Acta Medica Okayama6512002Littlewood's multiple formula for spin characters of symmetric groups19ENHiroshiMizukawaHiro-fumiYamada<p>This paper deals with some character values of the symmetric group S<sub>n</sub> as well as its double cover ~S<sub>n</sub>.<br />
 Let x<sup>λ</sup>(p) be the irreducible character of S<sub>n</sub>, indexed by the partition λ and evaluated at the conjugacy class p. Comparing the character tables of S<sub>2</sub> and S<sub>4</sub>, one observes that<br />
<br />
x<sup>(4)</sup>(2p)=x<sup>(2)</sup>(p)<br />
x<sup>(2<sup>2</sup>)</sup>(2p)=x<sup>(2)</sup>(p)+x<sup>(1<sup>(2)</sup>)</sup>(p)<br />
<br />
for p = (2), 2p = (4) and p = (1<sup>2</sup>), 2p = (2<sup>2</sup>). 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.<br />
 Our main objective is to obtain the spin character version of Littlewood's multiple formula (Theorem 2.3). Let ζ<sup>λ</sup>(p) be the irreducible negative character of ~S<sub>n</sub> (cf. [1]),
indexed by the strict partition λ and evaluated at the conjugacy class p. One finds
character tables (ζ<sup>λ</sup>(p)) in [1] for n ≤ 14. This time we evidently see that<br />
<br />
ζ<sup>3λ</sup>(3p) = ζ<sup>λ</sup>(p)<br />
<br />
for λ = (4),(3; 1) and p = (3,1),(1<sup>4</sup>). 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.</p>
No potential conflict of interest relevant to this article was reported.Academic Press Inc Elsevier ScienceActa Medica Okayama0196-88584042008Mixed expansion formula for the rectangular Schur functions and the affine Lie algebra A(1)((1))514535ENTakeshiIkedaHiroshiMizukawaTatsuhiroNakajimaHiro-FumiYamadaFormulas are obtained that express the Schur S-functions indexed by Young diagrams of rectangular shape as linear combinations of "mixed" products of Schur's S- and Q-functions. The proof is achieved by using representations of the affine Lie algebra of type A(1)((1)). A realization of the basic representation that is of "D-2((2))"-type plays the central role.No potential conflict of interest relevant to this article was reported.