Asymptotic representation theory of the symmetric group and by S. V. Kerov

By S. V. Kerov

This booklet reproduces the doctoral thesis written through a notable mathematician, Sergei V. Kerov. His premature demise at age fifty four left the mathematical neighborhood 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 thought. The booklet is a distinct resource of data at the vital subject of present examine. Asymptotic illustration idea of symmetric teams offers with difficulties of 2 forms: asymptotic houses of representations of symmetric teams of enormous order and representations of the proscribing item, i.e., the endless symmetric crew. 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 style, 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 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 team. This ends up in the learn 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 popular within the combinatorics of finite younger diagrams. In flip, this building presents a very new description of the relation among the classical second difficulties of Hausdorff and Markov. The booklet is appropriate for graduate scholars and study mathematicians drawn to illustration idea and combinatorics.

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C) If (a) holds for all z € Zq, then it holds for all z G Zq+1. We prove (c). Let x G Ml,y G Ms be such that Eiy = 0. By the assumption of (c), we have that (a) holds for z = x® F^y. By the earlier part of the proof, it follows that (a) holds for z\ = [q + \}%x F^q+1^y + v*~2qFiXF^y. Again by the assumption of (c), we have that (a) holds for It follows that (a) holds for z3 = [q + l]iX®F^+1)y; = v*~2qFiX®Flq)y. since [q + l]i ^ 0, it also holds for Thus (c) is proved. It remains to prove (b).

Assume first that m = 0. If x G Mn(0), then by definition, (c—s n )x = 0. Assume now that m > 0 and that the result is already known for m - 1. If x G M n ( m ) , then EiX G Mn+2(m - 1). By the induction hypothesis, we have (c — s n + 2 ) ( c — s n +4) • • • (c — sn+2Tn)EiX = 0. Since = 0. cEi = EiC, it follows that Ei(c - s n +2)(c - s n +4) • • • (c - sn+2m)x Applying Fit we get F i F i ( c - s n + 2 ) ( c - s n + 4 ) • • • ( c - s n + 2 r n ) x = 0. Hence, by the definition of c, we have (c —s n )(c —s n + 2 )(c —5 n+ 4) * • • ( c — s n + 2 m ) x = 0.

Let (U 0 U)f be the completion of the vector space U ® U with respect to the descending sequence of vector spaces for N = 1,2, Note that each Hn is a left ideal in U 0 U ; moreover, for any u G U 0 U, we can find r > 0 such that C TY^v for all N > 0. It follows that the Q(^)-algebra structure on U 0 U extends by continuity to a Q(t;)-algebra structure on (U® U)f. ;)-algebra homomorphism given by A(x) = A(x) for all x G U. 2. (a) There is a unique family of elements G U~ ® U+ B^ G (U ® U)f satisfies (Wtfi z/ G N [ / ] j swc/i that 0 O = 1 0 1 and 0 = a basis of for (b) B be a Q(v)-basis off such that By = BCis any v.

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