Three Theorems on the Uniqueness of the Plancherel Measure from Different Viewpoints

A. M. Vershik

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebraic topology, combinatorics, and mathematical physics, Tome 305 (2019), pp. 71-85.

Résumé

We consider three uniqueness theorems: one from the theory of meromorphic functions, another from asymptotic combinatorics, and the third concerns representations of the infinite symmetric group. The first theorem establishes the uniqueness of the function $\exp z$ in a class of entire functions. The second is about the uniqueness of a random monotone nondegenerate numbering of the two-dimensional lattice $\mathbb Z^2_+$, or of a nondegenerate central measure on the space of infinite Young tableaux. And the third theorem establishes the uniqueness of a representation of the infinite symmetric group $\mathfrak S_\mathbb N$ whose restrictions to finite subgroups have vanishingly few invariant vectors. However, in fact all the three theorems are, up to a nontrivial rephrasing of conditions from one area of mathematics in terms of another area, the same theorem! Up to now, researchers working in each of these areas have not been aware of this equivalence. The parallelism of these uniqueness theorems on the one hand and the difference of their proofs on the other call for a deeper analysis of the nature of uniqueness and suggest transferring the methods of proof between the areas. More exactly, each of these theorems establishes the uniqueness of the so-called Plancherel measure, which is the main object of our paper. In particular, we show that this notion is general for all locally finite groups.

Bibliographie

[1] Aissen M., Schoenberg I.J., Whitney A.M., “On the generating functions of totally positive sequences. I”, J. anal. math., 2 (1952), 93–103 | DOI | Zbl

[2] I. N. Bernshtein, I. M. Gel'fand, and S. I. Gel'fand, “Models of representations of compact Lie groups”, Funct. Anal. Appl., 9:4 (1975), 322–324 | DOI

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

[4] Buchstaber V.M., Glutsyuk A.A., “Total positivity, Grassmannian and modified Bessel functions”, Functional analysis and geometry: Selim Grigorievich Krein centennial, Contemp. Math., 733, Amer. Math. Soc., Providence, RI, 2019, 97–107 ; arXiv: 1708.02154 [math.DS] | DOI

[5] Bufetov A., Gorin V., “Stochastic monotonicity in Young graph and Thoma theorem”, Int. Math. Res. Not., 2015:23 (2015), 12920–12940 | Zbl

[6] J. Dixmier, Les $C^*$-algèbres et leurs represéntations, Gauthier-Villars, Paris, 1969 | Zbl

[7] Edrei A., “On the generating functions of totally positive sequences. II”, J. anal. math., 2 (1952), 104–109 | DOI | Zbl

[8] Fomin S., “Duality of graded graphs”, J. Algebr. Comb., 3:4 (1994), 357–404 | DOI | Zbl

[9] Fomin S., Zelevinsky A., “Total positivity: tests and parametrizations”, Math. Intell., 22:1 (2000), 23–33 ; arXiv: math/9912128 [math.RA] | DOI | Zbl

[10] Gantmacher F.R., Krein M.G., Oscillation matrices and kernels and small vibrations of mechanical systems, Rev. ed., AMS Chelsea Publ., Providence, RI, 2002 | Zbl

[11] Karlin S., Total positivity, v. 1, Stanford Univ. Press, Stanford, CA, 1968 | Zbl

[12] Kerov S.V., “Generalized Hall–Littlewood symmetric functions and orthogonal polynomials”, Representation theory and dynamical systems, Adv. Sov. Math., 9, Amer. Math. Soc., Providence, RI, 1992, 67–94

[13] Kerov S.V., Asymptotic representation theory of the symmetric group and its applications in analysis, Transl. Math. Monogr., 219, Amer. Math. Soc., Providence, RI, 2003 | DOI | Zbl

[14] A. A. Klyachko, “Centralizers of involutions and models of the symmetric and full linear groups”, Studies in Number Theory, v. 7, Saratov. Gos. Univ., Saratov, 1978, 59–64

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

[16] Lusztig G., “A survey of total positivity”, Milan J. Math., 76 (2008), 125–134 ; arXiv: 0705.3842 [math.RT] | DOI | Zbl

[17] Matveev K., “Macdonald-positive specializations of the algebra of symmetric functions: Proof of the Kerov conjecture”, Ann. Math. Ser. 2, 189:1 (2019), 277–316 | DOI | Zbl

[18] A. Yu. Okounkov, “Thoma's theorem and representations of the infinite bisymmetric group”, Funct. Anal. Appl., 28:2 (1994), 100–107 | DOI | Zbl

[19] A. Okounkov and G. Olshanski, “Shifted Schur functions”, St. Petersburg Math. J., 9:2 (1998), 239–300 | Zbl

[20] F. V. Petrov, “The asymptotics of traces of paths in the Young and Schur graphs”, Zap. Nauchn. Semin. POMI, 468 (2018), 126–137

[21] Romik D., Śniady P., “Jeu de taquin dynamics on infinite Young tableaux and second class particles”, Ann. Probab., 43:2 (2015), 682–737 | DOI | Zbl

[22] Schoenberg I.J., “Some analytical aspects of the problem of smoothing”, Studies and essays: Courant anniversary volume, Interscience, New York, 1948, 351–370

[23] J.-P. Serre, Représentations linéaires des groupes finis, Hermann, Paris, 1967

[24] Śniady P., “Robinson–Schensted–Knuth algorithm, jeu de taquin, and Kerov–Vershik measures on infinite tableaux”, SIAM J. Discrete Math., 28:2 (2014), 598–630 | DOI | Zbl

[25] Stanley R.P., “Differential posets”, J. Amer. Math. Soc., 1:4 (1988), 919–961 | DOI | Zbl

[26] Thoma E., “Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe”, Math. Z., 85 (1964), 40–61 | DOI | Zbl

[27] A. M. Vershik, “A statistical sum associated with Young diagrams”, J. Sov. Math., 47:2 (1989), 2379–2386 | DOI | Zbl

[28] Vershik A.M., “Asymptotic combinatorics and algebraic analysis”, Proc. Int. Congr. Math. (Zürich, 1994), v. II, Birkhäuser, Basel, 1995, 1384–1394 | DOI | Zbl

[29] A. M. Vershik, “Statistical mechanics of combinatorial partitions, and their limit shapes”, Funct. Anal. Appl., 30:2 (1996), 90–105 | DOI | DOI | Zbl

[30] A. M. Vershik, “The problem of describing central measures on the path spaces of graded graphs”, Funct. Anal. Appl., 48:4 (2014), 256–271 | DOI | DOI | Zbl

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

[32] A. M. Vershik and S. V. Kerov, “Characters and factor representations of the infinite symmetric group”, Sov. Math. Dokl., 23 (1982), 389–392 | Zbl

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

[34] A. M. Vershik and S. V. Kerov, “Locally semisimple algebras. Combinatorial theory and the $K_0$-functor”, J. Sov. Math., 38:2 (1987), 1701–1733 | DOI | Zbl

[35] Vershik A.M., Kerov S.V., “The characters of the infinite symmetric group and probability properties of the Robinson–Schensted–Knuth algorithm”, SIAM J. Algebr. Discrete Methods, 7 (1986), 116–124 | DOI | Zbl

[36] A. M. Vershik and A. A. Shmidt, “Limit measures arising in the asymptotic theory of symmetric groups. I”, Theory Probab. Appl., 22:1 (1977), 70–85 | DOI | Zbl

[37] A. M. Vershik and A. A. Shmidt, “Limit measures arising in the asymptotic theory of symmetric groups. II”, Theory Probab. Appl., 23:1 (1978), 36–49 | DOI | Zbl | Zbl

[38] Whitney A.M., “A reduction theorem for totally positive matrices”, J. anal. math., 2 (1952), 88–92 | DOI | Zbl