On Spin Models, Triply Regular Association Schemes, and Duality
On Spin Models, Triply Regular Association Schemes, and Duality
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DOI：
10.1023/a:1022429530062
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发表时间：
1995-04
影响因子：
0.8
通讯作者：
F. Jaeger
中科院分区：
文献类型：
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作者：
F. Jaeger
基于三维空间中链路不变量的构造，我们研究了所有边权（视为矩阵）都属于某种关联格式的Bose-Mesner代数的图上的自旋模型。我们证明了对于序列-并行图，配分函数的计算可以通过适当地将图的序列-并行约简与Bose-Mesner代数中的运算相结合来完成。然后，我们通过引入星形三角形变换将这种方法扩展到所有平面图，并将我们的注意力限制在一类特殊的玻色-梅斯纳代数上，我们称之为完全三正则代数。我们还介绍了玻色-梅斯纳代数的以下两个性质。平面对偶性（在自对偶情况下定义）用对偶图的配分函数来表示任意平面图的配分函数，平面可逆性断言任意平面图的配分函数等于对向图的配分函数。如果只考虑序列-平行图而不考虑任意平面图，这两个性质对任何玻色-梅斯纳代数都成立。我们将这些概念与连杆不变量的自旋模型联系起来，并在其他结果中证明了阿贝尔群玻色-梅斯纳代数具有平面对偶性质，并且对于自对偶玻色-梅斯纳代数，平面对偶意味着平面可变性。我们还证明了对于完全三正则玻色-梅斯纳代数，为了检验上述性质之一，在四个顶点的完全图上检验它是足够的。讨论了一些应用、实例和开放性问题。
Motivated by the construction of invariants of links in 3-space, we study spin models on graphs for which all edge weights (considered as matrices) belong to the Bose-Mesner algebra of some association scheme. We show that for series-parallel graphs the computation of the partition function can be performed by using series-parallel reductions of the graph appropriately coupled with operations in the Bose-Mesner algebra. Then we extend this approach to all plane graphs by introducing star-triangle transformations and restricting our attention to a special class of Bose-Mesner algebras which we call exactly triply regular. We also introduce the following two properties for Bose-Mesner algebras. The planar duality property (defined in the self-dual case) expresses the partition function for any plane graph in terms of the partition function for its dual graph, and the planar reversibility property asserts that the partition function for any plane graph is equal to the partition function for the oppositely oriented graph. Both properties hold for any Bose-Mesner algebra if one considers only series-parallel graphs instead of arbitrary plane graphs. We relate these notions to spin models for link invariants, and among other results we show that the Abelian group Bose-Mesner algebras have the planar duality property and that for self-dual Bose-Mesner algebras, planar duality implies planar reversibility. We also prove that for exactly triply regular Bose-Mesner algebras, to check one of the above properties it is sufficient to check it on the complete graph on four vertices. A number of applications, examples and open problems are discussed.