By S. V. Kerov
This ebook reproduces the doctoral thesis written through a extraordinary 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 specific monograph. In it, he supplies a transparent and lucid account of effects and techniques of asymptotic illustration conception. The e-book is a distinct resource of knowledge at the very important subject of present examine. Asymptotic illustration conception of symmetric teams bargains with difficulties of 2 kinds: asymptotic houses of representations of symmetric teams of enormous order and representations of the proscribing item, i.e., the endless symmetric staff. the writer contributed considerably within the improvement of either instructions. His e-book provides an account of those contributions, in addition to these of alternative researchers. one of the difficulties of the 1st sort, 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 appreciate to the Plancherel degree) younger diagram. He additionally reviews stochastic homes of the deviations of random diagrams from the restricting curve. one of the difficulties of the second one sort, Kerov reviews a big challenge of computing irreducible characters of the countless symmetric crew. This ends up in the research of a continual analog of the proposal of younger diagram, and particularly, to a continuing analogue of the hook stroll set of rules, that is 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 publication is appropriate for graduate scholars and learn mathematicians drawn to illustration idea and combinatorics.
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Additional resources for Asymptotic Representation Theory of the Symmetric Group and its Application in Analysis
The key idea of these applications consists in systematically studying, along with ordinary Young diagrams (regarded as piecewise linear functions), their uniform limits, called continuous diagrams. We will show that many constructions and algorithms, typical for the combinatorics of Young diagrams, can be extended by continuity and applied to arbitrary continuous diagrams. In this book we consider in detail the transform that associates with a Young diagram its transition probability in the Markov process of Plancherel growth.
THEOREM 12 . 5) P(A) = / n s h ( a ;a ) d p ( a ;13). E Harm(y) on the Youn,g E Y, where s x ( a ;/3) is the extended Schur function, and p is a Bore1 probability measure on the space of frequencies A. The distribution p is the weak limit of the measures i(Mn), where Mn(X) = d(X) . p(X) for X E y,. Since the problem of describing the characters of 6 , and the problem of describing totally positive series are equivalent (at the level of statements), the proof of the approximation theorem provides a new proof of the Schoenberg-Edrei theorem which does not use Nevanlinna's theory.
One of the most important parts of the problem of computing cliaracters is to find a parametrization of the set &(G) of extreme characters of tlie group G. Thoma's answer for the infinite symmetric group G = G, is the following. THEOREM 8 . Consider the space of pairs of sequences with the componentwise convergence topology. Then the set of extreme characters E(6,) is homeomorphic to A. 4) xa,0(wnt1, x ~ ; P ( w )= Z where w = w,,w,, permutations. . 2. Totally positive sequences and series.
Asymptotic Representation Theory of the Symmetric Group and its Application in Analysis by S. V. Kerov