Paths and partitions: Combinatorial descriptions of the parafermionic states

Abstract

The Z k parafermionic conformal field theories, despite the relative complexity of their modes algebra, offer the simplest context for the study of the bases of states and their different combinatorial representations. Three bases are known. The classic one is given by strings of the fundamental parafermionic operators whose sequences of modes are in correspondence with restricted partitions with parts at distance k − 1 differing at least by 2. Another basis is expressed in terms of the ordered modes of the k − 1 different parafermionic fields, which are in correspondence with the so-called multiple partitions. Both types of partitions have a natural (Bressoud) path representation. Finally, a third basis, formulated in terms of different paths, is inherited from the solution of the restricted solid-on-solid model of Andrews-Baxter-Forrester. The aim of this work is to review, in a unified and pedagogical exposition, these four different combinatorial representations of the states of the Z k parafermionic models.The first part of this article presents the different paths and partitions and their bijective relations; it is purely combinatorial, self-contained and elementary; it can be read independently of the conformal-field-theory applications. The second part links this combinatorial analysis with the bases of states of the Z k parafermionic theories. With the prototypical example of the parafermionic models worked out in detail, this analysis contributes to fix some foundations for the combinatorial study of more complicated theories. Indeed, as we briefly indicate in ending, generalized versions of both the Bressoud and the Andrews-Baxter-Forrester paths emerge naturally in the description of the minimal models.

Decomposition of fractional quantum Hall model states: Product rule symmetries and approximations

“…k0 r−1 k]. The Jack wave functions can be related to WA k−1 conformal field theories and can be classified in terms of symmetric polynomial categories [19][20][21][22][23][24][25] . Moreover, the quasiparticle excitations of the trial state systems can also be written as coherent state superpositions of Jacks 26,27 .…”

Unimodality, Log-concavity, Real-rootedness And Beyond

“…Bressoud [24] was the first to actually set up such a link. Since then, this connection has been explored much further and extended in various directions, particularly so in the physics literature, see [7,29,34,93,122] and the references therein.…”

Path representation of states II: Operator construction of the fermionic character and spin-–RSOS factorization

“…Quite interestingly, the paths can generally be decomposed into some basic building blocks, often called (path-)particles [12,39,31]. In some cases, these seem to correspond to the massless versions of the off-critical massive particles.…”

“…In principle, given this operator construction, we should be in position to work out the generating function by following the standard strategy [39,31] used in the level-1 case [29], namely, q-enumerate sequences of operators from a minimal-weight configuration for a fixed operator content, out of which all other configurations are generated by operator displacements, followed by a summation over the number of operators of each type. However, a frontal attack along this line proves to be difficult.…”