A generalized Berele-Schensted algorithm and conjectured Young tableaux for intermediate symplectic groups
HTML articles powered by AMS MathViewer
- by Robert A. Proctor PDF
- Trans. Amer. Math. Soc. 324 (1991), 655-692 Request permission
Abstract:
The Schensted and Berele algorithms combinatorially mimic the decompositions of ${ \otimes ^k}V$ with respect to ${\operatorname {GL} _N}$ and ${\operatorname {Sp} _{2n}}$. Here we present an algorithm which is a common generalization of these two algorithms. "Intermediate symplectic groups" ${\operatorname {Sp} _{2n,m}}$ are defined. These groups interpolate between ${\operatorname {GL} _N}$ and ${\operatorname {Sp} _N}$. We conjecture that there is a decomposition of ${ \otimes ^k}V$ with respect to ${\operatorname {Sp} _{2n,m}}$ which is described by the output of the new algorithm.References
- Allan Berele, A Schensted-type correspondence for the symplectic group, J. Combin. Theory Ser. A 43 (1986), no. 2, 320–328. MR 867655, DOI 10.1016/0097-3165(86)90070-1
- Allan Berele, Construction of $\textrm {Sp}$-modules by tableaux, Linear and Multilinear Algebra 19 (1986), no. 4, 299–307. MR 860718, DOI 10.1080/03081088608817725 H. Boerner, Representations of groups, North-Holland, Amsterdam, 1963. I. Gessel and G. Viennot, Determinants, paths, and plane partitions, in preparation.
- Phil Hanlon and David Wales, On the decomposition of Brauer’s centralizer algebras, J. Algebra 121 (1989), no. 2, 409–445. MR 992775, DOI 10.1016/0021-8693(89)90076-8
- James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
- R. C. King, Weight multiplicities for the classical groups, Group theoretical methods in physics (Fourth Internat. Colloq., Nijmegen, 1975) Lecture Notes in Phys., Vol. 50, Springer, Berlin, 1976, pp. 490–499. MR 0480895
- R. C. King, Modification rules and products of irreducible representations of the unitary, orthogonal, and symplectic groups, J. Mathematical Phys. 12 (1971), 1588–1598. MR 287816, DOI 10.1063/1.1665778
- Kazuhiko Koike and Itaru Terada, Young-diagrammatic methods for the representation theory of the classical groups of type $B_n,\;C_n,\;D_n$, J. Algebra 107 (1987), no. 2, 466–511. MR 885807, DOI 10.1016/0021-8693(87)90099-8 D. E. Littlewood, The theory of group characters, 2nd ed., Oxford Univ. Press, London, 1950.
- I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979. MR 553598
- Robert A. Proctor, Odd symplectic groups, Invent. Math. 92 (1988), no. 2, 307–332. MR 936084, DOI 10.1007/BF01404455 —, Four Littlewood-Richardson proofs, unpublished notes.
- Robert A. Proctor, A Schensted algorithm which models tensor representations of the orthogonal group, Canad. J. Math. 42 (1990), no. 1, 28–49. MR 1043509, DOI 10.4153/CJM-1990-002-1 —, Young tableaux, Gelfand patterns, and branching rules for classical groups, preprint.
- Arun Ram and Hans Wenzl, Matrix units for centralizer algebras, J. Algebra 145 (1992), no. 2, 378–395. MR 1144939, DOI 10.1016/0021-8693(92)90109-Y
- C. Schensted, Longest increasing and decreasing subsequences, Canadian J. Math. 13 (1961), 179–191. MR 121305, DOI 10.4153/CJM-1961-015-3 S. Sundaram, On the combinatorics of representations of $\operatorname {Sp} (2n,\,{\mathbf {C}})$, Ph.D. Thesis, M.I.T., 1986.
- Glânffrwd P. Thomas, On a construction of Schützenberger, Discrete Math. 17 (1977), no. 1, 107–118. MR 450084, DOI 10.1016/0012-365X(77)90024-3
- Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158 D. P. Zhelobenko, The classical groups. Spectral analysis of their finite dimensional representations, Russian Math. Surveys 17 (1962), 1-94.
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 324 (1991), 655-692
- MSC: Primary 20G05; Secondary 05A15, 20C15
- DOI: https://doi.org/10.1090/S0002-9947-1991-0989583-X
- MathSciNet review: 989583