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 Zap. Nauchn. Sem. POMI, 2015, Volume 437, Pages 81–99 (Mi znsl6174)

A study of the growth of maximal and typical normalized dimensions of strict Young diagrams

N. N. Vasiliev, V. S. Duzhin

St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia

Abstract: In this paper, we investigate the asymptotics of the normalized dimensions of strict Young diagrams (i.e., the numbers of paths to vertices in the Schur graph). We describe the results of corresponding computer experiments. The strict Young diagrams parametrize the projective representations of the symmetric group $S_n$. So, the asymptotics of the normalized dimensions of diagrams gives us the asymptotics of the dimensions of projective representations as well. Sequences of strict diagrams of high dimension consisting of up to one million cells were built. It was proved by an exhaustive search that the first 250 diagrams of all these sequences have the maximum possible dimensions. Presumably, these sequences contain infinitely many diagrams of maximum dimension, and thus give the correct asymptotics of their growth. Also, we investigate the behavior of the normalized dimensions of typical diagrams with respect to the Plancherel measure on the Schur graph. The calculations strongly agree with A. M. Vershik's hypothesis on the convergence of the normalized dimensions of maximal and Plancherel typical diagrams not only for the standard Young graph, but also for the Schur graph.

Key words and phrases: Young diagrams, Plancherel process, Schur graph, projective representations, symmetric group.

 Funding Agency Grant Number Russian Science Foundation 14-11-00581

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English version:
Journal of Mathematical Sciences (New York), 2016, 216:1, 53–64

Bibliographic databases:

UDC: 517.987

Citation: N. N. Vasiliev, V. S. Duzhin, “A study of the growth of maximal and typical normalized dimensions of strict Young diagrams”, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXVI. Representation theory, dynamical systems, combinatorial methods, Zap. Nauchn. Sem. POMI, 437, POMI, St. Petersburg, 2015, 81–99; J. Math. Sci. (N. Y.), 216:1 (2016), 53–64

Citation in format AMSBIB
\Bibitem{VasDuz15} \by N.~N.~Vasiliev, V.~S.~Duzhin \paper A study of the growth of maximal and typical normalized dimensions of strict Young diagrams \inbook Representation theory, dynamical systems, combinatorial and algoritmic methods. Part~XXVI. Representation theory, dynamical systems, combinatorial methods \serial Zap. Nauchn. Sem. POMI \yr 2015 \vol 437 \pages 81--99 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6174} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3499909} \transl \jour J. Math. Sci. (N. Y.) \yr 2016 \vol 216 \issue 1 \pages 53--64 \crossref{https://doi.org/10.1007/s10958-016-2887-x} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84969784610} 

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This publication is cited in the following articles:
1. Duzhin V.S., Vasilyev N.N., “Asymptotic behavior of normalized dimensions of standard and strict Young diagrams — growth and oscillations”, J. Knot Theory Ramifications, 25:12, SI (2016), 1642002
2. V. Duzhin, N. Vassiliev, “Randomized Schützenberger's jeu de taquin and approximate calculation of co-transition probabilities of a central Markov process on the 3D Young graph”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye metody. XXXI, Zap. nauchn. sem. POMI, 485, POMI, SPb., 2019, 90–106
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