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 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 S_{2} and S_{4}, 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 = (1^{2}), 2p = (2^{2}). 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),(1^{4}). 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