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Schubert Calculus and Schur Functions

  • Hung-Hsi Wu
Part of the The University Series in Mathematics book series (USMA)

Abstract

Given two Schubert cycles of a Grassmann manifold,σ a and σ b , we have the product formula
$${{\sigma }_{a}}\centerdot {{\sigma }_{b}}=\sum\limits_{c}{\delta \left( a,b,c \right)}{{\sigma }_{c}}.$$
The method for calculating δ(a,b,c) has already been given in a previous work [5] with the aid of Schur functions in the representation theory of the unitary group. On this basis of [5], making use of the similarity between the Schubert calculus and Schur functions and the branching formula of group representation theory, we investigate the question as to which of those can make σ c take δ(a,b,c) as coefficients in σ a σ b for given σ c ,σ b .

Keywords

Irreducible Representation Unitary Group Irreducible Character Young Tableau Product Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    H. Boemer, Representations of Groups, North-Holland, Amsterdam (1963).Google Scholar
  2. 2.
    P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, New York (1978).MATHGoogle Scholar
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    S. Kleiman, Problem 15, Rigorous Foundation of Schubert’s numerative Calculus, Proc. Sympos. Pure Math., Vol. 28, American Mathematical Society, Providence, RI (1976).Google Scholar
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    I. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford (1979).MATHGoogle Scholar
  5. 5.
    J.-Q. Zhong, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations,Vol. III, pp. 1697–1708, Science Press, Beijing, China, 1982.Google Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • Hung-Hsi Wu
    • 1
  1. 1.University of CaliforniaBerkeleyUSA

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