# the unitary minimal series M(ν, ν + 1). Exact results.

#### Abstract

The Hilbert space of an RSOS-model, introduced by Andrews, Baxter, and Forrester, can be viewed as a space of sequences (paths) {a0, a1, . . . , aL}, with aj-integers restricted by 1 ≤ aj ≤ ν, | aj − aj+1 |= 1, a0 ≡ s, aL ≡ 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(FB) = 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 r,s=1 (q)) and Fermionic Sum stands for the companion representation recently conjectured by the McCoy group [3].

### Cite this paper

@inproceedings{Berkovich1994theUM, title={the unitary minimal series M(ν, ν + 1). Exact results.}, author={Alexander Berkovich and Mikhail Nirenberg}, year={1994} }