Acta Medica Okayama 65 1 2002 Littlewood's multiple formula for spin characters of symmetric groups 1 9 EN Hiroshi Mizukawa Hiro-fumi Yamada <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 /> &#8195;Let x<sup>&#955;</sup>(p) be the irreducible character of S<sub>n</sub>, indexed by the partition &#955; 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 . We include a proof that is based on an `inflation' of the variables in a Schur function. This is different from one given in , and we claim that it is more complete than the one given there.<br /> &#8195;Our main objective is to obtain the spin character version of Littlewood's multiple formula (Theorem 2.3). Let &#950;<sup>&#955;</sup>(p) be the irreducible negative character of ~S<sub>n</sub> (cf. ), indexed by the strict partition &#955; and evaluated at the conjugacy class p. One finds character tables (&#950;<sup>&#955;</sup>(p)) in  for n &#8804; 14. This time we evidently see that<br /> <br /> &#950;<sup>3&#955;</sup>(3p) = &#950;<sup>&#955;</sup>(p)<br /> <br /> for &#955; = (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 Science Acta Medica Okayama 0196-8858 40 4 2008 Mixed expansion formula for the rectangular Schur functions and the affine Lie algebra A(1)((1)) 514 535 EN Takeshi Ikeda Hiroshi Mizukawa Tatsuhiro Nakajima Hiro-Fumi Yamada Formulas 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.