Transformations of Border Strips and Schur Function Determinants
William Y.C. Chen
, Guo-Guang Yan
and Arthur L.B. Yang
Center for Combinatorics, LPMC Nankai University Tianjin 300071 P. R. China
DOI: 10.1007/s10801-005-3018-9
Abstract
We introduce the notion of the cutting strip of an outside decomposition of a skew shape, and show that cutting strips are in one-to-one correspondence with outside decompositions for a given skew shape. Outside decompositions are introduced by Hamel and Goulden and are used to give an identity for the skew Schur function that unifies the determinantal expressions for the skew Schur functions including the Jacobi-Trudi determinant, its dual, the Giambelli determinant and the rim ribbon determinant due to Lascoux and Pragacz. Using cutting strips, one obtains a formula for the number of outside decompositions of a given skew shape. Moreover, one can define the basic transformations which we call the twist transformation among cutting strips, and derive a transformation theorem for the determinantal formula of Hamel and Goulden. The special case of the transformation theorem for the Giambelli identity and the rim ribbon identity was obtained by Lascoux and Pragacz. Our transformation theorem also applies to the supersymmetric skew Schur function.
Pages: 379–394
Keywords: keywords Young diagram; border strip; outside decomposition; Schur function determinants; Jacobi-trudi identity; giambelli identity; lascoux-pragacz identity
Full Text: PDF
References
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2. \ddot O.N. E\check gecio\check glu and J.B. Remmel, “A combinatorial proof of the Giambelli identity for Schur functions,” Adv. Math. 70 (1988), 59-86.
3. M. Fulmek and C. Krattenthaler, “Lattice path proofs for determinantal formulas for symplectic and orthogonal characters,” J. Combin. Theory. Ser. A 77 (1997), 3-50.
4. I.P. Goulden and C. Greene, “A new tableau representation for supersymmetric Schur functions,” J. Alg. 170 (1994), 687-703.
5. I.M. Gessel and G. Viennot, “Binomial determinants, paths, and hook length formulae,” Adv. Math. 58 (1985), 300-321.
6. I.M. Gessel and G. Viennot, “Determinants, paths, and plane partitions,” preprint.
7. A.M. Hamel and I.P. Goulden, “Planar decompositions of tableaux and Schur function determinants,” Europ. J. Combin. 16 (1995), 461-477.
8. A. Lascoux and P. Pragacz, “ Équerres et fonctuons de Schur,” C. R. Acad. Sci. Paris, Ser. I. Math. 299 (1984), 955-958.
9. A. Lascoux and P. Pragacz, “Ribbon Schur functions,” Europ. J. Combin. 9 (1988), 561-574.
10. I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford University Press, Oxford, 1995.
11. I.G. Macdonald, Schur Functions: Theme and variations, Actes 28e Séminaire Lotharingien, Publ. I.R.M.A. Strasbourg, 1992, pp. 5-39. CHEN, YAN AND YANG
12. B.E. Sagan, The Symmetric Group, 2nd edition, Springer-Verlag Press, 2001.
13. M.P. Sch\ddot utzenberger, La correspondance de Robinson, in Combinatorics et représentation du groupe symmétrique (Table Ronde, Strasbourg 1976, D. Foata, ed.), Lecture Notes in Math., vol. 579, Springer, 1977, pp. 59-113.
14. R.P. Stanley, Enumerative Combinatorics, Vol. II, Cambridge University Press, 1999.
15. J.R. Stembridge, “Nonintersecting paths, Pfaffians, and plane partitions,” Adv. Math. 83 (1990), 96-131.
16. K. Ueno, “Lattice path proof of the ribbon determinant formula for Schur functions,” Nagoya Math. J. 124 (1991), 55-59.
17. A. Zelevinsky, “A generalization of the Littlewood-Richardson rule,” J. Algebra 69 (1981), 82-94.