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Representations of the infinite symmetric group. (English) Zbl 1364.20001

Cambridge Studies in Advanced Mathematics 160. Cambridge: Cambridge University Press (ISBN 978-1-107-17555-6/hbk; 978-1-316-79857-7/ebook). vii, 160 p. (2016).
Let \(G\) be a finite group. An ordinary representation for \(G\) is a homomorphism \(T\) from \(G\) to \(\mathrm{GL}^n(C)\), the multiplicative group of \(n\) by \(n\) invertible matrices over the complex field \(C\). The character afforded by \(T\) is a function \(\chi\) from \(G\) to \(C\) defined by \(\chi(g)=\mathrm{tr}(T(g))\) for all \(g\in G\). If there is no non-zero and proper subspace of \(C^n\) invariant under \(T\), then \(T\), as well as \(\chi\), is called an irreducible representation, as well as an irreducible character of \(G\). The theory of group representations came to existance when character theory of abelian groups in particular was used for the first time by Gaus and Dirichlet in the context of number theory. But later the theory was generalized by Frobenius to any finite abelian group. The representation theory of finite groups emerged around the turn of the century by the works of Frobenius, Schur, Burnside and Brauer. For a brief history of the subject, see [C. W. Curtis, Pioneers of representation theory: Frobenius, Burnside, Schur, and Brauer. Providence, RI: American Mathematical Society; London: London Mathematical Society (1999; Zbl 0939.01007)]. The representation theory and consequently the character theory of the finite symmetric group \(S_n\) is simple but an important special case. Alfred Young found a natural class of all the irreducible representations of \(S_n\) in terms of Young tableaux. For an account of irreducible characters of \(S_n\), one is referred to [D. E. Littlewood, the theory of group characters and matrix representations of groups. 2nd ed. London: Geoffrey Cumberledge; Oxford University Press (1950; Zbl 0038.16504)] and [B. E. Sagan, The symmetric group. Representations, combinatorial algorithms, and symmetric functions. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software (1991; Zbl 0823.05061)].
In the book under review, representations and characters of the infinite symmetric group are investigated. The infinite symmetric group is denoted by \(S(\infty)\) and is defined as the group of all finite permutations of the set of natural numbers \(\{1,2,3,\dots\}\). The character theory for \(S(\infty)\) was independently discovered by E. Thoma [Math. Z. 85, 40–61 (1964; Zbl 0192.12402)].
As the authors of the book mention, the aim of this book is to provide a detailed introduction to the representation theory of \(S(\infty)\) in such a way that would be accessible to graduate and advanced undergraduate students. At the end of each section of the book, there are excersices and notes which are helpful for students who choose the book for the course. The book has two parts: Part one on symetric functions and Thoma’s theorem, and Part two on unitary representations. In the first part of the book, characters of \(S(\infty)\) are described. Characters are not defined as for a finite group, but a character of a group \(G\) in general is a function \(\chi:G\to C\) that is positive definitive, constant on conjugacy classes and normalized to take value 1 at the identity element of \(G\). In the second part of the book, the authors move to representation based on the works [the second author, “Unitary representations of \((G, K)\)-pairs connected with the infinite symmetric group \(S(\infty)\)”, Algebra Anal. 1, No. 4, 178–209 (1989); S. Kerov et al., Invent. Math. 158, No. 3, 551–642 (2004; Zbl 1057.43005)].

MSC:

20-02 Research exposition (monographs, survey articles) pertaining to group theory
20C32 Representations of infinite symmetric groups
20C15 Ordinary representations and characters
20B07 General theory for infinite permutation groups
05E10 Combinatorial aspects of representation theory
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