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Home > Vol 14, No 3 (2001)

Itaru Terada

We interpret geometrically a variant of the Robinson-Schensted correspondence which links Brauer diagrams with updown tableaux, in the spirit of Steinberg”s result [32] on the original Robinson-Schensted correspondence. Our result uses the variety of all ( N, w, V) (N,ω,V) where V V is a complete flag in \mathbb C ^{2 n} , w \mathbb{C}^{2n} ,ω is a nondegenerate alternating bilinear form on \mathbb C ^{2 n} \mathbb{C}^{2n} and N is a nilpotent element of the Lie algebra of the simultaneous stabilizer of both V V instead of Steinberg”s variety of ( N, V, V") (N,V,V”) where V \text and V" V {\text{and}} V” are two complete flags in \mathbb C ⁿ \mathbb{C}^n and N is a nilpotent element of the Lie algebra of the simultaneous stabilizer of both V \text and V" V {\text{and}} V” .

1. G. Benkart and A. Ram, A letter dated October 28 (1990).
2. A. Berele, “A Schensted-type correspondence for the symplectic group,” J. Combin. Theory Ser. A 43 (1986), 320-328.
3. A. Borel, Linear Algebraic Groups, 2nd Edition, Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, New York, 1991.
4. R. Brauer, “On algebras which are connected with the semisimple continuous groups,” Ann. of Math. 38 (1937), 857-872.
5. M.-P. Delest, S. Dulucq, and L. Favreau, “An analogue to Robinson-Schensted correspondence for oscillating tableaux,” Actes du Séminaire, Séminaire Lotharingien de Combinatoire, 20e Session, Alghero, 1988, pp. 57-78, Publ. Inst. Rech. Math. Av. 372/S-20, IRMA Strasbourg/Univ. Louis Pasteur, Strasbourg, 1988.
6. P. Edelman and C. Greene, “Balanced tableaux,” Advances in Math. 63 (1987), 42-99.
7. L. Favreau, “Combinatoire des tableaux oscillants et des polyn\hat omes de Bessel,” Ph.D. Thesis, LaBRI/Univ. Bordeaux I, 1991, Publications du LACIM, Vol. 7, LACIM/Univ. du Québec `a Montréal, Montréal, 1991.
8. S.V. Fomin, “Generalized Robinson-Schensted-Knuth correspondence,” Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI ) 155 (1986), 156-175, 195 (Russian); English transl., J. Soviet Math. 41 (1988), 979-991.
9. E.R. Gansner, “Acyclic digraphs, Young tableaux and nilpotent matrices,” SIAM J. Algebraic Discrete Methods 2 (1981), 429-440.
10. I.M. Gessel, “Counting paths in Young's lattice,” J. Statist. Plann. Inference 34 (1993), 125-134.
11. C. Greene, “An extension of Schensted's theorem,” Advances in Math. 14 (1974), 254-265.
12. J.E. Humphreys, Introduction to Linear Algebraic Groups, Graduate Texts in Mathematics, Vol. 21, Springer- Verlag, New York, 1975.
13. W. Kraskiewicz, “Reduced decompositions in hyperoctahedral groups,” C.R. Acad. Sci. Paris, Ser. I Math. 309 (1989), 903-907.
14. C. Krattenthaler, “Oscillating tableaux and nonintersecting lattice paths,” J. Statist. Plann. Inference 54 (1996), 75-85.
15. T.-K. Lam, “A variation of Kraśkiewicz insertion and shifted tableaux,” in The 8th International Conference on Formal Power Series and Algebraic Combinatorics: Extended Abstracts, Minneapolis, 1996, Univ. of Minnesota, 1996, pp. 307-316.
16. T. Matsuki, “The orbits of affine symmetric spaces under the action of minimal parabolic subgroups,” J. Math. Soc. Japan 31 (1979), 331-357.
17. T. Matsuki and T. Oshima, “Embeddings of discrete series into principal series,” in The Orbit Method in Representation Theory, Progress in Mathematics, Vol. 82, Copenhagen, 1988, Birkh\ddot auser, Boston, 1990, pp. 147-175.
18. A. Postnikov, “Oscillating tableaux, Sm \times Sn-modules, and Robinson-Schensted-Knuth correspondence,” in The 8th International Conference on Formal Power Series and Algebraic Combinatorics: Extended Abstracts, Minneapolis, 1996, Univ. of Minnesota, 1996, pp. 391-402.
19. R.A. Proctor, “Reflection and algorithm proofs of some more Lie group dual pair identities,” J. Combin. Theory Ser. A 62 (1993), 107-127.
20. R.W. Richardson and T. A. Springer, “Combinatorics and geometry of K -orbits on the flag manifold,” in Linear Algebraic Groups and Their Representations, Contemporary Math. Vol. 153, Los Angeles, 1992, Amer. Math. Soc., Providence, 1993, pp. 109-142.
21. G. de B. Robinson, “On the representations of the symmetric group,” Amer. J. Math. 60 (1938), 746-760.
22. T.W. Roby, “Applications and extensions of Fomin's generalization of the Robinson-Schensted correspondence to differential posets,” Ph.D. Thesis, M.I.T., 1991.
23. W. Rossmann, “The structure of semisimple symmetric spaces,” Canad. J. Math. 131 (1979), 157-180.
24. B.E. Sagan, “Shifted tableaux, Schur Q-functions, and a conjecture of R. Stanley,” J. Combin. Theory Ser. A 45 (1987), 62-103.
25. B.E. Sagan, The Symmetric Group, Wadsworth & Brooks/Cole, Pacific Grove, 1991; 2nd Edition, Graduate Texts in Mathematics, Vol. 203, Springer-Verlag, New York, 2001.
26. M. Saks, “Dilworth numbers, incidence maps and product partial orders,” SIAM J. Algebraic Discrete Methods 1 (1980), 211-215.
27. M. Saks, “Some sequences associated with combinatorial structures,” Discrete Math. 59 (1986), 125-166.
28. C. Schensted, “Longest increasing and decreasing subsequences,” Canad. J. Math. 13 (1961), 179-191.
29. N. Spaltenstein, “On the fixed point set of a unipotent transformation on the flag manifold,” Nederl. Akad. Wetensch. Indag. Math. 38 (1976), 452-458 (same as: Nederl. Akad. Wetensh. Proc. Ser. A 79).
30. T.A. Springer, “Some results on algebraic groups with involutions,” in Algebraic Groups and Related Topics, Advanced Studies in Pure Mathematics, Vol. 6, Kyoto/Nagoya, 1983, Kinokuniya, Tokyo, 1985, pp. 525-543.
31. T.A. Springer, “Schubert varieties and generalizations,” in Representation Theories and Algebraic Geometry, Montréal, 1997, A. Broer and A. Daigneault (Eds.), Kluwer, Dortrecht, 1998, pp. 413-440.
32. R. Steinberg, “An occurrence of the Robinson-Schensted correspondence,” J. Algebra 113 (1988), 523-528.
33. S. Sundaram, “On the combinatorics of representations of Sp(2n, C),” Ph.D. Thesis, M.I.T., 1986.
34. S. Sundaram, “The Cauchy identity for Sp(2n),” J. Combin. Theory Ser. A 53 (1990), 209-238.
35. I. Terada, “A Robinson-Schensted-type correspondence for a dual pair on spinors,” J. Combin. Theory Ser. A 63 (1993), 90-109.
36. G.P. Tesler, “Semi-primary lattices and tableau algorithms,” Ph.D. Thesis, M.I.T., 1995.
37. P.E. Trapa, “Generalized Robinson-Schensted algorithms for real groups,” Internat. Math. Res. Notices 1999 (1999), 803-832.
38. M.A.A. van Leeuwen, “A Robinson-Schensted algorithm in the geometry of flags for classical groups,” Ph.D. Thesis, De Rijksuniversiteit te Utrecht, 1989.