## 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, - 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, - 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
—, - 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
—, - 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, - 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,

*Representations of groups*, North-Holland, Amsterdam, 1963. I. Gessel and G. Viennot,

*Determinants, paths, and plane partitions*, in preparation.

*The theory of group characters*, 2nd ed., Oxford Univ. Press, London, 1950.

*Four Littlewood-Richardson proofs*, unpublished notes.

*Young tableaux, Gelfand patterns, and branching rules for classical groups*, preprint.

*On the combinatorics of representations of*$\operatorname {Sp} (2n,\,{\mathbf {C}})$, Ph.D. Thesis, M.I.T., 1986.

*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