By S. V. Kerov

This ebook reproduces the doctoral thesis written through a outstanding mathematician, Sergei V. Kerov. His premature loss of life at age fifty four left the mathematical group with an intensive physique of labor and this unique monograph. In it, he offers a transparent and lucid account of effects and techniques of asymptotic illustration idea. The publication is a different resource of knowledge at the vital subject of present study. Asymptotic illustration concept of symmetric teams offers with difficulties of 2 forms: asymptotic houses of representations of symmetric teams of huge order and representations of the proscribing item, i.e., the countless symmetric staff. the writer contributed considerably within the improvement of either instructions. His ebook provides 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 appreciate to the Plancherel degree) younger diagram. He additionally reviews stochastic houses of the deviations of random diagrams from the restricting curve. one of the difficulties of the second one kind, Kerov stories an immense challenge of computing irreducible characters of the countless symmetric crew. This results in the learn of a continual analog of the proposal of younger diagram, and specifically, to a continual analogue of the hook stroll set of rules, that's popular within the combinatorics of finite younger diagrams. In flip, this development presents a very new description of the relation among the classical second difficulties of Hausdorff and Markov. The ebook is appropriate for graduate scholars and study mathematicians drawn to illustration concept and combinatorics.

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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.

Corresponding to w, = (1 - q7') / (1 - t n ) ) : (SO) If X = ( X I , . . ,An) is a Young diagram with n nonzero rows and :r = (21,. . , x,, 0 , . 5) PA(x; w) = 21 2 2 . . 2, PA* (2;w), where A, = (A1 - 1 , . . , A n - 1). We show that this property does not hold for more general sequences w. T H E O R E20 M ([119], Theorem 2). 9) wn=(Y, wkm = a , TLXl,2, . . , W, = 1 if n $ 0 mod rn. 7). The latter are obtained as follows. 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 .

An) is a Young diagram with n nonzero rows and :r = (21,. . , x,, 0 , . 5) PA(x; w) = 21 2 2 . . 2, PA* (2;w), where A, = (A1 - 1 , . . , A n - 1). We show that this property does not hold for more general sequences w. T H E O R E20 M ([119], Theorem 2). 9) wn=(Y, wkm = a , TLXl,2, . . , W, = 1 if n $ 0 mod rn. 7). The latter are obtained as follows. 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 .