An explicit formula for the characters of the symmetric group

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 [5] 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 [5, 9]. The purpose of this paper is to give such an explicit expression in the general case. The irreducible representations of the symmetric group Sn of n letters are labelled by partitions λ of n (i.e. weakly decreasing sequences of positive integers summing to n). Their characters χ are evaluated at a conjugacy class of Sn, labelled by a partition μ giving the cycle-type of the class. Let χμ be the value of the character χ λ at a permutation of cycle-type μ. We shall give an explicit formula for the normalized character χ̂μ = χμ/dimλ. This result was announced in [20]. It should be first emphasized that our formula gives the dependence of χ̂μ with respect to λ in terms of the “contents” of this partition. More precisely the normalized character χ̂μ is expressed as some (unique) symmetric function evaluated on the contents of λ. This description of characters by content evaluation was proved in [13] and [3]. Previously the importance of contents had been apparent from the works of Jucys [11] and Murphy [24]. The fact had been noticed by Suzuki [29], Lascoux and Thibon [15] and Garsia [6].