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Lie Groups pp 407-417

# Random Matrix Theory

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

## Abstract

In this chapter, we will work not with $$\mathrm{GL}(n, \mathbb{C})$$ but with its compact subgroup U(n). As in the previous chapters, we will consider elements of $$\mathcal{R}_{k}$$ as generalized characters on S k . If $$\mathbf{f} \in \mathcal{R}_{k}$$, then $$f ={ \mathrm{ch}}^{(n)}(\mathbf{f}) \in \varLambda _{k}^{(n)}$$ is a symmetric polynomial in n variables, homogeneous of weight k. Then $$\psi _{f}: \mathrm{U}(n)\longrightarrow \mathbb{C}$$, defined by (), is the function on U(n) obtained by applying f to the eigenvalues of g ∈U(n). We will denote $$\psi _{f} ={ \mathrm{Ch}}^{(n)}(\mathbf{f})$$. Thus, Ch(n) maps the additive group of generalized characters on S k to the additive group of generalized characters on U(n). It extends by linearity to a map from the Hilbert space of class functions on S k to the Hilbert space of class functions on U(n).

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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