Vexillary Elements in the Hyperoctahedral Group
Sara Billey
and Tao Kai Lam2
2Current address: 78 Montgomery St, #1, Boston, MA 02116
DOI: 10.1023/A:1008633710118
Abstract
In analogy with the symmetric group, we define the vexillary elements in the hyperoctahedral group to be those for which the Stanley symmetric function is a single Schur Q-function. We show that the vexillary elements can be again determined by pattern avoidance conditions. These results can be extended to include the root systems of types A, B, C, and D. Finally, we give an algorithm for multiplication of Schur Q-functions with a superfied Schur function and a method for determining the shape of a vexillary signed permutation using jeu de taquin.
Pages: 139–152
Keywords: vexillary; Stanley symmetric function; reduced word; hyperoctahedral group
Full Text: PDF
References
1. S. Billey, “Transition equations for isotropic flag manifolds,” Discrete Math., to appear. Special Issue in honor of Adriano Garsia, 1998.
2. S. Billey and M. Haiman, “Schubert polynomials for the classical groups,” J. Amer. Math. Soc. 8 (1995), 443-482.
3. S. Billey, W. Jockusch, and R. Stanley, “Some combinatorial properties of Schubert polynomials,” J. Alg. Combin. 2 (1993), 345-374.
4. M. Bona, “Exact enumeration of 1342-avoiding permutations; a close link with labeled trees and planar maps,” J. Combin. Theory Ser. A 80(2) (1997), 257-272.
5. M. Bona, “The number of permutations with exactly r 132-subsequences is P-recursive in the size!,” Adv. in Appl. Math. 18 (1997), 510-522.
6. P. Edelman and C. Greene, “Balanced tableaux,” Adv. Math. 63 (1987), 42-99.
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8. S. Fomin and R.P. Stanley, “Schubert polynomials and the NilCoxeter algebra,” Adv. Math. 103 (1994), 196-207.
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10. W. Kraśkiewicz, “Reduced decompositions in hyperoctahedral group,” C.R. Acad. Sci. Paris Sér. I Math. 309 (1989), 903-904.
11. V. Lakshmibai and B. Sandhya, “Criterion for smoothness of Schubert varieties in S L(n)/B,” Proc. Indian Acad. Sci. Math. Sci. 100(1) (1990), 45-52.
12. T.K. Lam, “B and D analogues of stable Schubert polynomials and related insertion algorithms,” Ph.D. Thesis, MIT, 1995.
13. T.K. Lam, “Bn Stanley symmetric functions,” Discrete Math. 157 (1996), 241-270.
14. T.K. Lam, “Superfication of the Stanley symmetric functions,” in preparation, 1996.
15. A. Lascoux and M.-P. Sch\ddot utzenberger, “Structure de hopf de l'anneau de cohomologie et de l'anneau de grothendieck d'une variete de drapeaux,” C.R. Acad. Sci. Paris Sér. I Math. 295 (1982), 629-633.
16. A. Lascoux and M.-P. Sch\ddot utzenberger, “Schubert polynomials and the Littlewood-Richardson rule,” Lett. Math. Phys. 10 (1985), 111-124. P1: JVS/NKD P2: VBI Journal of Algebraic Combinatorics KL600-02-Billey July 7, 1998 11:12 152 BILLEY AND LAM
17. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1979.
18. I.G. Macdonald, Notes on Schubert Polynomials, Vol. 6, Publications du LACIM, Université du Québec `a Montréal, 1991.
19. D. Monk, “The geometry of flag manifolds,” Proc. London Math. Soc. 3(9) (1959), 253-286.
20. J. Noonan, “The number of permutations containing exactly one increasing subsequence of length three,” Discrete Math. 152 (1996), 307-313.
21. V. Reiner and M. Shimozono, “Plactification,” J. Alg. Combin. 4 (1995), 331-351.
22. R. Simion and F.W. Schmidt, “Restricted permutations,” European. J. Combin. 6 (1985), 383-406.
23. R. Stanley, “On the number of reduced decompositions of elements of Coxeter groups,” European. J. Combin. 5 (1984), 359-372.
24. R. Stanley, “Unimodality and Lie superalgebras,” Stud. Appl. Math. 72 (1985), 263-281.
25. J. Stembridge, “Shifted tableaux and the projective representations of symmetric group,” Adv. Math. 74 (1989), 87-134.
26. J. Stembridge, personal communication, 1993.
27. J. Stembridge, “Some combinatorial aspects of reduced words in finite Coxeter groups,” Trans. Amer. Math. Soc. 349 (1997), 1285-1332.
28. J. West, “Permutations with forbidden sequences; and, stack-sortable permutations,” Ph.D. Thesis, MIT, 1990.
29. D.R. Worley, “A theory of shifted Young tableaux,” Ph.D. Thesis, MIT, 1984.
30. D.R. Worley, “The shifted analog of the Littlewood-Richardson rule,” preprint, 1987.
2. S. Billey and M. Haiman, “Schubert polynomials for the classical groups,” J. Amer. Math. Soc. 8 (1995), 443-482.
3. S. Billey, W. Jockusch, and R. Stanley, “Some combinatorial properties of Schubert polynomials,” J. Alg. Combin. 2 (1993), 345-374.
4. M. Bona, “Exact enumeration of 1342-avoiding permutations; a close link with labeled trees and planar maps,” J. Combin. Theory Ser. A 80(2) (1997), 257-272.
5. M. Bona, “The number of permutations with exactly r 132-subsequences is P-recursive in the size!,” Adv. in Appl. Math. 18 (1997), 510-522.
6. P. Edelman and C. Greene, “Balanced tableaux,” Adv. Math. 63 (1987), 42-99.
7. S. Fomin and A.N. Kirillov, “Combinatorial Bn analogues of Schubert polynomials,” Trans. Amer. Math. Soc. 348 (1996), 3591-3620.
8. S. Fomin and R.P. Stanley, “Schubert polynomials and the NilCoxeter algebra,” Adv. Math. 103 (1994), 196-207.
9. M. Haiman, “Dual equivalence with applications, including a conjecture of Proctor,” Discrete Math. 99 (1992), 79-113.
10. W. Kraśkiewicz, “Reduced decompositions in hyperoctahedral group,” C.R. Acad. Sci. Paris Sér. I Math. 309 (1989), 903-904.
11. V. Lakshmibai and B. Sandhya, “Criterion for smoothness of Schubert varieties in S L(n)/B,” Proc. Indian Acad. Sci. Math. Sci. 100(1) (1990), 45-52.
12. T.K. Lam, “B and D analogues of stable Schubert polynomials and related insertion algorithms,” Ph.D. Thesis, MIT, 1995.
13. T.K. Lam, “Bn Stanley symmetric functions,” Discrete Math. 157 (1996), 241-270.
14. T.K. Lam, “Superfication of the Stanley symmetric functions,” in preparation, 1996.
15. A. Lascoux and M.-P. Sch\ddot utzenberger, “Structure de hopf de l'anneau de cohomologie et de l'anneau de grothendieck d'une variete de drapeaux,” C.R. Acad. Sci. Paris Sér. I Math. 295 (1982), 629-633.
16. A. Lascoux and M.-P. Sch\ddot utzenberger, “Schubert polynomials and the Littlewood-Richardson rule,” Lett. Math. Phys. 10 (1985), 111-124. P1: JVS/NKD P2: VBI Journal of Algebraic Combinatorics KL600-02-Billey July 7, 1998 11:12 152 BILLEY AND LAM
17. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, Oxford, 1979.
18. I.G. Macdonald, Notes on Schubert Polynomials, Vol. 6, Publications du LACIM, Université du Québec `a Montréal, 1991.
19. D. Monk, “The geometry of flag manifolds,” Proc. London Math. Soc. 3(9) (1959), 253-286.
20. J. Noonan, “The number of permutations containing exactly one increasing subsequence of length three,” Discrete Math. 152 (1996), 307-313.
21. V. Reiner and M. Shimozono, “Plactification,” J. Alg. Combin. 4 (1995), 331-351.
22. R. Simion and F.W. Schmidt, “Restricted permutations,” European. J. Combin. 6 (1985), 383-406.
23. R. Stanley, “On the number of reduced decompositions of elements of Coxeter groups,” European. J. Combin. 5 (1984), 359-372.
24. R. Stanley, “Unimodality and Lie superalgebras,” Stud. Appl. Math. 72 (1985), 263-281.
25. J. Stembridge, “Shifted tableaux and the projective representations of symmetric group,” Adv. Math. 74 (1989), 87-134.
26. J. Stembridge, personal communication, 1993.
27. J. Stembridge, “Some combinatorial aspects of reduced words in finite Coxeter groups,” Trans. Amer. Math. Soc. 349 (1997), 1285-1332.
28. J. West, “Permutations with forbidden sequences; and, stack-sortable permutations,” Ph.D. Thesis, MIT, 1990.
29. D.R. Worley, “A theory of shifted Young tableaux,” Ph.D. Thesis, MIT, 1984.
30. D.R. Worley, “The shifted analog of the Littlewood-Richardson rule,” preprint, 1987.