Volume of a compact Lie group

introduction

• Define \(J_\lambda(x)= h(\lambda)^{-1}\det(e^{\lambda_i x_j})\), where \( h(\lambda)=\prod_{i<j}(\lambda_i-\lambda_j)\).
• For each \(x\), \(J_\lambda(x)\) is an analytic function of \(\lambda\); in particular, \(J_0(x)=\left(\prod_{j=1}^{n-1} j!\right) h(x)\).
• The functions \(J_\lambda(x)\) play a central role in random matrix theory.
• For example, if \(\Lambda\) and \(X\) are Hermitian matrices with eigenvalues given by \(\lambda\) and \(x\), respectively, then

\begin{equation}\label{iz} \int_{U(n)}e^{\rm tr \Lambda U X U^*} dU=\frac{J_\lambda(x)}{J_0(x)}, \end{equation} where the integral is with respect to normalised Haar measure on the unitary group.

• This is known as the Harish-Chandra, or Itzykson-Zuber, formula.

related items

• Random matrix

expositions

• Diaconis, Persi, and Peter J. Forrester. “A. Hurwitz and the Origins of Random Matrix Theory in Mathematics.” arXiv:1512.09229 [math-Ph], December 31, 2015. http://arxiv.org/abs/1512.09229.
• https://terrytao.wordpress.com/2013/02/08/the-harish-chandra-itzykson-zuber-integral-formula/
• Zhang, Lin. “Volumes of Orthogonal Groups and Unitary Groups.” arXiv:1509.00537 [math-Ph, Physics:quant-Ph], September 1, 2015. http://arxiv.org/abs/1509.00537.
• Bernardoni, Fabio, Sergio L. Cacciatori, Bianca L. Cerchiai, and Antonio Scotti. “Mapping the Geometry of the E6 Group.” Journal of Mathematical Physics 49, no. 1 (2008): 012107. doi:10.1063/1.2830522.
• Boya, Luis J., E. C. G. Sudarshan, and Todd Tilma. “Volumes of Compact Manifolds.” Reports on Mathematical Physics 52, no. 3 (December 2003): 401–22. doi:10.1016/S0034-4877(03)80038-1. http://arxiv.org/abs/math-ph/0210033v3

articles

• Shu, Fu-Wen, and You-Gen Shen. “Several Integrals of Quaternionic Field on Hyperbolic Matrix Space.” arXiv:1511.01385 [gr-Qc, Physics:math-Ph], November 4, 2015. http://arxiv.org/abs/1511.01385.
• Hashimoto, Y. “On Macdonald’s Formula for the Volume of a Compact Lie Group.” Commentarii Mathematici Helvetici 72, no. 4 (April 3, 2014): 660–62. doi:10.1007/s000140050040.
• Macdonald, I. G. “The Volume of a Compact Lie Group.” Inventiones Mathematicae 56, no. 2 (February 1980): 93–95. doi:10.1007/BF01392542.
• Itzykson, C., and J. B. Zuber. “The Planar Approximation. II.” Journal of Mathematical Physics 21, no. 3 (1980): 411–21. doi:10.1063/1.524438.