"Volume of a compact Lie group"의 두 판 사이의 차이
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==introduction== | ==introduction== | ||
| − | * Define | + | * Define <math>J_\lambda(x)= h(\lambda)^{-1}\det(e^{\lambda_i x_j})</math>, where <math> h(\lambda)=\prod_{i<j}(\lambda_i-\lambda_j)</math>. |
| − | * For each | + | * For each <math>x</math>, <math>J_\lambda(x)</math> is an analytic function of <math>\lambda</math>; in particular, <math>J_0(x)=\left(\prod_{j=1}^{n-1} j!\right) h(x)</math>. |
| − | * The functions | + | * The functions <math>J_\lambda(x)</math> play a central role in random matrix theory. |
| − | * For example, if | + | * For example, if <math>\Lambda</math> and <math>X</math> are Hermitian matrices with eigenvalues given by <math>\lambda</math> and <math>x</math>, respectively, then |
\begin{equation}\label{iz} | \begin{equation}\label{iz} | ||
\int_{U(n)}e^{\rm tr \Lambda U X U^*} dU=\frac{J_\lambda(x)}{J_0(x)}, | \int_{U(n)}e^{\rm tr \Lambda U X U^*} dU=\frac{J_\lambda(x)}{J_0(x)}, | ||
2020년 11월 13일 (금) 06:38 기준 최신판
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.
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.