셀베르그 적분(Selberg integral)

수학노트
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개요

\[ \begin{align} S_{n} (\alpha, \beta, \gamma) & = \int_0^1 \cdots \int_0^1 \prod_{i=1}^n t_i^{\alpha-1}(1-t_i)^{\beta-1} \prod_{1 \le i < j \le n} |t_i - t_j |^{2 \gamma}\,dt_1 \cdots dt_n \\ & = \prod_{j = 0}^{n-1} \frac {\Gamma(\alpha + j \gamma) \Gamma(\beta + j \gamma) \Gamma (1 + (j+1)\gamma)} {\Gamma(\alpha + \beta + (n+j-1)\gamma) \Gamma(1+\gamma)} \end{align},\] 여기서 $$ \Re(\alpha)>0, \Re(\beta)>0, \Re(\gamma)>\max\{-\frac{1}{n},-\frac{\Re{\alpha}}{n-1},-\frac{\Re{\beta}}{n-1}\} $$

  • n=1 인 경우

\[S_{1} (\alpha, \beta,\gamma)=B(\alpha,\beta) = \int_0^1t^{\alpha-1}(1-t)^{\beta-1}\,dt\]


메모

  • Algebra (Coxeter groups, double affine Hecke algebras)
  • Conformal field theory (KZ equations)
  • Gauge theory (supersymmetry, AGT conjecture)
  • Geometry (hyperplane arrangements)
  • Number theory (moments $\zeta(s)$
  • Orthogonal polynomials (Generalised Jacobi polynomials)
  • Random matrices
  • Statistics
  • Statistical physics

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사전 형태의 자료


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관련논문

  • Peter J. Forrester, Volumes for ${\rm SL}_N(\mathbb R)$, the Selberg integral and random lattices, arXiv:1604.07462 [math-ph], April 25 2016, http://arxiv.org/abs/1604.07462
  • Rosengren, Hjalmar. “Selberg Integrals, Askey-Wilson Polynomials and Lozenge Tilings of a Hexagon with a Triangular Hole.” arXiv:1503.00971 [math], March 3, 2015. http://arxiv.org/abs/1503.00971.
  • Patterson, Samuel J. “Selberg Sums - a New Perspective.” arXiv:1411.7600 [math], November 27, 2014. http://arxiv.org/abs/1411.7600.
  • Rains, Eric M. “Multivariate Quadratic Transformations and the Interpolation Kernel.” arXiv:1408.0305 [math], August 1, 2014. http://arxiv.org/abs/1408.0305.
  • Mironov, S., A. Morozov, and Y. Zenkevich. ‘Generalized Jack Polynomials and the AGT Relations for the SU(3) Group’. JETP Letters 99, no. 2 (1 March 2014): 109–13. doi:10.1134/S0021364014020076.
  • Zhang, Hong, and Yutaka Matsuo. ‘Selberg Integral and SU(N) AGT Conjecture’. Journal of High Energy Physics 2011, no. 12 (December 2011). doi:10.1007/JHEP12(2011)106.
  • Mironov, A., Al Morozov, and And Morozov. ‘Matrix Model Version of AGT Conjecture and Generalized Selberg Integrals’. Nuclear Physics B 843, no. 2 (February 2011): 534–57. doi:10.1016/j.nuclphysb.2010.10.016.
  • Warnaar, S. Ole. “The $\mathfrak{sl}_3$ Selberg Integral.” Advances in Mathematics 224, no. 2 (2010): 499–524. doi:10.1016/j.aim.2009.11.011.
  • Warnaar, S. Ole. “A Selberg Integral for the Lie Algebra $A_n$.” Acta Mathematica 203, no. 2 (2009): 269–304. doi:10.1007/s11511-009-0043-x.
  • Warnaar, S. Ole. ‘The Mukhin--Varchenko Conjecture for Type A’. DMTCS Proceedings 0, no. 1 (22 December 2008). http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAJ0108.
  • Luque, Jean-Gabriel, and Jean-Yves Thibon. “Hankel Hyperdeterminants and Selberg Integrals.” Journal of Physics A: Mathematical and General 36, no. 19 (May 16, 2003): 5267. doi:10.1088/0305-4470/36/19/306.
  • Tarasov, V., and A. Varchenko. ‘Selberg-Type Integrals Associated with SL3’. Letters in Mathematical Physics 65, no. 3 (1 September 2003): 173–85. doi:10.1023/B:MATH.0000010712.67685.9d.
  • Gustafson, Robert A. “A Generalization of Selberg’s Beta Integral.” Bulletin (New Series) of the American Mathematical Society 22, no. 1 (January 1990): 97–105.
  • Selberg, Atle. “Remarks on a Multiple Integral.” Norsk Mat. Tidsskr. 26 (1944): 71–78.