Path generating transforms
Abstract
We study combinatorial aspects of qweighted, lengthL ForresterBaxter paths, P^{p, p'}_{a, b, c}(L), where p, p', a, b, c \in Z_{+}, 0 < p < p', 0 < a, b, c < p', c = b \pm 1, L+ab \equiv 0 (mod 2), and p and p' are coprime. 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 qweights of P^{p, p'}_{a, b, c}(L). Using the new weights, and defining s_0 and r_0 to be the smallest nonnegative integers for which p s_0  p' r_0=1, we restrict the discussion to P^{p, p'}_{s_0} \equiv P^{p, p'}_{s_0,s_0,s_0+1}(L), and introduce two combinatorial transforms: 1. A Baileytype transform B: P^{p, p'}_{s_0}(L) > P^{p, p'+p}_{s_0 + r_0}(L'), L \leq L', 2. A dualitytype transform D: P^{p, p'}_{s_0}(L) > P^{p'p, p'}_{s_0}(L). We study the action of B and D, as qpolynomial transforms on the P^{p, p'}_{s_0}(L) generating functions, \chi^{p, p'}_{s_0}(L). In the limit L > \infinity, \chi^{p, p'}_{s_0}(L) reduces to the Virasoro characters, \chi^{p, p'}_{r_0, s_0}, of minimal conformal field theories M^{p, p'}, or equivalently, to the onepoint functions of regimeIII ForresterBaxter models. As an application of the B and D transforms, we rederive the constantsign expressions for \chi^{p, p'}_{r_0, s_0}, first derived by Berkovich and McCoy.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 1998
 DOI:
 10.48550/arXiv.math/9810043
 arXiv:
 arXiv:math/9810043
 Bibcode:
 1998math.....10043F
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Combinatorics;
 Mathematics  Mathematical Physics;
 Mathematics  Representation Theory;
 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 qSeries from a contemporary perspective (South Hadley, MA, 1998), 157186, Contemp. Math. 254, Amer. Math. Soc., Providence, RI, 2000.