Between "very large" and "infinite" : The asymptotic representation theory

• Autorzy: Vershik A.
• Języki publikacji: EN

Abstrakty

EN

I illustrate the historical roots of the theory which I called later “Asymptotic Representation Theory” – the theory which can be considered as a part functional analysis, representation theory, and more general – probability theory, asymptotic combinatorics, the theory of random matrices, dynamics, etc. The first and very concrete example is a remarkable (and forgotten) paper by J. von Neumann, which I try here to connect with the modern theory of random matrices; the second example is a quote of an important thought of H.Weyl about the theory of symmetric groups. In the last section I give a short review of the ideas of the asymptotic representation theory, which was developed starting from the 1970s, and now became very popular. I mention several important problems, and give a list (incomplete) of references. But the reader must remember that this is just a synopsis of the “baby talk”.

Słowa kluczowe

EN

matrix algebras; invariant subspaces; random matrices; characters

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2013
• Tom: Vol. 33, Fasc. 2
• Strony: 467--476
• Opis fizyczny: Bibliogr. 28 poz.

Twórcy

autor

Vershik A.

• St. Petersburg Department of Steklov Institute of Mathematics of the Russian Academy of Sciences, 191023 St. Petersburg, Russia

Bibliografia

• [1] T. Ceccherini-Silberstein, F. Scarabotti, and F. Tolli, Representation Theory of the Symmetric Groups: The Okounkov-Vershik Approach, Character Formulas and Partition Algebras, Cambridge Stud. Adv. Math., Vol. 121, Cambridge University Press, 2010.
• [2] J. Glimm, Type I C∗-algebras, Ann. of Math. 73 (1961), pp. 572-612.
• [3] V. Gorin, S. Kerov, and A. Vershik, Finite traces and representations of the group of infinite matrices over a finite field, PDMI preprint 1/2013 (2013). arXiv:1209.4945.
• [4] V. Ivanov and S. Kerov, The algebra of conjugacy classes in symmetric groups, and partial permutations, J. Math. Sci. 107 (5) (2001), pp. 4212-4230.
• [5] V. Ivanov and G. Olshanski, Kerov’s central limit theorem for the Plancherel measure on Young diagrams, in: Symmetric Functions 2001: Surveys of Developments and Perspectives, S. Fomin (Ed.), Kluwer Academic Publishers, Dordrecht 2002, pp. 93-151.
• [6] S. Kerov, Asymptotic Representation Theory of the Symmetric Group and Its Applications in Analysis, Transl. Math. Monogr., Vol. 219, Amer. Math. Soc., 2003.
• [7] J. von Neumann, Approximative properties of matrices of high finite order, Portugal. Math. 3 (1942), pp. 1-62. Collected Works, Vol. IV, Pergamon Press, 1962, p. 271.
• [8] J. von Neumann, Selected Papers on Functional Analysis, Vols. 1 and 2, Nauka, Moscow 1987 (in Russian). Comments to [7] by A. Vershik, Vol. 1, pp. 372-374.
• [9] A. Okounkov, On the representations of the infinite symmetric group, J. Math. Sci. 96 (5) (1995), pp. 3550-3589.
• [10] A. Okounkov and A. Vershik, A new approach to representation theory of symmetric groups, Selecta Math. 2 (4) (1996), pp. 581-605. A new approach to representation theory of symmetric groups. II, Zap. Nauchn. Sem. S.-Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 307 (2004), pp. 57-98. English translation: J. Soviet. Math. 131 (2) (2006), pp. 5471-5494.
• [11] E. Thoma, Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen symmetrischen Gruppe, Math. Z. 85 (1) (1964), pp. 40-61.
• [12] N. Tsilevich and A. Vershik, Infinite-dimensional Schur-Weyl duality and the Coxeter-Laplace operator, PDMI preprint 16/2012 (2012). arXiv:1209.4800.
• [13] A. Vershik, Local algebras and a new version of Young’s orthogonal form, in: Topics in Algebra. Parts 1, 2, Banach Center Publ. 26 (2) (1990), pp. 467-473. Local stationary algebras, in: Algebra and Analysis, Amer. Math. Soc. Transl. Ser. 2, 148 (1991), pp. 1-13.
• [14] A. Vershik, Asymptotic combinatorics and algebraic analysis, in: Proceedings of the International Congress of Mathematicians (Zürich, 1994), Vol. II, Birkhäuser, Basel, 1995, pp. 1384-1394.
• [15] A.Vershik, A new approach to the representation theory of the symmetric group. III, Induced representations and Frobenius-Young correspondence, Moscow Math. J. 6 (3) (2006), pp. 567-585.
• [16] A. Vershik, The life and fate of functional analysis in the twentieth century, in: Mathematical Events of the Twentieth Century, Springer, Berlin 2006, pp. 437-447.
• [17] A. Vershik, Long history of the Monge-Kantorovich transportation problem, Math. Intelligencer 35 (4) (2013).
• [18] A. Vershik, Smooth and non-smooth AF-algebras and problem on invariant measures, arXiv:1304.2193 (2013).
• [19] A. Vershik and S. Kerov, Characters and factor representations of the infinite symmetric group, Soviet Math. Dokl. 23 (1981), pp. 389-392.
• [20] A. Vershik and S. Kerov, Asymptotic theory of characters of the symmetric groups, Funct. Anal. Appl. 15 (1982), pp. 246-255. Asymptotics of maximal and typical dimensions of irreducible representations of a symmetric group, Funct. Anal. Appl. 19 (1985), pp. 21-31.
• [21] A. Vershik and S. Kerov, The K-functor (Grothendieck group) of the infinite symmetric group, J. Soviet Math. 28 (1985), pp. 549-568.
• [22] A. Vershik and S. Kerov, On an infinite-dimensional group over a finite field, Funct. Anal. Appl. 32 (3) (1998), pp. 147-152.
• [23] A. Vershik and S. Kerov, Four drafts on the representation theory of the group of Infinite matrices over a finite field, J. Math. Sci. 147 (6) (2007), pp. 7129-7144.
• [24] A. Vershik and N. Nessonov, Stable states and representations of the infinite symmetric group, Dokl. Math. 86 (1) (2012), pp. 450-453.
• [25] A. Vershik and A. Sergeev, A new approach to the representation theory of the symmetric group. IV, Z − 2-graded groups and algebras, Moscow Math. J. 8 (4) (2008), pp. 813-842.
• [26] A. Vershik and A. Shmidt, Symmetric group of higher degree, Soviet Math. Dokl. 206 (2) (1972), pp. 269-272.
• [27] A. Vershik and A. Shmidt, Limit measures arising in the asymptotic theory of symmetric groups I, Teor. Veroyatnost. i Primenen. 22 (1977), pp. 72-88. English translation: Theory Probab. Appl. 22 (1) (1977), pp. 70-85. Limit measures arising in the asymptotic theory of symmetric groups II, Teor. Veroyatnost. i Primenen. 23 (1) (1978), pp. 42-54. English translation: Theory Probab. Appl. 23 (1978), pp. 36-49.
• [28] H. Weyl, Philosophy of Mathematics and Natural Sciences, Princeton University Press, Princeton 1949.

Uwagi

Dedicated to the 110th anniversary of von Neumann’s birthday

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).