Entropy of Schur-Weyl measures
Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 678-713.

Les dimensions relatives des composants isotypiques des représentations tensorielles du Nième ordre du groupe symétrique sur n lettres induisent une mesure du type Plancherel sur l’espace des diagrammes de Young avec n cellules et au plus N rangs. G. Olshanski a conjecturé que ces dimensions, après renormalisation, convergent vers une constante sous cette famille de mesures du type Plancherel dans la limite où N n converge vers une constante. Le principal résultat de cet article est la preuve de cette conjecture.

Relative dimensions of isotypic components of Nth order tensor representations of the symmetric group on n letters give a Plancherel-type measure on the space of Young diagrams with n cells and at most N rows. It was conjectured by G. Olshanski that dimensions of isotypic components of tensor representations of finite symmetric groups, after appropriate normalization, converge to a constant with respect to this family of Plancherel-type measures in the limit when N n converges to a constant. The main result of the paper is the proof of this conjecture.

DOI : https://doi.org/10.1214/12-AIHP519
Classification : 05D40,  05E10,  20C30,  60C05
Mots clés : asymptotic representation theory, Schur-Weyl duality, Plancherel measure, Schur-Weyl measure, Vershik-Kerov conjecture
@article{AIHPB_2014__50_2_678_0,
     author = {Mkrtchyan, Sevak},
     title = {Entropy of Schur-Weyl measures},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {678--713},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {2},
     year = {2014},
     doi = {10.1214/12-AIHP519},
     zbl = {1290.05148},
     mrnumber = {3189089},
     language = {en},
     url = {http://archive.numdam.org/item/AIHPB_2014__50_2_678_0/}
}
Mkrtchyan, Sevak. Entropy of Schur-Weyl measures. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 678-713. doi : 10.1214/12-AIHP519. http://archive.numdam.org/item/AIHPB_2014__50_2_678_0/

[1] P. Biane. Approximate factorization and concentration for characters of symmetric groups. Int. Math. Res. Not. 4 (2001) 179-192. | MR 1813797 | Zbl 1106.20304

[2] A. Borodin and J. Kuan. Asymptotics of Plancherel measures for the infinite-dimensional unitary group. Adv. Math. 219 (2008) 894-931. | MR 2442056 | Zbl 1153.60058

[3] A. Borodin, A. Okounkov and G. Olshanski. Asymptotics of Plancherel measures for symmetric groups. J. Amer. Math. Soc. 13 (2000) 481-515. | MR 1758751 | Zbl 0938.05061

[4] A. Borodin and G. Olshanski. Asymptotics of Plancherel-type random partitions. J. Algebra 313 (2007) 40-60. | MR 2326137 | Zbl 1117.60051

[5] A. Borodin and G. Olshanski. The boundary of the Gelfand-Tsetlin graph: A new approach. Adv. Math. 230 (2012) 1738-1779. | MR 2927353 | Zbl 1245.05131

[6] A. I. Bufetov. On the Vershik-Kerov conjecture concerning the Shannon-Macmillan-Breiman theorem for the Plancherel family of measures on the space of Young diagrams. Geom. Funct. Anal. 22 (2012) 938-979. | MR 2984121 | Zbl 1254.05024

[7] W. Fulton and J. Harris. Representation Theory. A First Course. Graduate Texts in Mathematics 129. Springer-Verlag, New York, 1991. | MR 1153249 | Zbl 0744.22001

[8] K. Johansson. Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. Math. 153 (2001) 259-296. | MR 1826414 | Zbl 0984.15020

[9] B. F. Logan and L. A. Shepp. A variational problem for random Young tableaux. Adv. Math. 26 (1977) 206-222. | MR 1417317 | Zbl 0363.62068

[10] S. Mkrtchyan. Asymptotics of the maximal and the typical dimensions of isotypic components of tensor representations of the symmetric group. European J. Combin. 33 (2012) 1631-1652. | MR 2923474 | Zbl 1248.20012

[11] A. Okounkov. Symmetric functions and random partitions. In Symmetric Functions 2001: Surveys of Developments and Perspectives 223-252. NATO Sci. Ser. II Math. Phys. Chem. 74. Kluwer Acad. Publ., Dordrecht, 2002. | MR 2059364 | Zbl 1017.05103

[12] A. Okounkov and G. Olshanski. Asymptotics of Jack polynomials as the number of variables goes to infinity. Int. Math. Res. Not. 13 (1998) 641-682. | MR 1636541 | Zbl 0913.33004

[13] G. Olshanski. Difference operators and determinantal point processes. Funct. Anal. Appl. 42 (2008) 317-329. | MR 2492429 | Zbl 1157.60319

[14] G. Olshanski. Asymptotic representation theory: Lectures at Independent University of Moscow II, Lecture Notes, 2009, available at http://www.iitp.ru/en/userpages/88/.

[15] A. Soshnikov. Determinantal random point fields. Uspekhi Mat. Nauk 55 (2000) 107-160. English translation: Russian Math. Surveys 55 (2000) 923-975. | MR 1799012 | Zbl 0991.60038

[16] A. M. Vershik and S. V. Kerov. Asymptotics of the Plancherel measure of the symmetric group. Soviet Math. Dokl. 18 (1977) 527-531. | Zbl 0406.05008

[17] A. M. Vershik and S. V. Kerov. Characters and factor representations of the infinite unitary group. Soviet Math. Dokl. 26 (1982) 570-574. | MR 681202 | Zbl 0524.22017

[18] A. M. Vershik and S. V. Kerov. Asymptotic behavior of the maximum and generic dimensions of irreducible representations of the symmetric group. Funktsional. Anal. i Prilozhen. 19 (1985) 25-36. | MR 783703 | Zbl 0592.20015

[19] A. M. Vershik and D. Pavlov. Some numerical and algorithmical problems in the asymptotic representation theory. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 373 (2009) 77-93, 346-347. | Zbl 1288.20012

[20] D. Voiculescu. Représentations factorielles de type II1 de U(infty). J. Math. Pures Appl. 55 (1976) 1-20. | MR 442153 | Zbl 0352.22014

[21] H. Weyl. The Classical Groups: Their Invariants and Representations. Princeton Univ. Press, Princeton, NJ, 1939. | MR 1488158 | Zbl 1024.20502