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Determinantal Forms for Symplectic and Orthogonal Schur Functions

Published online by Cambridge University Press:  20 November 2018

A. M. Hamel
A. M. Hamel*
Affiliation:
Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand e-mail: physamh@phys.canterbury.ac.nz
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Abstract

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Symplectic and orthogonal Schur functions can be defined combinatorially in a manner similar to the classical Schur functions. This paper demonstrates that they can also be expressed as determinants. These determinants are generated using planar decompositions of tableaux into strips and the equivalence of these determinants to symplectic or orthogonal Schur functions is established by Gessel-Viennot lattice path techniques. Results for rational (also called composite) Schur functions are also obtained.

Keywords

05E0505E1020C33
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 2 , 01 April 1997 , pp. 263 - 282
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Balantekin, A.B. and Bars, I., Representations of supergroups, J. Math. Phys. 22(1981), 1810–1818.Google Scholar
2. Benkart, G. and Stroomer, J., Tableaux and insertions schemes for spinor representations of the orthogonal, Lie algebra so(2r + 1,C), J. Combin. Theory Ser. A 57(1991), 211–237.Google Scholar
3. Berele, A., A Schensted-type correspondence for the symplectic group, J.Combin. Theory Ser.A 43(1984), 320–328.Google Scholar
4. Cummins, C.J. andR.C.King,Composite Young diagrams, supercharacters ofU(M/N) and modification rules, J. Phys. A 20(1987), 3121–3133.Google Scholar
5. El Samra, N. and King, R.C., Reduced determinantal forms for characters of the classical Lie groups, J. Phys. A 12(1979), 2305–2315.Google Scholar
6. Gessel, I. and Viennot, G.X., Determinants, paths, and plane partitions, preprint.Google Scholar
7. Goulden, I.P. and Jackson, D.M., Combinatorial Enumeration, Wiley, New York, 1983.Google Scholar
8. Hamel, A.M. and Goulden, I.P., Planar Decompositions of Tableaux and Schur Function Determinants, European J. Combin. 16(1995), 461–477.Google Scholar
9. Hamel, A.M., Algebraic and Combinatorial Methods for Symmetric and Supersymmetric Functions, Ph.D. Dissertation, University of Waterloo, Waterloo, Ontario, Canada. April, 1994.Google Scholar
10. Hamel, A.M., Pfaffians and determinants for Schur Q-functions, University of Canterbury Research Report No. 111, 1994.Google Scholar
11. King, R.C., Generalized Young tableaux and the general linear group, J.Math. Phys. 11(1970), 280–293.Google Scholar
12. King, R.C., Modification rules and products of irreducible representations of the unitary, orthogonal, and symplectic groups, J. Math. Phys. 12(1971), 1588–1598.Google Scholar
13. King, R.C., Weight multiplicities for the classical groups, in Lecture Notes in Physics 50, Springer-Verlag, New York, 1975. 490–499.Google Scholar
14. Koike, K. and Terada, I., Young-diagramatic methods for the restriction of representations of complex classical Lie groups to reductive subgroups of maximal rank, Adv. in Math 79(1990), 104–135.Google Scholar
15. Littlewood, D.E., The Theory of Group Characters, 2nd ed., Oxford Univ. Press, Oxford, 1950.Google Scholar
16. Macdonald, I.G., Symmetric Functions and Hall Polynomials, Oxford Univ. Press, Oxford, 1979.Google Scholar
17. Okada, S., A Robinson-Schensted-type algorithm for SO(2n, , C), J. Algebra 143(1991), 334–372.Google Scholar
18. Okada, S., Robinson-Schensted-type algorithms for the spinor representations of the orthogonal Lie algebra so(2n, C), J. Algebra 158(1993), 155–200.Google Scholar
19. Proctor, R., Odd symplectic groups, Invent. Math. 92(1988), 307–332.Google Scholar
20. Proctor, R., A Schensted algorithm which models tensor representations of the orthogonal group, Canad. J. Math. 42(1990), 28–49.Google Scholar
21. Proctor, R., A generalized Berele-Schensted algorithm and conjectured Young tableaux for intermediate symplectic groups, Trans. Amer.Math. Soc. 324(1991), 655–692.Google Scholar
22. Proctor, R., Young tableaux, Gelfand patterns, and branching rules for classical groups, J. Algebra 164(1994), 299–360.Google Scholar
23. Sagan, B.E., The ubiquitous Young tableau. In: Invariant Theory and Tableau, IMA Volumes in Mathematics and its Applications 19, (ed. Dennis Stanton), Springer-Verlag, New York, 1990. 262–298.Google Scholar
24. Sagan, B.E., The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Wadsworth & Brooks/Cole, Pacific Grove, California, 1991.Google Scholar
25. Stembridge, J.R., Rational tableaux and the tensor algebra of gln, J. Combin. Theory Ser. A 46(1987), 79–120.Google Scholar
26. Stembridg, J.R., Nonintersecting paths, pfaffians and plane partitions, Adv. in Math. 83(1990), 96–131.Google Scholar
27. Sundaram, S., The Cauchy identity for Sp(2n, ), J. Combin. Theory Ser. A 53(1990), 209–238.Google Scholar
28. Sundaram, S., Orthogonal tableaux and an insertion algorithm for SO(2n, + 1), J. Combin. Theory Ser. A 53(1990), 239–285.Google Scholar
29. Weyl, H., The Classical Groups, 2nd ed., Princeton Univ. Press, Princeton, 1946.Google Scholar