By S. V. Kerov
This e-book reproduces the doctoral thesis written by way of a striking mathematician, Sergei V. Kerov. His premature dying at age fifty four left the mathematical group with an intensive physique of labor and this one of a kind monograph. In it, he offers a transparent and lucid account of effects and strategies of asymptotic illustration concept. The ebook is a distinct resource of knowledge at the very important subject of present study. Asymptotic illustration thought of symmetric teams offers with difficulties of 2 kinds: asymptotic houses of representations of symmetric teams of huge order and representations of the proscribing item, i.e., the endless symmetric staff. 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 kind, the writer discusses the homes 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 homes of the deviations of random diagrams from the restricting curve. one of the difficulties of the second one style, Kerov reviews a massive challenge of computing irreducible characters of the countless symmetric team. This results in the examine of a continuing analog of the suggestion of younger diagram, and particularly, to a continuing analogue of the hook stroll set of rules, that's renowned within the combinatorics of finite younger diagrams. In flip, this building offers a very new description of the relation among the classical second difficulties of Hausdorff and Markov. The publication is acceptable for graduate scholars and examine mathematicians drawn to illustration concept and combinatorics.
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Extra info for Asymptotic Representation Theory of the Symmetric Group and its Application in Analysis
For the Thoma used the zeros and poles of parametrization of a character x E &(em), the power series where UJ, = ( 1 , 2 , . . , n) E 6, is a permutation with a single nontrivial cycle of length n. Vershik and the author  found a more adequate interpretation of the Thoma parameters which is related to the approximation of extreme characters of the group 6, by the characters of irreducible representations of the finite symmetric slibgroups 6 , . Recall that the irreducible representations of the group 6, and their characters have a standard parametrization by partitions of the number n.
1. The Plancherel measure and the partial fraction expansion. 2. Recall that we denote by X I , x2,. . , xd the points of minima, and by yl, . . , yd-1, the points of maxima of a diagram v = w(u). Condition (3) is immaterial for what follows, so we drop it. A function v = w(u) that satisfies only the two conditions (1) wl(u) = &I, and (2) there exists c E R such that w(u) = lu - cl for sufficiently large / u ( , will be called a rectangular diagram. The point c = C x k - C yk is called the centre of the diagram, and the number the area of the diagram.
Substitute two variables x = ( x l , 2 2 ) into Px(x; q, t ) , then set 21x2 = 1, and consider PAas a polynomial in one variable y = xl r 2 . + 53. T H E PLANCHEREL MEASURE O F 6, 31 This observation allows us to relate the characterization of the true Macdonald polynomials among the generalized ones to an old problem of Fejkr's [go]. This problem is stated as follows: among all polynomials of the form where bo, bl, . . is a scalar sequence, find orthogonal polynomials with respect to a certain measure.