"Volume of a compact Lie group"의 두 판 사이의 차이
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| + | ==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== | ==related items== | ||
* [[Random matrix]] | * [[Random matrix]] | ||
2016년 1월 1일 (금) 05:43 판
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.