Littlewood's multiple formula for spin characters of symmetric groups

This paper deals with some character values of the symmetric group S_n as well as its double cover ~S_n.
 Let x^λ(p) be the irreducible character of S_n, 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 ζ^λ(p) be the irreducible negative character of ~S_n (cf. [1]), indexed by the strict partition λ and evaluated at the conjugacy class p. One finds character tables (ζ^λ(p)) in [1] for n ≤ 14. This time we evidently see that

ζ^3λ(3p) = ζ^λ(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.

Digital Object Identifier:10.1112/S0024610701002800
Published with permission from the copyright holder. This is the institute's copy, as published in Journal of the London Mathematical Society, Feb. 2002, Volume 65, Issue 1, Pages 1-9.
Publisher URL:http://dx.doi.org/10.1112/S0024610701002800
Copyright © 2002 London Mathematical Society. All rights reserved.