Path generating transforms

Abstract

We study combinatorial aspects of q-weighted, length-L Forrester-Baxter paths, P^{p, p&#39;}_{a, b, c}(L), where p, p&#39;, a, b, c \in Z_{+}, 0 < p < p&#39;, 0 < a, b, c < p&#39;, c = b \pm 1, L+a-b \equiv 0 (mod 2), and p and p&#39; are co-prime. We obtain a bijection between P^{p, p&#39;}_{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&#39;}_{a, b, c}(L). Using the new weights, and defining s_0 and r_0 to be the smallest non-negative integers for which |p s_0 - p&#39; r_0|=1, we restrict the discussion to P^{p, p&#39;}_{s_0} \equiv P^{p, p&#39;}_{s_0,s_0,s_0+1}(L), and introduce two combinatorial transforms: 1. A Bailey-type transform B: P^{p, p&#39;}_{s_0}(L) -> P^{p, p&#39;+p}_{s_0 + r_0}(L&#39;), L \leq L&#39;, 2. A duality-type transform D: P^{p, p&#39;}_{s_0}(L) -> P^{p&#39;-p, p&#39;}_{s_0}(L). We study the action of B and D, as q-polynomial transforms on the P^{p, p&#39;}_{s_0}(L) generating functions, χ^{p, p&#39;}_{s_0}(L). In the limit L -> \infinity, χ^{p, p&#39;}_{s_0}(L) reduces to the Virasoro characters, χ^{p, p&#39;}_{r_0, s_0}, of minimal conformal field theories M^{p, p&#39;}, or equivalently, to the one-point functions of regime-III Forrester-Baxter models. As an application of the B and D transforms, we re-derive the constant-sign expressions for χ^{p, p&#39;}_{r_0, s_0}, first derived by Berkovich and McCoy.