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Zbl 0534.20005
Cartier; Pierre
La théorie classique et moderne des fonctions symétriques. (French)
[A] Sémin. Bourbaki, 35e année, Vol. 1982/83, Exp. No.597, Astérisque 105-106, 1-23 (1983).

[For the entire collection see Zbl 0514.00009.] \par The report is a broad (with particular emphasis on broadness) survey of different aspects of the theory of symmetric functions. Classical facts and their, often quite recent, applications and generalizations are presented in a concise and systematic manner. The report consists of 5 Sections. In the first generating series and their q-analogs appearing in the theory of partitions are surveyed. The second section gives a modern theory of symmetric functions. It considers the algebra of formal power series in an infinite number of variables (over any ring K) and its graded subalgebra F of symmetric formal power series. Different bases of F, each indexed by partitions, are introduced and the relations between them are investigated to some length. F turns out to be a Hopf bigebra. Some q-variations are also described. In Section 3 the theory of representations over ${\bbfC}$ of the symmetric groups $S\sb n$ (and, at the end, of general linear groups $G\sb n=GL\sb n({\bbfF}\sb q)$ over a fixed finite field ${\bbfF}\sb q)$, $n=1,2,..$. is related to the theory of symmetric functions. Namely, the space $R:=\oplus\sb{n\ge 0}{\bbfC}[S\sb n]$ is given using induction (and restriction) from (and to) $S\sb m\times S\sb{n-m}$ a structure of a bigebra which turns out to be isomorphic to F above. For the groups $G\sb n$ the situation was only recently described by A. Zelevinsky. He constructs a Hopf algebra similar to R above and observes close analogy with the case of groups $S\sb n$. Section 4 describes recent study of certain problems in the theory of symmetric functions by M. Schützenberger; he uses monoids constructed on an ordered alphabet. Although his constructions and results are too involved to be dealt with here, we mention that they are often of classical interest. In the concluding Section 5 the author gives applications of the theory of symmetric functions to characteristic classes, formal groups, and stochastic sequences.
[B.Weisfeiler]

MSC 2000:: *20C30 Representations of finite symmetric groups
05A15 Combinatorial enumeration problems
16W50 Graded associative rings and modules
16W30 Hopf algebras (assoc. rings and algebras)
05A17 Partitions of integres (combinatorics)
20C05 Group rings of finite groups and their modules (group theory)

Keywords: representations of symmetric groups; survey; symmetric functions; generating series; partitions; symmetric formal power series; Hopf bigebra; general linear groups

Citations: Zbl 0514.00009

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