04.024
Shunsuke Tsuchioka, A modular branching rule for the generalized symmetric groups, J. Algebra 316 (2007), no. 1, 459–470. MR 2354872, DOI 10.1016/j.jalgebra.2006.12.007
W. Turner, Rock blocks, to appear in Mem. Amer. Math. Soc.; arXiv:0710.5462.
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
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, DOI 10.1215/S0012-7094-99-10010-X
M. Varagnolo and E. Vasserot, Canonical bases and Khovanov-Lauda algebras, arXiv:0901.3992.
M. Vazirani, Parameterizing Hecke algebra modules: Bernstein-Zelevinsky multisegments, Kleshchev multipartitions, and crystal graphs, Transform. Groups 7 (2002), no. 3, 267–303. MR 1923974, DOI 10.1007/s00031-002-0014-1
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, DOI 10.1093/imrn/rnn128
Weiqiang Wang, Spin Hecke algebras of finite and affine types, Adv. Math. 212 (2007), no. 2, 723–748. MR 2329318, DOI 10.1016/j.aim.2006.11.007
Hans Wenzl, Hecke algebras of type $A_n$ and subfactors, Invent. Math. 92 (1988), no. 2, 349–383. MR 936086, DOI 10.1007/BF01404457
Adrian Williams, Symmetric group decomposition numbers for some three-part partitions, Comm. Algebra 34 (2006), no. 5, 1599–1613. MR 2229479, DOI 10.1080/00927870500542523
Maozhi Xu, On $p$-series and the Mullineux conjecture, Comm. Algebra 27 (1999), no. 11, 5255–5265. MR 1713034, DOI 10.1080/00927879908826755
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, DOI 10.2969/aspm/02810401
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, DOI 10.1006/jabr.1999.8049
Xavier Yvonne, Canonical bases of higher-level $q$-deformed Fock spaces, J. Algebraic Combin. 26 (2007), no. 3, 383–414. MR 2348103, DOI 10.1007/s10801-007-0062-7