We present fermionic sum representations of the characters χ (p,p ′ ) r,s of the minimal M (p, p ′ ) models for all relatively prime integers p ′ > p for some allowed values of r and s. Our starting point is binomial (q-binomial) identities derived from a truncation of the state counting equations of the XXZ spin 1 2 chain of anisotropy −∆ = − cos(π p p ′ ). We use the Takahashi-Suzuki method to express the allowed values of r (and s) in terms of the continued fraction decomposition of { p ′ p } (and p p ′ ) where {x} stands for the fractional part of x. These values are, in fact, the dimensions of the hermitian irreducible representations of SU q − (2) (and SU q + (2)) with q − = exp(iπ{ p ′ p }) (and q + = exp(iπ p p ′ )). We also establish the duality relation M (p, p ′ ) ↔ M (p ′ − p, p ′ ) and discuss the action of the Andrews-Bailey transformation in the space of minimal models. Many new identities of the Rogers-Ramanujan type are presented.

The Hilbert space of an RSOS-model, introduced by Andrews, Baxter, and Forrester, can be viewed as a space of sequences (paths) {a 0 , a 1 , . . . , a L }, with a j -integers restricted by 1 ≤ a j ≤ ν, | a j − a j+1 |= 1, a 0 ≡ s, a L ≡ r. In this paper we introduce different basis which, as shown here, has the same dimension as that of an RSOS-model. This basis appears naturally in the Bethe ansatz calculations of the spin ν−1 2 XXZ-model. Following McCoy et al, we call this basis -fermionic (FB). Our first theorem Dim(F B) = Dim(RSOS − basis) can be succinctly expressed in terms of some identities for binomial coefficients. Remarkably, these binomial identities can be qdeformed. Here, we give a simple proof of these q-binomial identities in the spirit of Schur's proof of the Rogers-Ramanujan identities. Notably, the proof involves only the elementary recurrences for the q-binomial coefficients and a few creative observations. Finally, taking the limit L → ∞ in these q-identities, we derive an expression for the character formulas of the unitary minimal series M(ν, ν +1) "Bosonic Sum ≡ Fermionic Sum". Here, Bosonic Sum denotes Rocha-Caridi representation (χ ν,ν+1 r,s=1 (q)) and Fermionic Sum stands for the companion representation recently conjectured by the McCoy group [3].

We present and prove Rogers-Schur-Ramanujan (Bose/Fermi) type identities for the

2016-12-26T14:59:35

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