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Characters and factor representations of the infinite symmetric group. (English. Russian original) Zbl 0534.20008

Sov. Math., Dokl. 23, 389-392 (1981); translation from Dokl. Akad. Nauk SSSR 257, 1037-1040 (1981).
From the introduction: ”The infinite symmetric group \({\mathfrak S}_{\infty}\) of all finite permutations of the natural numbers is a nontrivial example in the theory of locally finite groups or, more generally, inductive limits of locally compact groups. Its group algebra is a typical example of a locally semisimple algebra. The representations of locally finite groups and locally semisimple algebras are at present the subject of intensive study.”
”Most interesting of all are the factor representations of type \(II_ 2\) (and \(II_{\infty})\). They are given by group characters or, more precisely, finite or semifinite traces on the group algebra. On the group \({\mathfrak S}_{\infty}\) there are rather many characters, but they allow of an effective description. They were first obtained by E. Thoma [Math. Z. 85, 40-61 (1964; Zbl 0192.12402)] as solutions of functional equations. However, the corresponding factor representations of type \(II_ 1\) have not, to date, been effectively described.”
”It is the fundamental aim of this note to give natural realizations of the representations that correspond to the characters of \({\mathfrak S}_{\infty}\) and certain other groups. From these realizations another interpretation of the parameters will emerge that is connected with a generalization of the transformation of Robinson-Schensted-Knuth.”
Reviewer: J.Libicher

MSC:

20C32 Representations of infinite symmetric groups

Citations:

Zbl 0192.12402
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