ALGEBRAIC COMBINATORICS

no. 6
Vogan classes in type B n
Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1033-1057.

Kazhdan and Lusztig have shown how to partition a Coxeter group into cells. In this paper, we use the theory of Vogan classes to obtain a first characterisation of the left cells of type B n with respect to a certain choice of weight function.

Received: 2018-04-18
Revised: 2019-03-06
Accepted: 2019-03-14
Published online: 2019-12-04
DOI: https://doi.org/10.5802/alco.74
Keywords: Coxeter groups, Iwahori–Hecke algebras, Kazhdan–Lusztig cells 
@article{ALCO_2019__2_6_1033_0,
     author = {Howse, Edmund},
     title = {Vogan classes in type $B\_n$},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {2},
     number = {6},
     year = {2019},
     pages = {1033-1057},
     doi = {10.5802/alco.74},
     mrnumber = {4049837},
     zbl = {1428.05324},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2019__2_6_1033_0/}
}
Howse, Edmund. Vogan classes in type $B_n$. Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1033-1057. doi : 10.5802/alco.74. https://alco.centre-mersenne.org/item/ALCO_2019__2_6_1033_0/

[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 1855588 | Zbl 0986.05097

[2] Bonnafé, Cédric Two-sided cells in type B (asymptotic case), J. Algebra, Volume 304 (2006) no. 1, pp. 216-236 | Article | MR 2255826 | Zbl 1150.20004

[3] Bonnafé, Cédric Semicontinuity properties of Kazhdan–Lusztig cells, New Zealand J. Math., Volume 39 (2009), pp. 171-192 | MR 2772408 | Zbl 1231.20004

[4] Bonnafé, Cédric On Kazhdan–Lusztig cells in type B, J. Algebraic Combin., Volume 31 (2010) no. 1, pp. 53-82 | Article | MR 2577478 | Zbl 1198.20034

[5] Bonnafé, Cédric Erratum to: On Kazhdan–Lusztig cells in type B, J. Algebraic Combin., Volume 35 (2012) no. 3, pp. 515-517 | Article | MR 2892987 | Zbl 1376.20039

[6] Bonnafé, Cédric; Geck, Meinolf Hecke algebras with unequal parameters and Vogan’s left cell invariants, Representations of reductive groups (Prog. Math.) Volume 312, Birkhäuser/Springer, Cham, 2015, pp. 173-187 | Article | MR 3495796 | Zbl 1343.20004

[7] Bonnafé, Cédric; Geck, Meinolf; Iancu, Lacrimioara; Lam, Thomas On domino insertion and Kazhdan–Lusztig cells in type B n , Representation theory of algebraic groups and quantum groups (Progr. Math.) Volume 284, Birkhäuser/Springer, New York, 2010, pp. 33-54 | Article | MR 2761947 | Zbl 1223.20003

[8] Bonnafé, Cédric; Iancu, Lacrimioara Left cells in type B n with unequal parameters, Represent. Theory, Volume 7 (2003), pp. 587-609 | Article | MR 2017068 | Zbl 1070.20004

[9] Garfinkle, Devra On the classification of primitive ideals for complex classical Lie algebras. I, Compositio Math., Volume 75 (1990) no. 2, pp. 135-169 | Numdam | MR 1065203 | Zbl 0737.17003

[10] Garfinkle, Devra On the classification of primitive ideals for complex classical Lie algebra. II, Compositio Math., Volume 81 (1992) no. 3, pp. 307-336 | MR 1149172 | Zbl 0762.17007

[11] Garfinkle, Devra On the classification of primitive ideals for complex classical Lie algebras. III, Compositio Math., Volume 88 (1993) no. 2, pp. 187-234 | Numdam | MR 1237920 | Zbl 0798.17007

[12] Geck, Meinolf On the induction of Kazhdan–Lusztig cells, Bull. London Math. Soc., Volume 35 (2003) no. 5, pp. 608-614 | Article | MR 1989489 | Zbl 1045.20004

[13] Geck, Meinolf Computing Kazhdan–Lusztig cells for unequal parameters, J. Algebra, Volume 281 (2004) no. 1, pp. 342-365 | Article | MR 2091976 | Zbl 1064.20006

[14] Geck, Meinolf Relative Kazhdan–Lusztig cells, Represent. Theory, Volume 10 (2006), p. 481-524) | Article | MR 2266700 | Zbl 1139.20003

[15] Geck, Meinolf PyCox: computing with (finite) Coxeter groups and Iwahori–Hecke algebras, LMS J. Comput. Math., Volume 15 (2012), pp. 231-256 | Article | MR 2988815 | Zbl 1296.20009

[16] Geck, Meinolf; Pfeiffer, Götz Characters of finite Coxeter groups and Iwahori–Hecke algebras, London Mathematical Society Monographs. New Series, Volume 21, The Clarendon Press, Oxford University Press, New York, 2000, xvi+446 pages | MR 1778802 | Zbl 0996.20004

[17] Jantzen, Jens C. Einhüllende Algebren halbeinfacher Lie-Algebren, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Volume 3, Springer-Verlag, Berlin, 1983, ii+298 pages | Article | MR 721170 | Zbl 0541.17001

[18] Kazhdan, David; Lusztig, George Representations of Coxeter groups and Hecke algebras, Invent. Math., Volume 53 (1979) no. 2, pp. 165-184 | Article | MR 560412 | Zbl 0499.20035

[19] Knuth, Donald E. The art of computer programming. Volume 3, Addison–Wesley Publishing Co., Reading, Mass.–London–Don Mills, Ont., 1973, xi+722 pp. (1 foldout) pages (Sorting and searching, Addison–Wesley Series in Computer Science and Information Processing) | MR 0445948 | Zbl 0302.68010

[20] Lusztig, George Left cells in Weyl groups, Lie group representations, I (College Park, Md., 1982/1983) (Lecture Notes in Math.) Volume 1024, Springer, Berlin, 1983, pp. 99-111 | Article | MR 727851 | Zbl 0537.20019

[21] Lusztig, George Hecke algebras with unequal parameters, CRM Monograph Series, Volume 18, American Mathematical Society, Providence, RI, 2003, vi+136 pages | MR 1974442 | Zbl 1051.20003

[22] Lusztig, George Hecke algebras with unequal parameters (2014) (http://arxiv.org/pdf/math/0208154.pdf)

[23] Pietraho, Thomas Knuth relations for the hyperoctahedral groups, J. Algebraic Combin., Volume 29 (2009) no. 4, pp. 509-535 | Article | MR 2506719 | Zbl 1226.05261

[24] Schensted, Craige Longest increasing and decreasing subsequences, Canad. J. Math., Volume 13 (1961), pp. 179-191 | Article | MR 0121305 | Zbl 0097.25202

[25] Vogan, David A. Jr. A generalized τ-invariant for the primitive spectrum of a semisimple Lie algebra, Math. Ann., Volume 242 (1979) no. 3, pp. 209-224 | Article | MR 545215 | Zbl 0387.17007