Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Integration formulas for Brownian motion on classical compact Lie groups

Antoine Dahlqvist

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Combinatorial formulas for the moments of the Brownian motion on classical compact Lie groups are obtained. These expressions are deformations of formulas of B. Collins and P. Śniady for moments of the Haar measure and yield a proof of the First Fundamental Theorem of invariant theory and of classical Schur–Weyl dualities based on stochastic calculus.

Résumé

On obtient ici des formules combinatoires pour les moments du mouvement Brownien sur les groupes de Lie classiques compacts. Elles sont des déformations de celles obtenues par B. Collins et P. Śniady pour les moments de la mesure de Haar et permettent de donner une preuve fondée sur le calcul stochastique du premier théorème fondamental de la théorie des invariants et de la dualité de Schur–Weyl.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 1971-1990.

Dates
Received: 3 December 2014
Revised: 7 July 2016
Accepted: 11 July 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1511773734

Digital Object Identifier
doi:10.1214/16-AIHP779

Mathematical Reviews number (MathSciNet)
MR3729643

Zentralblatt MATH identifier
06847070

Subjects
Primary: 46L53: Noncommutative probability and statistics 60J65: Brownian motion [See also 58J65] 60H30: Applications of stochastic analysis (to PDE, etc.) 60G15: Gaussian processes 14L35: Classical groups (geometric aspects) [See also 20Gxx, 51N30] 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24] 15A72: Vector and tensor algebra, theory of invariants [See also 13A50, 14L24] 16W22: Actions of groups and semigroups; invariant theory 05E10: Combinatorial aspects of representation theory [See also 20C30]

Keywords
Brownian motion on Compact Lie groups Weingarten calculus Schur–Weyl duality First Fundamental Theorem of invariant theory

Citation

Dahlqvist, Antoine. Integration formulas for Brownian motion on classical compact Lie groups. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 1971--1990. doi:10.1214/16-AIHP779. https://projecteuclid.org/euclid.aihp/1511773734


Export citation

References

  • [1] N. Alon and M. Tarsi. Colorings and orientations of graphs. Combinatorica 12 (2) (1992) 125–134.
  • [2] P. Biane. Free Brownian motion, free stochastic calculus and random matrices. In Free Probability Theory (Waterloo, ON, 1995) 1–19. Fields Inst. Commun. 12. Amer. Math. Soc., Providence, RI, 1997.
  • [3] R. Brauer. On algebras which are connected with the semisimple continuous groups. Ann. of Math. (2) 38 (4) (1937) 857–872.
  • [4] T. Ceccherini-Silberstein, F. Scarabotti and F. Tolli. Representation Theory of the Symmetric Groups. The Okounkov–Vershik Approach, Character Formulas, and Partition Algebras. Cambridge Studies in Advanced Mathematics 121. Cambridge University Press, Cambridge, 2010.
  • [5] B. Collins. Moments and cumulants of polynomial random variables on unitary groups, the Itzykson–Zuber integral, and free probability. Int. Math. Res. Not. IMRN 17 (2003) 953–982.
  • [6] B. Collins and S. Matsumoto. On some properties of orthogonal Weingarten functions. J. Math. Phys. 50 (11) (2009) 113516.
  • [7] B. Collins and P. Śniady. Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Comm. Math. Phys. 264 (2006) 773–795.
  • [8] A. A. Drisko. On the number of even and odd Latin squares of order $p+1$. Adv. Math. 128 (1) (1997) 20–35.
  • [9] D. G. Glynn. The conjectures of Alon–Tarsi and Rota in dimension prime minus one. SIAM J. Discrete Math. 24 (2) (2010) 394–399.
  • [10] R. W. Goodman and N. R. Wallach. Symmetry, Representations, and Invariants. Springer, Dordchect, 2009.
  • [11] T. Halverson. Characters of the centralizer algebras of mixed tensor representations of ${\operatorname{Gl}}(r,\mathbb{C})$ and the quantum group $\mathcal{U}_{q}(\mathfrak{gl}(r,\mathbb{C}))$. Pacific J. Math. 174 (2) (1996) 359–410.
  • [12] A. A. Jucys. Symmetric polynomials and the center of the symmetric group ring. Rep. Math. Phys. 5 (1) (1974) 107–112.
  • [13] K. Koike. On the decomposition of tensor products of the representations of the classical groups: By means of the universal characters. Adv. Math. 74 (1) (1989) 57–86.
  • [14] S. Kumar and J. M. Landsberg. Connections between conjectures of Alon–Tarsi, Hadamard–Howe, and integrals over the special unitary group. Discrete Math. 338 (7) (2015) 1232–1238.
  • [15] T. Lévy. Schur–Weyl duality and the heat kernel measure on the unitary group. Adv. Math. 218 (2) (2008) 537–575.
  • [16] M. Liao. Lévy Processes in Lie Groups. Cambridge Tracts in Mathematics 162. Cambridge University Press, Cambridge, 2004.
  • [17] M. L. Mehta. Random Matrices, 3rd edition. Pure and Applied Mathematics (Amsterdam) 142. Elsevier/Academic Press, Amsterdam, 2004.
  • [18] A. Okounkov and A. Vershik. A new approach to representation theory of symmetric groups. Selecta Math. (N.S.) 2 (4) (1996) 581–605.
  • [19] C. Procesi. Lie Groups. An Approach Through Invariants and Representations. Universitext. Springer, New York, 2007.
  • [20] I. M. Singer. Functional Analysis on the Eve of the 21st Century. Vol. I. Progress in Mathematics 131. S. Gindikin, J. Lepowsky and R. L. Wilson (Eds). Birkhäuser, Boston, MA, 1995. In honor of the eightieth birthday of I. M. Gel’fand. Papers from the conference held at Rutgers University, New Brunswick, New Jersey, October 24–27, 1993.
  • [21] V. G. Turaev. Operator invariants of tangles, and $R$-matrices. Izv. Ross. Akad. Nauk Ser. Mat. 53 (5) (1989) 1073–1107, 1135.
  • [22] D. Weingarten. Asymptotic behavior of group integrals in the limit of infinite rank. J. Math. Phys. 19 (5) (1978) 999–1001.
  • [23] F. Xu. A random matrix model from two-dimensional Yang–Mills theory. Comm. Math. Phys. 190 (2) (1997) 287–307.
  • [24] P. Zinn-Justin. Jucys–Murphy elements and Weingarten matrices. Lett. Math. Phys. 91 (2010) 119–127.