The partition algebra and the Potts model transfer matrix spectrum in high dimensions
The partition algebra and the Potts model transfer matrix spectrum in high dimensions
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高维划分代数和 Potts 模型传递矩阵谱
DOI：
10.1088/0305-4470/33/19/304
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
Paul Martin
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文献类型：
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作者：
Paul Martin
我们构造了分区代数 Pn(Q) 的泛化 Pmn(Q) (Martin P P 1996 J. Algebra 183 319)，促进了对具有磁场和源项的高维 Q 状态 Potts 模型的 n 位转移矩阵谱的表示理论方法（以及相应的二色多项式）。对于每个 Q ∊ ℂ，我们描述代数序列 P*(Q) = {Pn(Q) ⊂ Pn1(Q) ⊂ Pn + 1(Q)| 的不可约表示论n = 0,1,2,...} 接近大 n 极限。对于每个正整数 Q，我们将 Pn(Q) 的 Potts 模型表示 ρn 扩展到 P1n(Q) 的表示。我们展示了这些 Potts 表示如何嵌入到划分代数的表示理论中。这些结果共同提供了一种检查物理相关函数性质的工具。对于大 n，Potts 表示的不可约内容可以总结为 ℂSQ ≅ EndPn(Q)({VQ⊗n}) 和 ℂSQ-1 ≅ EndPn1(Q)({VQ⊗n})，其中 SQ 是对称群，VQ 是 Potts 自旋的状态空间。我们展示了划分代数形式如何匹配 Potts 模型的相关函数及其传递矩阵的相应绝对谱简并性。
We construct generalizations Pmn(Q) of the partition algebra Pn(Q) (Martin P P 1996 J. Algebra 183 319), facilitating a representation theoretic approach to the n-site transfer matrix spectrum of a high-dimensional Q-state Potts model with magnetic field and source terms (and to corresponding dichromatic polynomials). For each Q ∊ ℂ we describe the irreducible representation theory of the sequence of algebras P*(Q) = {Pn(Q) ⊂ Pn1(Q) ⊂ Pn + 1(Q)| n = 0,1,2,...} approaching the large-n limit. For each positive integer Q we extend the Potts model representation ρn of Pn(Q) to a representation of P1n(Q). We show how these Potts representations embed in the representation theory of the partition algebras. These results together provide a tool with which to examine the nature of physical correlation functions. For large n the irreducible content of the Potts representations can be summarized by ℂSQ ≅ EndPn(Q)({VQ⊗n}) and ℂSQ-1 ≅ EndPn1(Q)({VQ⊗n}), where SQ is the symmetric group, and VQ is the space of states of a Potts spin. We show how the partition algebra formalism matches up the correlation functions of the Potts model and the corresponding absolute spectrum degeneracies of its transfer matrix.