start-ver=1.4 cd-journal=joma no-vol=65 cd-vols= no-issue=1 article-no= start-page=1 end-page=9 dt-received= dt-revised= dt-accepted= dt-pub-year=2002 dt-pub=20022 dt-online= en-article= kn-article= en-subject= kn-subject= en-title= kn-title=Littlewood's multiple formula for spin characters of symmetric groups en-subtitle= kn-subtitle= en-abstract= kn-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

ζ(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.

en-copyright= kn-copyright= en-aut-name=MizukawaHiroshi en-aut-sei=Mizukawa en-aut-mei=Hiroshi kn-aut-name= kn-aut-sei= kn-aut-mei= aut-affil-num=1 ORCID= en-aut-name=YamadaHiro-fumi en-aut-sei=Yamada en-aut-mei=Hiro-fumi kn-aut-name= kn-aut-sei= kn-aut-mei= aut-affil-num=2 ORCID= affil-num=1 en-affil= kn-affil=Hokkaido University affil-num=2 en-affil= kn-affil=Okayama University END start-ver=1.4 cd-journal=joma no-vol=40 cd-vols= no-issue=4 article-no= start-page=514 end-page=535 dt-received= dt-revised= dt-accepted= dt-pub-year=2008 dt-pub=200805 dt-online= en-article= kn-article= en-subject= kn-subject= en-title= kn-title=Mixed expansion formula for the rectangular Schur functions and the affine Lie algebra A(1)((1)) en-subtitle= kn-subtitle= en-abstract= kn-abstract=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. en-copyright= kn-copyright= en-aut-name=IkedaTakeshi en-aut-sei=Ikeda en-aut-mei=Takeshi kn-aut-name= kn-aut-sei= kn-aut-mei= aut-affil-num=1 ORCID= en-aut-name=MizukawaHiroshi en-aut-sei=Mizukawa en-aut-mei=Hiroshi kn-aut-name= kn-aut-sei= kn-aut-mei= aut-affil-num=2 ORCID= en-aut-name=NakajimaTatsuhiro en-aut-sei=Nakajima en-aut-mei=Tatsuhiro kn-aut-name= kn-aut-sei= kn-aut-mei= aut-affil-num=3 ORCID= en-aut-name=YamadaHiro-Fumi en-aut-sei=Yamada en-aut-mei=Hiro-Fumi kn-aut-name= kn-aut-sei= kn-aut-mei= aut-affil-num=4 ORCID= affil-num=1 en-affil= kn-affil=Department of Applied Mathematics, Okayama University of Science affil-num=2 en-affil= kn-affil=Department of Mathematics, National Defense Academy affil-num=3 en-affil= kn-affil=Faculty of Economics, Meikai University, affil-num=4 en-affil= kn-affil=Department of Mathematics, Okayama University en-keyword=Schur function kn-keyword=Schur function en-keyword=Schur's Q-function kn-keyword=Schur's Q-function en-keyword=Boson-fermion correspondence kn-keyword=Boson-fermion correspondence END