Inner tableau translation property of the weak order and related results

Taşkin, Müge

doi:10.1090/S0002-9939-2012-11415-7

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ISSN 1088-6826(online) ISSN 0002-9939(print)

Inner tableau translation property of the weak order and related results

Author: Müge Taşkin
Journal: Proc. Amer. Math. Soc. 141 (2013), 837-856
MSC (2010): Primary 05E10, 20C30
Posted: July 24, 2012
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Abstract: Let $SYT_{n}$ be the set of all standard Young tableaux with cells and $\leq _{weak}$ be Melnikov's weak order on . The aim of this paper is to introduce a conjecture on the weak order, named the inner tableau translation property. We prove the conjecture for some special cases and discuss its significance.

1. Dan Barbasch and David Vogan, Primitive ideals and orbital integrals in complex exceptional groups, J. Algebra 80 (1983), no. 2, 350–382. MR 691809 (84h:22038), http://dx.doi.org/10.1016/0021-8693(83)90006-6
2. A. Björner, Topological methods, Handbook of combinatorics, Vol. 1, 2, Elsevier, Amsterdam, 1995, pp. 1819–1872. MR 1373690 (96m:52012)
3. Anders Björner and Michelle L. Wachs, Permutation statistics and linear extensions of posets, J. Combin. Theory Ser. A 58 (1991), no. 1, 85–114. MR 1119703 (92m:06010), http://dx.doi.org/10.1016/0097-3165(91)90075-R
4. Dieter Blessenohl and Manfred Schocker, Noncommutative character theory of the symmetric group, Imperial College Press, London, 2005. MR 2424338 (2009i:20021)
5. David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR 1251060 (94j:17001)
6. Meinolf Geck, On the induction of Kazhdan-Lusztig cells, Bull. London Math. Soc. 35 (2003), no. 5, 608–614. MR 1989489 (2004d:20003), http://dx.doi.org/10.1112/S0024609303002236
7. Curtis Greene, An extension of Schensted’s theorem, Advances in Math. 14 (1974), 254–265. MR 0354395 (50 #6874)
8. David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412 (81j:20066), http://dx.doi.org/10.1007/BF01390031
9. Donald E. Knuth, The art of computer programming. Volume 3, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1973. Sorting and searching; Addison-Wesley Series in Computer Science and Information Processing. MR 0445948 (56 #4281)
10. Donald E. Knuth, Permutations, matrices, and generalized Young tableaux, Pacific J. Math. 34 (1970), 709–727. MR 0272654 (42 #7535)
11. Jean-Louis Loday and María O. Ronco, Order structure on the algebra of permutations and of planar binary trees, J. Algebraic Combin. 15 (2002), no. 3, 253–270. MR 1900627 (2003m:05213), http://dx.doi.org/10.1023/A:1015064508594
12. George Lusztig, Cells in affine Weyl groups. II, J. Algebra 109 (1987), no. 2, 536–548. MR 902967 (88m:20103a), http://dx.doi.org/10.1016/0021-8693(87)90154-2
13. George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472 (86j:20038)
14. Clauda Malvenuto and Christophe Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (1995), no. 3, 967–982. MR 1358493 (97d:05277), http://dx.doi.org/10.1006/jabr.1995.1336
15. Anna Melnikov, On orbital variety closures in 𝔰𝔩_{𝔫}. I. Induced Duflo order, J. Algebra 271 (2004), no. 1, 179–233. MR 2022481 (2004m:17009), http://dx.doi.org/10.1016/j.jalgebra.2003.09.012
16. Anna Melnikov, On orbital variety closures in 𝔰𝔩_{𝔫}. II. Descendants of a Richardson orbital variety, J. Algebra 271 (2004), no. 2, 698–724. MR 2025547 (2004k:17013), http://dx.doi.org/10.1016/j.jalgebra.2003.09.013
17. Anna Melnikov, Irreducibility of the associated varieties of simple highest weight modules in 𝔰𝔩(𝔫), C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 1, 53–57 (English, with English and French summaries). MR 1198749 (93k:17012)
18. Anna Melnikov, On orbital variety closures in 𝔰𝔩_{𝔫}. III. Geometric properties, J. Algebra 305 (2006), no. 1, 68–97. MR 2262520 (2007j:17011), http://dx.doi.org/10.1016/j.jalgebra.2006.01.010
19. Stéphane Poirier and Christophe Reutenauer, Algèbres de Hopf de tableaux, Ann. Sci. Math. Québec 19 (1995), no. 1, 79–90 (French, with English and French summaries). MR 1334836 (96g:05146)
20. Astrid Reifegerste, Permutation sign under the Robinson-Schensted correspondence, Ann. Comb. 8 (2004), no. 1, 103–112. MR 2061380 (2005a:05214), http://dx.doi.org/10.1007/s00026-004-0208-4
21. Bruce E. Sagan, The symmetric group, 2nd ed., Graduate Texts in Mathematics, vol. 203, Springer-Verlag, New York, 2001. Representations, combinatorial algorithms, and symmetric functions. MR 1824028 (2001m:05261)
22. C. Schensted, Longest increasing and decreasing subsequences, Canad. J. Math. 13 (1961), 179–191. MR 0121305 (22 #12047)
23. M. P. Schützenberger, Quelques remarques sur une construction de Schensted, Math. Scand. 12 (1963), 117–128 (French). MR 0190017 (32 #7433)
24. M.-P. Schützenberger, La correspondance de Robinson, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976), Springer, Berlin, 1977, pp. 59–113. Lecture Notes in Math., Vol. 579 (French). MR 0498826 (58 #16863)
25. Müge Taşkin, Properties of four partial orders on standard Young tableaux, J. Combin. Theory Ser. A 113 (2006), no. 6, 1092–1119. MR 2244136 (2007h:05162), http://dx.doi.org/10.1016/j.jcta.2005.09.008
26. David A. Vogan Jr., Ordering of the primitive spectrum of a semisimple Lie algebra, Math. Ann. 248 (1980), no. 3, 195–203. MR 575938 (81k:17006), http://dx.doi.org/10.1007/BF01420525

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