By S. V. Kerov

This publication reproduces the doctoral thesis written by means of a awesome mathematician, Sergei V. Kerov. His premature dying at age fifty four left the mathematical group with an in depth physique of labor and this distinctive monograph. In it, he offers a transparent and lucid account of effects and strategies of asymptotic illustration idea. The e-book is a special resource of data at the very important subject of present study. Asymptotic illustration idea of symmetric teams bargains with difficulties of 2 varieties: asymptotic homes of representations of symmetric teams of huge order and representations of the restricting item, i.e., the endless symmetric workforce. the writer contributed considerably within the improvement of either instructions. His ebook provides 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 proscribing 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 restricting curve. one of the difficulties of the second one variety, Kerov reviews an immense challenge of computing irreducible characters of the endless symmetric staff. This results in the learn of a continuing analog of the inspiration of younger diagram, and specifically, to a continual 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 publication is appropriate for graduate scholars and study mathematicians attracted to illustration concept and combinatorics.

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Gn . Thus every element a e A(G) can be written uniquely: n Now define a multiplication in A{G) as follows: if a = Σ ; = 1 / ^ · , then n ad = ^fifjgii. gn = the g r o u P element gigj . It is a routine matter to check the validity of the rules: {aa)a" = a(a'a")\ {a + a)a" = aa" + a a"; a"(a + a) = a" a + a" a 58 and agx = gxa, 59 14. THE GROUP ALGEBRA ifg 1 is the identity of G. Thus A(G) is an algebra, with an identity, of rank n over F . It is the group algebra of G over F. 2) Lemma. Every representation σ of A(G) induces a representation è of G and conversely.

Vr tl e Ä r t . Pi(at) and the proof is complete. 11. Intertwining Numbers In this section all modules considered a r e F - i ? modules with a finite F-basfe. This includes the ring R itself, as a module. Hence R is an algebra. This means that R has a basis consisting of a finite number of elements of R, and every element of R is a unique linear combination of these basis elements with coefficients fromF. As a consequence the double chain condition holds for ideals in R. We assume again that R has an identity.

Mr s , for some s]. Thus, since rn Φ 0, f'\ — T = 0, therefore proving that rank Ο(μ) = 1. Let R± be an indecomposable component of the ring R, R = Rx 0 · · · . 2(a)]. Suppose that the representation module M of a representation μ has the composition series: M = M0^> M1D -D Mt = 0. Then the factors M^JMi are representation modules for the irreducible constituents of μ. We have: 52 II. 10) Theorem. t(Rx, M ) = qk where q is the number of factors of M g^. RJR'i and k is the rank of the skew field CfôJRl).