路径与分区：仿费米子态的组合描述
Paths and partitions: combinatorial descriptions of the parafermionic states
摘要：
Z_k亚费米子共形场论，尽管其模代数相对复杂，但为研究态基及其不同组合表示提供了最简单的背景。已知有三个基地。经典的一个是由字符串的基本parafermionic运营商的序列的模式是在对应的限制分区与部分在距离k-1不同至少2。另一个基是用k-1个不同的仿费米子场的有序模来表示的，这与所谓的多重划分相对应。这两种类型的分区都有一个自然（Bressoud）路径表示。最后，第三个基础上，制定不同的路径，继承了有限的固体对固体模型的Andrews-Baxter-Forrester的解决方案。本文的目的是在一个统一的和教学性的论述中，回顾Z_k副费米模型的这四种不同的态的组合表示。本文的第一部分介绍了不同的道路和分区和他们的双射关系;它是纯粹的组合，自包含和基本的;它可以独立于共形场论的应用程序阅读。第二部分将这种组合分析与Z_k亚费米子理论的态基联系起来。通过对仿费米子模型的典型实例的详细分析，为更复杂理论的组合研究奠定了一定的基础。事实上，正如我们在结尾处简要指出的那样，Bressoud和Andrews-Baxter-Forrester路径的广义版本都自然地出现在对极小模型的描述中。
英文摘要：
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.
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