Lie Groups pp 395-406 | Cite as

The Cauchy Identity

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


Suppose that \(\alpha _{1},\ldots,\alpha _{n}\) and \(\beta _{1},\ldots,\beta _{m}\) are two sets of variables. The Cauchy identity asserts that
$$\displaystyle{ \prod _{i=1}^{n}\prod _{ j=1}^{m}{(1 - \alpha _{ i}\beta _{j})}^{-1} =\sum _{ \lambda }s_{\lambda }(\alpha _{1},\ldots,\alpha _{n})s_{\lambda }(\beta _{1},\ldots,\beta _{m}), }$$
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.


Irreducible Representation Regular Function Formal Power Series Clifford Algebra Automorphic Form 
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© Springer Science+Business Media New York 2013

Authors and Affiliations

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

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