PRINCIPALLY SPECIALIZED CHARACTERS OF ŝl(m|1)-MODULES

In this paper, we calculate a series of principally specialized characters of the ŝl(m|1)-modules of level 1. In particular, we show that the principally specialized characters of the basic modules L(Λ0) is expressed as an infinite product. In addition, we deduce the specialized character formula of “quasiparticle” type. 0. INTRODUCTION Character formulas of basic representation of ŝl(m|n) are given in [KW2], based on their explicit construction in terms of bosonic and fermionic fields. In this paper, we calculate the principally specialized characters of some ŝl(m|1)modules. In §1, we describe that the principally specialized characters of the basic ŝl(m|1)-modules L(Λ0) is expressed as an infinite product. In §2, we deduce the specialized character formula of “quasiparticle” type. We follow notation and terminologies from [KW2] without repeating their explanation. 1. SPECIALIZED CHARACTER FORMULA FOR SOME SERIES OF ŝl(m|1)-MODULES Throughout this paper, we assume that m ≥ 2 and let α0, . . . , αm denote the set of simple roots for ŝl(m|1) where α0 and αm are odd and αi(i = 1, . . . , m − 1) are even. Provided that all si are positive integers, the sequence s = (s0, . . . , sm) defines a homomorphism Fs : C[[e0 , . . . , em ]] −→ C[[q]] by Fs(ei) = qi(i = 0, . . . , m), called the specialization of type s. In this paper, we consider the specialization of type s = (1, . . . , 1, 0) which makes sense for the characters of integrable representations, and we write simply F for this specialization Fs when no confusion can arise. Since the set of simple roots for the even part of ĝl(m|1) is Π̂ = {α 0 = α0 + αm, α1, . . . , αm−1}, this specialization is the principal specialization with respect to the even part of ĝl(m|1); namely F(e−α′0) = F(e1) = · · · = F(em−1) = q. We recall the Fock space and its charge decomposition: F = ⊕s∈ZFs from §3 of [KW2]. Date: February 7, 2008. 1 Lemma 1.1. Let m ≥ 2, s ∈ Z. Then we have: (a) q sm 2 F(echFs) + q sm 2 F(echF−s) = 2 ∏∞ i=1(1 + q ) φ(qm)2 . (b) F(e0chFs) = F(e0chFm−1−s). Here and further φ(q) = ∏∞ j=1(1− q). Proof. In the case ĝl(m|1), the formula (3.15) in [KW2] gives the following: chF = e0 ∞ ∏ k=1 ∏m i=1(1 + ze ǫi−(k− 1 2 )(1 + zei 1 2 ) (1− zem+1 1 2 )δ)(1− z−1em+1 12 ) , (1.1) where z is the “charge” variable. Since α0 = δ − ǫ1 + ǫm+1 and αi = ǫi − ǫi+1, (i = 1, . . . , m), our principal specialization F = Fs is written in terms of ei as follows: F(ei) = q(i = 1, . . . , m), F(em+1) = 1. (1.2) Thus, we obtain F(e0chF ) = ∞ ∏ k=1 ∏m i=1(1 + zq q 1 2 )(1 + zqq 1 2 ) (1− zq 12 ))(1− z−1q 1 2 ) = ∞ ∏ k=1 (1 + zq m 2 )(1 + zq m 2 ) (1− zq 12 ))(1− z−1q 12 ) = ∞ ∏ k=1 (1 + (zq 1−m 2 )q 1 2 )(1 + (zq m−1 2 )q 1 2 ) (1 + (−z−1)qm(k− 1 2 )(1 + (−z)qm(k− 12 ) . (1.3) In order to compute the coefficient of z, we use the Jacobi triple product identity: ∞ ∏ n=1 (1 + zq 1 2 )(1 + zq 1 2 ) = 1 φ(q) ∑ j∈Z zq 1 2 j2, (1.4) and also the following well-known identity (see (5.26) in [KP] and §5.8 in [K4]): ∞ ∏ k=1 (1 + zq 1 2 )(1 + zq 1 2 ) = 1 φ(q)2 ∑ m∈Z (−1) q 1 2 m(m+1) 1 + zq 1 2