By S. V. Kerov
This booklet reproduces the doctoral thesis written by way of a impressive mathematician, Sergei V. Kerov. His premature loss of life at age fifty four left the mathematical neighborhood with an intensive physique of labor and this exceptional monograph. In it, he offers a transparent and lucid account of effects and strategies of asymptotic illustration idea. The publication is a distinct resource of knowledge at the very important subject of present examine. Asymptotic illustration conception of symmetric teams offers with difficulties of 2 kinds: asymptotic homes of representations of symmetric teams of enormous order and representations of the restricting item, i.e., the endless symmetric workforce. the writer contributed considerably within the improvement of either instructions. His e-book offers an account of those contributions, in addition to these of alternative researchers. one of the difficulties of the 1st kind, the writer discusses the houses of the distribution of the normalized cycle size in a random permutation and the restricting form of a random (with admire to the Plancherel degree) younger diagram. He additionally experiences stochastic houses of the deviations of random diagrams from the proscribing curve. one of the difficulties of the second one style, Kerov stories a major challenge of computing irreducible characters of the countless symmetric workforce. This results in the examine of a continuing analog of the inspiration of younger diagram, and particularly, to a continuing analogue of the hook stroll set of rules, that is renowned within the combinatorics of finite younger diagrams. In flip, this development offers a very new description of the relation among the classical second difficulties of Hausdorff and Markov. The e-book is appropriate for graduate scholars and examine mathematicians attracted to illustration thought and combinatorics.
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Additional resources for Asymptotic Representation Theory of the Symmetric Group and its Application in Analysis
0. Important results on the asymptotics of the shape of Young diagrams in the course of the Plancherel growth process were obtained by Vershik and the author , ,and independently by Logan and Shepp . Most problems considered in this book are motivated by these results. It will be more convenient to discuss the limit shape of Young diagrams if we slightly modify their original description. Following the combinatorial tradition, we have been believing up t o now that Young diagrams are merely graphic representations of partitions of positive integers*).
Denote by X X the character of the irreducible representation of the group 6, associated with a Young diagram X E Yn, and let X: be its value on the class of conjugate permutations with cycle structure p = (Ir1,2rZ,.. ). For a fixed p E y,, the character is a random variable defined on the set Y, with the Plancherel measure M,. We are interested in the asymptotic behaviour of the distribution of this variable as n --, co. To obtain a nontrivial limiting distribution, we must normalize the values of characters in an appropriate way.
The tensors from the subspace V,,,(X) are said to have symmetry type A. n(A) = dim Vn,m(X) dim V,,, , XEYn,,, the relative dimension of the primary component V,,,(X). The weights h/ln,,(X) determine a probability measure on y,,,, the distribution of tensors into symmetry types. It turns out that this distribution obeys the arcsine law as n , m + oo. 0. 4). Assume that n, m lim $ = y > 0. Then + cc so that In the case when n + cc and m is fixed, the asymptotics is completely different. Let us define the reduced row lengths of a Young diagram X E yn,, by Then the vector x = ( x l , .