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A new approach to the representation theory of the symmetric groups. III: Induced representations and the Frobenius-Young correspondence. (English) Zbl 1152.20013

Summary: We give a new (inductive) proof of the classical Frobenius-Young correspondence between irreducible complex representations of the symmetric group and Young diagrams, using the new approach, suggested by A. Okounkov and the author [part I, Sel. Math., New Ser. 2, No. 4, 581-605 (1996; Zbl 0959.20014), part II, Zap. Nauchn. Semin. POMI 307, 57-98 (2004); translation in J. Math. Sci., New York 131, No. 2, 5471-5490 (2005; Zbl 1083.20502)], to determining this correspondence. We also give linear relations between Kostka numbers that follow from the decomposition of the restrictions of induced representations to the previous symmetric subgroup. We consider a realization of representations induced from Young subgroups in polylinear forms and describe its relation to Specht modules.

MSC:

20C30 Representations of finite symmetric groups
05E10 Combinatorial aspects of representation theory
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