RSK bases and Kazhdan-Lusztig cells
Annales de l'Institut Fourier, Volume 62 (2012) no. 2, pp. 525-569.

From the combinatorial characterizations of the right, left, and two-sided Kazhdan-Lusztig cells of the symmetric group, “ RSK bases” are constructed for certain quotients by two-sided ideals of the group ring and the Hecke algebra. Applications to invariant theory, over various base rings, of the general linear group and representation theory, both ordinary and modular, of the symmetric group are discussed.

À partir des caractérisations combinatoires des cellules de Kazhdan-Lusztig du groupe symétrique, on construit des bases “RSK” pour certains quotients du l’algèbre du groupe et de l’algèbre de Hecke. On étudie des applications à la théorie des invariants du groupe linéaire général sur divers anneaux de base et à la théorie des réprésentations, soit ordinaire ou modulaire, du groupe symétrique.

DOI: 10.5802/aif.2687
Classification: 05E10,  05E15,  20C08,  20C30
Keywords: Symmetric group, Hecke algebra, Kazhdan-Lusztig basis, RSK correspondence
Raghavan, K. N. 1; Samuel, Preena 1; Subrahmanyam, K. V. 2

1 Institute of Mathematical Sciences C. I. T. Campus Chennai 600 113 (India)
2 Chennai Mathematical Institute Plot No. H1, SIPCOT IT Park Padur Post, Siruseri 603 103 Tamilnadu (India)
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Raghavan, K. N.; Samuel, Preena; Subrahmanyam, K. V. RSK bases and Kazhdan-Lusztig cells. Annales de l'Institut Fourier, Volume 62 (2012) no. 2, pp. 525-569. doi : 10.5802/aif.2687. http://archive.numdam.org/articles/10.5802/aif.2687/

[1] Ariki, Susumu Robinson-Schensted correspondence and left cells, Combinatorial methods in representation theory (Kyoto, 1998) (Adv. Stud. Pure Math.), Volume 28, Kinokuniya, Tokyo, 2000, pp. 1-20 | MR

[2] Bourbaki, N. Éléments de mathématique, Fasc. XXIII, Hermann, Paris, 1973 (Livre II: Algèbre. Chapitre 8: Modules et anneaux semi-simples, Nouveau tirage de l’édition de 1958, Actualités Scientifiques et Industrielles, No. 1261) | MR | Zbl

[3] Bourbaki, Nicolas Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002 (Translated from the 1968 French original by Andrew Pressley) | MR | Zbl

[4] de Concini, C.; Procesi, C. A characteristic free approach to invariant theory, Advances in Math., Volume 21 (1976) no. 3, pp. 330-354 | DOI | MR | Zbl

[5] Datt, Sumanth; Kodiyalam, Vijay; Sunder, V. S. Complete invariants for complex semisimple Hopf algebras, Math. Res. Lett., Volume 10 (2003) no. 5-6, pp. 571-586 | MR

[6] Dipper, Richard; James, Gordon Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. (3), Volume 52 (1986) no. 1, pp. 20-52 | DOI | MR | Zbl

[7] Dipper, Richard; James, Gordon Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc. (3), Volume 54 (1987) no. 1, pp. 57-82 | DOI | MR | Zbl

[8] Dipper, Richard; James, Gordon The q-Schur algebra, Proc. London Math. Soc. (3), Volume 59 (1989) no. 1, pp. 23-50 | DOI | MR | Zbl

[9] Doty, Stephen; Nyman, Kathryn Annihilators of permutation modules, Quart. J. Math., Volume 00 (2009), pp. 1-16 (http://qjmath.oxfordjournals.org/content/early/2009/06/04/qmath.hap020.abstract)

[10] Du, J.; Parshall, B.; Scott, L. Cells and q-Schur algebras, Transform. Groups, Volume 3 (1998) no. 1, pp. 33-49 | DOI | MR | Zbl

[11] Fayers, Matthew; Lyle, Sinéad Some reducible Specht modules for Iwahori-Hecke algebras of type A with q=-1 (to appear)

[12] Frame, J. S.; Robinson, G. de B.; Thrall, R. M. The hook graphs of the symmetric groups, Canadian J. Math., Volume 6 (1954), pp. 316-324 | DOI | MR | Zbl

[13] Fulton, William Young tableaux, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997 (With applications to representation theory and geometry) | MR | Zbl

[14] Garsia, A. M.; McLarnan, T. J. Relations between Young’s natural and the Kazhdan-Lusztig representations of S n , Adv. in Math., Volume 69 (1988) no. 1, pp. 32-92 | DOI | MR | Zbl

[15] Geck, Meinolf Kazhdan-Lusztig cells and the Murphy basis, Proc. London Math. Soc. (3), Volume 93 (2006) no. 3, pp. 635-665 | DOI | MR

[16] Geck, Meinolf; Pfeiffer, Götz Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, 21, The Clarendon Press Oxford University Press, New York, 2000 | MR

[17] Graham, J. J.; Lehrer, G. I. Cellular algebras, Invent. Math., Volume 123 (1996) no. 1, pp. 1-34 (http://dx.doi.org/10.1007/BF01232365) | DOI | MR | Zbl

[18] James, G. D. On a conjecture of Carter concerning irreducible Specht modules, Math. Proc. Cambridge Philos. Soc., Volume 83 (1978) no. 1, pp. 11-17 | DOI | MR | Zbl

[19] James, Gordon; Mathas, Andrew A q-analogue of the Jantzen-Schaper theorem, Proc. London Math. Soc. (3), Volume 74 (1997) no. 2, pp. 241-274 | DOI | MR | Zbl

[20] Kazhdan, David; Lusztig, George Representations of Coxeter groups and Hecke algebras, Invent. Math., Volume 53 (1979) no. 2, pp. 165-184 | DOI | MR | Zbl

[21] Lusztig, G. Hecke algebras with unequal parameters, CRM Monograph Series, 18, American Mathematical Society, Providence, RI, 2003 | MR

[22] Lusztig, George On a theorem of Benson and Curtis, J. Algebra, Volume 71 (1981) no. 2, pp. 490-498 | DOI | MR | Zbl

[23] Mathas, Andrew Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, 15, American Mathematical Society, Providence, RI, 1999 | MR | Zbl

[24] McDonough, T. P.; Pallikaros, C. A. On relations between the classical and the Kazhdan-Lusztig representations of symmetric groups and associated Hecke algebras, J. Pure Appl. Algebra, Volume 203 (2005) no. 1-3, pp. 133-144 | DOI | MR

[25] Murphy, G. E. The representations of Hecke algebras of type A n , J. Algebra, Volume 173 (1995) no. 1, pp. 97-121 | DOI | MR | Zbl

[26] Naruse, Hiroshi On an isomorphism between Specht module and left cell of 𝔖 n , Tokyo J. Math., Volume 12 (1989) no. 2, pp. 247-267 | DOI | MR | Zbl

[27] Procesi, C. The invariant theory of n×n matrices, Advances in Math., Volume 19 (1976) no. 3, pp. 306-381 | DOI | MR | Zbl

[28] Raghavan, K N; Samuel, Preena; Subrahmanyam, K V KRS bases for rings of invariants and for endomorphism spaces of irreducible modules (2009) (http://arxiv.org/abs/0902.2842v1)

[29] Razmyslov, Ju. P. Identities with trace in full matrix algebras over a field of characteristic zero, Izv. Akad. Nauk SSSR Ser. Mat., Volume 38 (1974), pp. 723-756 | MR | Zbl

[30] Sagan, Bruce E. The symmetric group, Graduate Texts in Mathematics, 203, Springer-Verlag, New York, 2001 (Representations, combinatorial algorithms, and symmetric functions) | MR

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