Entropy of Schur-Weyl measures

Mkrtchyan, Sevak

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 $N$ième 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 $\frac{N}{\sqrt{n}}$ converge vers une constante. Le principal résultat de cet article est la preuve de cette conjecture.

Relative dimensions of isotypic components of $N$th 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 $\frac{N}{\sqrt{n}}$ converges to a constant. The main result of the paper is the proof of this conjecture.

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

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

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

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

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

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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

[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 | Zbl

[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 | Zbl

[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

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

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