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An explicit formula for the characters of the symmetric group. (English) Zbl 1182.20011
From the introduction: The characters of the irreducible representations of the symmetric group play an important role in many areas of mathematics. However, since the early work of Frobenius in 1900, no explicit formula was found for them. The characters of the symmetric group were computed through various recursive algorithms, but explicit formulas were only known for about ten particular cases. The purpose of this paper is to give such an explicit expression in the general case.
The irreducible representations of the symmetric group \(S_n\) of \(n\) letters are labelled by partitions \(\lambda\) of \(n\) (i.e. weakly decreasing sequences of positive integers summing to \(n\)). Their characters \(\chi^\lambda\) are evaluated at a conjugacy class of \(S_n\), labelled by a partition \(\mu\) giving the cycle-type of the class. Let \(\chi^\lambda_\mu\) be the value of the character \(\chi^\lambda\) at a permutation of cycle-type \(\mu\). We shall give an explicit formula for the normalized character \(\widehat\chi^\lambda_\mu=\chi^\lambda_\mu/\!\dim\lambda\). This result was announced in [the author, C. R., Math., Acad. Sci. Paris 341, No. 9, 529-534 (2005; Zbl 1081.20014)].
It should be first emphasized that our formula gives the dependence of \(\widehat\chi^\lambda_\mu\) with respect to \(\lambda\) in terms of the “contents” of this partition. More precisely the normalized character \(\widehat\chi^\lambda_\mu\) is expressed as some (unique) symmetric function evaluated on the contents of \(\lambda\).
However the symmetric function expressing \(\widehat\chi^\lambda_\mu\) remained quite obscure, even in the very elementary situation of a partition \(\mu\) having only one non-unary part. The purpose of this paper is to give an explicit expression.
It is a second remarkable fact that this symmetric function can only be written by using a new family of positive integers, which we have introduced in [Ann. Comb. 6, No. 3-4, 399-405 (2002; Zbl 1017.05004)]. The connection of these integers with the symmetric group is still mysterious and certainly needs more investigation.
We emphasize that our method provides a very efficient algorithm, implemented on computer. Tables giving \(\widehat\chi^\lambda_\mu\) for \(|\mu|-l(\mu)\leq 12\) are available on a web page [M. Lassalle, available at http://igm.univ-mlv.fr/~lassalle/char.html].

MSC:
20C30 Representations of finite symmetric groups
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
05A17 Combinatorial aspects of partitions of integers
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