No abstract

Fermionic expressions for all minimal model Virasoro characters χ p,p ′ r,s are stated and proved. Each such expression is a sum of terms of fundamental fermionic form type. In most cases, all these terms are written down using certain trees which are constructed for s and r from the Takahashi lengths and truncated Takahashi lengths associated with the continued fraction of p ′ /p. In the remaining cases, in addition to such terms, the fermionic expression for χ p,p ′ r,s contains a different character χp ,p ′ r,ŝ , and is thus recursive in nature.Bosonic-fermionic q-series identities for all characters χ p,p ′ r,s result from equating these fermionic expressions with known bosonic expressions. In the cases for which p = 2r, p = 3r, p ′ = 2s or p ′ = 3s, Rogers-Ramanujan type identities result from equating these fermionic expressions with known product expressions for χ p,p ′ r,s . The fermionic expressions are proved by first obtaining fermionic expressions for the generating functions χ p,p ′ a,b,c (L) of length L Forrester-Baxter paths, using various combinatorial transforms. In the L → ∞ limit, the fermionic expressions for χ p,p ′ r,s emerge after mapping between the trees that are constructed for b and r from the Takahashi and truncated Takahashi lengths respectively.

Dedicated to Professor Richard Askey on the occasion of his 65th birthday.Abstract. We study combinatorial aspects of q-weighted, length-L Forrester-Baxter paths,, and p and p ′ are co-prime.We obtain a bijection between P p,p ′ a,b,c (L) and partitions with certain prescribed hook differences. Thereby, we obtain a new description of the q-weights of P p,p ′ a,b,c (L). Using the new weights, and defining s 0 and r 0 to be the smallest non-negative integers for which |ps 0 − p ′ r 0 | = 1, we restrict the discussion to P p,p ′ s 0

We provide further boson-fermion q-polynomial identities for the 'finitised' Virasoro characters χ p,p ′ r,s of the Forrester-Baxter minimal models M (p, p ′ ), for certain values of r and s. The construction is based on a detailed analysis of the combinatorics of the set P p,p ′ a,b,c (L) of q-weighted, length-L Forrester-Baxter paths, whose generating function χ p,p ′ a,b,c (L) provides a finitisation of χ p,p ′ r,s . In this paper, we restrict our attention to the case where the startpoint a and endpoint b of each path both belong to the set of 'Takahashi lengths'. In the limit L → ∞, these polynomial identities reduce to q-series identities for the corresponding characters.We obtain two closely related fermionic polynomial forms for each (finitised) character. The first of these forms uses the classical definition of the Gaussian polynomials, and includes a term that is a (finitised) character of a certain M (p,p ′ ) wherep ′ < p ′ . We provide a combinatorial interpretation for this form using the concept of 'particles'. The second form, which was first obtained using different methods by the Stony-Brook group, requires a modified definition of the Gaussian polynomials, and its combinatorial interpretation requires not only the concept of particles, but also the additional concept of 'particle annihilation'.

Using a summation formula due to Burge, and a combinatorial identity between partition pairs, we obtain an infinite tree of q-polynomial identities for the Virasoro characters χ p,p ′ r,s , dependent on two finite size parameters M and N , in the cases where: * foda@maths.mu.oz.au † ksml@maths.mu.oz.au ‡ trevor@maths.mu.oz.au 10 Roughly speaking, this is related to the fact that the objects that are counted are single partitions.11 Roughly speaking, this is related to the fact that the objects that are counted are are pairs of partitions. 12 To be more precise, the Burge transform involves four finite size parameters, say N , M , N ′ , and M ′ . However, the identities that we obtain can all be derived in terms of two parameters only: M = M ′ , and N = N ′ . The general case of four parameters is relevant to identities that correspond to the most general RSOS characters. We do not deal with the most general case in this work.13 This is the character that has the smallest conformal dimension in the model [18]. 14 We can say that the latter correspond to all possible 'tapered truncations' of the former. 15 There are two results in [12] that, to the best of our understanding, could only be derived using such an identity. 16 We find that though [12] is ingenious, it is also very succinctly written, and therefore not easy to read. In particular, certain consistency conditions that must be imposed on the partitions pairs are not explicitly stated. This difficulty is further compounded by the fact that it contains, unfortunately, a large number of misprints.