Andrews-Gordon identities from combinations of Virasoro characters

Abstract

For p ∈ 3, 4 and all p' > p, with p' coprime to p, we obtain fermionic expressions for the combination χ^p, p'₁, s + q^Δ χ^p, p'_p-1,s of Virasoro (W₂) characters for various values of s, and particular choices of Delta. Equating these expressions with known product expressions, we obtain q-series identities which are akin to the Andrews-Gordon identities. For p=3, these identities were conjectured by Bytsko. For p=4, we obtain identities whose form is a variation on that of the p=3 cases. These identities appear to be new. The case (p,p')=(3,14) is particularly interesting because it relates not only to W₂, but also to W₃ characters, and offers W₃ analogues of the original Andrews-Gordon identities. Our fermionic expressions for these characters differ from those of Andrews et al which involve Gaussian polynomials.