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Asymptotic Representation Theory of the Symmetric Group and by S. V. Kerov

By S. V. Kerov

This publication reproduces the doctoral thesis written by means of a striking mathematician, Sergei V. Kerov. His premature dying at age fifty four left the mathematical neighborhood with an in depth physique of labor and this extraordinary monograph. In it, he offers a transparent and lucid account of effects and techniques of asymptotic illustration thought. The booklet is a special resource of knowledge at the vital subject of present learn. Asymptotic illustration thought of symmetric teams bargains with difficulties of 2 varieties: asymptotic houses of representations of symmetric teams of enormous order and representations of the proscribing item, i.e., the limitless symmetric crew. the writer contributed considerably within the improvement of either instructions. His publication offers an account of those contributions, in addition to these of different researchers. one of the difficulties of the 1st style, 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 recognize to the Plancherel degree) younger diagram. He additionally reports stochastic houses of the deviations of random diagrams from the proscribing curve. one of the difficulties of the second one kind, Kerov experiences a massive challenge of computing irreducible characters of the countless symmetric workforce. This results in the research of a continuing analog of the idea of younger diagram, and specifically, to a continuing analogue of the hook stroll set of rules, that's popular within the combinatorics of finite younger diagrams. In flip, this building offers a totally new description of the relation among the classical second difficulties of Hausdorff and Markov. The booklet is appropriate for graduate scholars and learn mathematicians drawn to illustration concept and combinatorics.

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A remarkable description of totally positive series was suggested by Schoenberg and proved by Edrei [84]. THEOREM 9. The class of totally positive formal series normalized so that H ( 0 ) = 1 coincides with the class of the Taylor series of meromorphic functions of the form > > where y 0, a, 0, pi 2 0, and x(ai+ P i ) a nonzero radius of convergence. < m. I n particular, these series have The most difficult problem here is t o prove that H ( z ) = eYZ provided that a totally positive series H ( z ) defines an integral function without zeros.

Krein (see [47]), so we call it the Krein correspondence. 5 above we mentioned a beautiful combinatorial algorithm, the hook walk, intended for the stochastic simulation of the transition probabilities of the Plancherel measure of the infinite symmetric group 6,. Here we will find its limiting version, the interval shrinkage algorithm, which can be applied to arbitrary continuous diagrams. This algorithm provides a new description of the Krein algorithm in probabilistic terms. The second important topic of Chapter 4 is the asymptotic behaviour of interlacing sequences.

The hook walk. It is shown in [36] that the hook walk algorithm can be adapted to generate more general central measures. It still seems somewhat mysterious that exactly the same class of central measures can be used for constructing topological invariants of knots in the Jones-Okneanu scheme, see [114], [93]. The link is provided by the characters of the infinite-dimensional Hecke algebra which satisfy the A. A. ) property. The description of all such characters in terms of frequencies was obtained by Vershik and the author [19].

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