Geodesic
The limit shape of a probability measure on a tensor product of modules of the $B_n$ algebra
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIX, Tome 468 (2018), pp. 82-97 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We study a probability measure on the integral dominant weights in the decomposition of the $N$th tensor power of the spinor representation of the Lie algebra $\mathrm{so}(2n+1)$. The probability of a dominant weight $\lambda$ is defined as the dimension of the irreducible component of $\lambda$ divided by the total dimension $2^{nN}$ of the tensor power. We prove that as $N\to\infty$, the measure weakly converges to the radial part of the $\mathrm{SO}(2n+1)$-invariant measure on $\mathrm{so}(2n+1)$ induced by the Killing form. Thus, we generalize Kerov's theorem for $\mathrm{su}(n)$ to $\mathrm{so}(2n+1)$. 
@article{ZNSL_2018_468_a7,
     author = {A. A. Nazarov and O. V. Postnova},
     title = {The limit shape of a~probability measure on a~tensor product of modules of the $B_n$ algebra},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {82--97},
     year = {2018},
     volume = {468},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_468_a7/}
}
TY  - JOUR
AU  - A. A. Nazarov
AU  - O. V. Postnova
TI  - The limit shape of a probability measure on a tensor product of modules of the $B_n$ algebra
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2018
SP  - 82
EP  - 97
VL  - 468
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2018_468_a7/
LA  - en
ID  - ZNSL_2018_468_a7
ER  - 
%0 Journal Article
%A A. A. Nazarov
%A O. V. Postnova
%T The limit shape of a probability measure on a tensor product of modules of the $B_n$ algebra
%J Zapiski Nauchnykh Seminarov POMI
%D 2018
%P 82-97
%V 468
%U http://geodesic.mathdoc.fr/item/ZNSL_2018_468_a7/
%G en
%F ZNSL_2018_468_a7
A. A. Nazarov; O. V. Postnova. The limit shape of a probability measure on a tensor product of modules of the $B_n$ algebra. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIX, Tome 468 (2018), pp. 82-97. http://geodesic.mathdoc.fr/item/ZNSL_2018_468_a7/

[1] S. V. Kerov, “On asymptotic distribution of symmetry types of high rank tensors”, Zap. Nauchn. Semin. LOMI, 155, 1986, 181–186 | MR | Zbl

[2] A. M. Vershik, S. V. Kerov, “Asymptotics of the Plancherel measure of the symmetric group and the limit form of Young tableaux”, Sov. Math. Dokl., 18 (1977), 527–531 | MR | Zbl

[3] A. M. Vershik, S. V. Kerov, “Asymptotics of maximal and typical dimensions of irreducible representations of a symmetric group”, Funct. Anal. Appl., 19:1 (1985), 21–31 | DOI | MR | Zbl

[4] B. F. Logan, L. A. Shepp, “A variational problem for random Young tableaux”, Adv. Math., 26:2 (1977), 206–222 | DOI | MR | Zbl

[5] A. Okounkov, “Random matrices and random permutations”, Int. Math. Res. Not., 2000:20 (2000), 1043–1095 | DOI | MR | Zbl

[6] A. Borodin, A. Okounkov, G. Olshanski, “Asymptotics of Plancherel measures for symmetric groups”, J. Amer. Math. Soc., 13:3 (2000), 481–515 | DOI | MR | Zbl

[7] P. Kulish, V. Lyakhovsky, O. Postnova, “Tensor power decomposition. $B_n$ case”, J. Phys. Conf. Series, 343 (2012), Paper 012095 | DOI

[8] B. V. Gnedenko, Theory of Probability, Routledge, 2017

[9] M. Balazs, B. Toth, Stirling's formula and de Moivre–Laplace central limit theorem, Lecture notes https://people.maths.bris.ac.uk/~mb13434/Stirling_DeMoivre_Laplace.pdf

[10] V. I. Bogachev, Weak Convergence of Measures, Institute of Computer Science, Moscow–Izhevsk, 2016 (in Russian) | MR

[11] P. P. Kulish, V. D. Lyakhovsky, O. Postnova, “Tensor powers for non-simply laced Lie algebras $\mathrm B_2$-case”, J. Phys. Conf. Series, 346 (2012), Paper 012012 | DOI

[12] P. P. Kulish, V. D. Lyakhovsky, O. V. Postnova, “Multiplicity function for tensor powers of modules of the $\mathrm A_n$ algebra”, Theoret. Math. Phys., 171:2 (2012), 666–674 | DOI | MR | Zbl

[13] P. Kulish, V. Lyakhovsky, O. Postnova, “Multiplicity functions for tensor powers. $\mathrm A_n$-case”, J. Phys. Conf. Series, 343 (2012), Paper 012070 | DOI

[14] I. G. Macdonald, “Some conjectures for root systems”, J. Math. Anal., 13:6 (1982), 988–1007 | MR | Zbl