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Lie Groups pp 395-406 | Cite as

The Cauchy Identity

  • Daniel Bump
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 225)

Abstract

Suppose that \(\alpha _{1},\ldots,\alpha _{n}\) and \(\beta _{1},\ldots,\beta _{m}\) are two sets of variables. The Cauchy identity asserts that
$$\displaystyle{ \prod _{i=1}^{n}\prod _{ j=1}^{m}{(1 - \alpha _{ i}\beta _{j})}^{-1} =\sum _{ \lambda }s_{\lambda }(\alpha _{1},\ldots,\alpha _{n})s_{\lambda }(\beta _{1},\ldots,\beta _{m}), }$$
(38.1)
where the sum is over all partitions λ (of all k). The series is absolutely convergent if all \(\vert \alpha _{i}\vert,\vert \beta _{i}\vert <1\). It can also be regarded as an equality of formal power series.

Keywords

Irreducible Representation Regular Function Formal Power Series Clifford Algebra Automorphic Form 
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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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Daniel Bump
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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