Signed enumeration of ribbon tableaux: an approach through growth diagrams

详细信息    查看全文

• 作者：Dominique Gouyou-Beauchamps (1) dgb@acces.lri.fr
  Philippe Nadeau (2) philippe.nadeau@univie.ac.at
• 关键词：Ribbon tableaux – ; Growth diagrams – ; Murnaghan– ; Nakayama rule – ; Garsia– ; Milne involution principle – ; RSK correspondence
• 刊名：Journal of Algebraic Combinatorics
• 出版年：2012
• 出版时间：August 2012
• 年：2012
• 卷：36
• 期：1
• 页码：67-102
• 全文大小：1.1 MB
• 参考文献：1. Bessenrodt, C., Olsson, J.: On the sequence A085642. In: The On-line Encyclopedia of Integer Sequences (2004)
  2. Chauve, C., Dulucq, S.: A geometric version of the Robinson–Schensted correspondence for skew oscillating tableaux. Discrete Math. 246(1–3), 67–81 (2002)
  3. Dulucq, S., Sagan, B.E.: The Robinson–Schensted correspondence for skew-oscillating tableaux. In: Proceedings of the 4th Conference on Formal Power Series and Algebraic Combinatorics, Amsterdam, The Netherlands, pp. 129–142. Amsterdam, Elsevier (1995)
  4. Dulucq, S., Sagan, B.E.: La correspondance de Robinson–Schensted pour les tableaux oscillants gauches. Discrete Math. 139(1–3), 129–142 (1995)
  5. Fomin, S.V.: The generalized Robinson–Schensted–Knuth correspondence. Zap. Nauč. Semin. LOMI 155(Differentsialnaya Geometriya, Gruppy Li i Mekh. VIII):156–175, 195 (1986)
  6. Fomin, S.: Duality of graded graphs. J. Algebr. Comb. 3(4), 357–404 (1994)
  7. Fomin, S.: Schensted algorithms for dual graded graphs. J. Algebr. Comb. 4(1), 5–45 (1995)
  8. Fomin, S.: Schur operators and Knuth correspondences. J. Comb. Theory, Ser. A 72(2), 277–292 (1995)
  9. Fomin, S., Stanton, D.: Rim Hook Lattices. Algebra Anal. 9(5), 140–150 (1997)
  10. Fulton, W.: Young Tableaux. London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge (1997). With applications to representation theory and geometry
  11. Garsia, A., Milne, S.: Method for constructing bijections for classical partition identities. Proc. Natl. Acad. Sci. USA 78(4, part 1), 2026–2028 (1981)
  12. Garsia, A., Milne, S.: A Rogers–Ramanujan bijection. J. Comb. Theory, Ser. A 31(3), 289–339 (1981)
  13. Isaacs, I.M.: Character Theory of Finite Groups. Dover, New York (1994). Corrected reprint of the 1976 original (Academic Press, New York; MR0460423 (57 #417))
  14. Kerber, A.: Applied Finite Group Actions. Algorithms and Combinatorics, vol. 19, 2nd edn. Springer, Berlin (1999)
  15. Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs, 2nd edn. Clarendon/Oxford University Press, New York (1995)
  16. Murnaghan, F.D.: On the representations of the symmetric group. Am. J. Math. 59(3), 437–488 (1937)
  17. Nakayama, T.: On some modular properties of irreducible representations of a symmetric group. I. Jpn. J. Math. 18, 89–108 (1941)
  18. Roby, T.: The connection between the Robinson–Schensted correspondence for skew oscillating tableaux and graded graphs. Discrete Math. 139(1–3), 481–485 (1995). Formal power series and algebraic combinatorics (Montreal, PQ, 1992)
  19. Sagan, B.E.: Shifted tableaux, Schur Q-functions, and a conjecture of R. Stanley. J. Comb. Theory, Ser. A 45(1), 62–103 (1987)
  20. Sagan, B.E.: The Symmetric Group. Graduate Texts in Mathematics, vol. 203, 2nd edn. Springer, New York (2001). Representations, combinatorial algorithms, and symmetric functions
  21. Sagan, B.E., Stanley, R.P.: Robinson–Schensted algorithms for Skew tableaux. J. Comb. Theory, Ser. A 55(2), 161–193 (1990)
  22. Schensted, C.: Longest increasing and decreasing subsequences. Can. J. Math. 13, 179–191 (1961)
  23. Shimozono, M., White, D.E.: Color-to-spin ribbon Schensted algorithms. Discrete Math. 246(1–3), 295–316 (2002). Formal power series and algebraic combinatorics (Barcelona, 1999)
  24. Sloane, N.: On-line encyclopedia of integer sequences. Accessible from N. Sloane’s homepage
  25. Stanley, R.P.: Differential posets. J. Am. Math. Soc. 1(4), 919–961 (1988)
  26. Stanley, R.P.: Variations on differential posets. In: Invariant Theory and Tableaux, Minneapolis, MN, 1988. IMA Vol. Math. Appl., vol. 19, pp. 145–165. Springer, New York (1990)
  27. Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge Studies in Advanced Mathematics, vol. 62. Cambridge University Press, Cambridge (1999). With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin
  28. Stanton, D.W., White, D.E.: A Schensted algorithm for rim hook tableaux. J. Comb. Theory, Ser. A 40(2), 211–247 (1985)
  29. Sundaram, S.: The Cauchy identity for Sp(2n). J. Comb. Theory, Ser. A 53(2), 209–238 (1990)
  30. van Leeuwen, M.A.A.: Edge sequences, ribbon tableaux, and an action of affine permutations. Eur. J. Comb. 20(2), 179–195 (1999)
  31. White, D.E.: A bijection proving orthogonality of the characters of S n . Adv. Math. 50(2), 160–186 (1983)
• 作者单位：1. Laboratoire de Recherche en Informatique, Universit茅 Paris Sud, 91405 Orsay, France2. Fakult盲t f眉r Mathematik, Universit盲t Wien, Garnisongasse 3, 1090 Vienna, Austria
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Convex and Discrete Geometry
  Order, Lattices and Ordered Algebraic Structures
  Computer Science, general
  Group Theory and Generalizations
  
• 出版者：Springer U.S.
• ISSN：1572-9192

文摘

We give an extension of the famous Schensted correspondence to the case of ribbon tableaux, where ribbons are allowed to be of different sizes. This is done by extending Fomin’s growth diagram approach of the classical correspondence, in particular by allowing signs in the enumeration. As an application, we give in particular a combinatorial proof, based on the Murnaghan–Nakayama rule, for the evaluation of the column sums of the character table of the symmetric group.