LITTLEWOOD'S MULTIPLE FORMULA FOR SPIN CHARACTERS OF SYMMETRIC GROUPS

Abstract

This paper deals with some character values of the symmetric group Sn as well as its double cover ~Sn. Let xλ(p) be the irreducible character of Sn, indexed by the partition λ and evaluated at the conjugacy class p. Comparing the character tables of S2 and S4, one observes that x(4)(2p)=x(2)(p) x(22)(2p)=x(2)(p)+x(1(2))(p) for p = (2), 2p = (4) and p = (12), 2p = (22). A number of such observations lead to what we call Littlewood's multiple formula (Theorem 1.1). This formula appears in Littlewood's book [2]. We include a proof that is based on an `inflation' of the variables in a Schur function. This is different from one given in [2], and we claim that it is more complete than the one given there. Our main objective is to obtain the spin character version of Littlewood's multiple formula (Theorem 2.3). Let ζλ(p) be the irreducible negative character of ~Sn (cf. [1]), indexed by the strict partition λ and evaluated at the conjugacy class p. One finds character tables (ζλ(p)) in [1] for n ≤ 14. This time we evidently see that ζ3λ(3p) = ζλ(p) for λ = (4),(3; 1) and p = (3,1),(14). The proof of Theorem 2.3 is achieved in a way that is similar to the case of ordinary characters. Instead of a Schur function, we deal with Schur's P-function, which is defined as a ratio of Pfaffians.