Fermionic counting of RSOS states and Virasoro character formulas for the unitary minimal series M( v, v + 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) { a 0, a 1,…, a L }, with a j -integers restricted by 1 ≤ a j ≤ v, | 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 ( v−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 q-deformed. 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( v, v + 1) “Bosonic Sum ≡ Fermionic Sum”. Here, Bosonic Sum denotes Rocha-Caridi representation ( X r, s=1 v, v+1 ( q)) and Fermionic Sum stands for the companion representation recently conjectured by the McCoy group.