Korean J. Math. Vol. 23 No. 3 (2015) pp.427-438
DOI: https://doi.org/10.11568/kjm.2015.23.3.427

The jeu de taquin on the shifted rim hook tableaux

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Published: Sep 30, 2015
Keywords:
partition, shifted rim hook tableau, Schensted algorithm, switching rule, jeu de taquin.
Mathematics Subject Classification:
05E10.

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Jaejin Lee
Department of Mathematics Hallym University Chunchon, Korea 200-702

Abstract

The Schensted algorithm first described by Robinson [5] is a remarkable combinatorial correspondence associated with the theory of symmetric functions. Sch\"{u}tzenberger's jeu de taquin [10] can be used to give alternative descriptions of both $P$- and $Q$-tableaux of the Schensted algorithm as well as the ordinary and dual Knuth relations. In this paper we describe the jeu de taquin on shifted rim hook tableaux using the switching rule, which shows that the sum of the weights of the shifted rim hook tableaux of a given shape and content does not depend on the order of the content if content parts are all odd.


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References

[1] E. A. Bender and D. E. Knuth, Enumeration of plane partitions, J. Combin. Theory (A) 13 (1972), 40–54. Google Scholar

[2] A. Garsia and S. Milne, A Rogers-Ramanujan bijection, J. Combin. Theory (A) 31 (1981), 289–339. Google Scholar

[3] C. Greene, An extension of Schensted’s theorem, Adv. in Math., 14 (1974), 254–265. Google Scholar

[4] D. E. Knuth, Permutations, matrices and generalized Young tableaux, Pacific J. Math., 34 (1970), 709–727. Google Scholar

[5] G. de B. Robinson, On the representations of the symmetric group , Amer. J. Math., 60, (1938), 745–760. Google Scholar

[6] B. E. Sagan, Shifted tableaux, Schur Q-functions and a conjecture of R. Stanley, J. Combin. Theory Ser. A 45 (1987), 62–103. Google Scholar

[7] C. Schensted, Longest increasing and decreasing subsequences, Canad. J. Math., 13 (1961), 179–191. Google Scholar

[8] J. R. Stembridge, Shifted tableaux and projective representations of symmetric groups, Advances in Math., 74 (1989), 87–134. Google Scholar

[9] D. W. Stanton and D. E. White, A Schensted algorithm for rim hook tableaux, J. Combin. Theory Ser. A 40 (1985), 211–247. Google Scholar

[10] M. P. Schu ̈tzenberger, Quelques remarques sur une construction de Schensted, Math., Scand. 12 (1963), 117–128. Google Scholar

[11] G. Viennot, Une forme g eom etrique de la correspondance de Robinson- Schensted, in Combiatoire et Repr esentation du Groupe Sym etrique, D. Foata ed., Lecture Notes in Math., Vol. 579, Springer-Verlag, New York, NY, 1977, 29-58. Google Scholar