The Potts model and the symmetric group

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 18 April 2014

Notes and References

This is an excerpt of the paper The Potts model and the symmetric group by V.F.R. Jones. It appeared in: Subfactors: Proceedings of the Taniguchi Symposium on Operator Algebras (Kyuzeso, 1993), River Edge, NJ, World Sci. Publishing, 1994, pp. 259–267.

Abstract

The symmetric group $S_k$ acts on a vector space $V$ of dimension $k$ by permuting the basis elements $v_1,v_2,\dots v_k\text{.}$ The groups $S_n$ acts on ${\otimes }^{n}V$ by permuting the tensor product factors. We show that the algebra of all matrices on ${\otimes }^{n}V$ commuting with $S_k$ is generated by $S_n$ and the operators $e_1$ and $e_2$ where $$\begin{array}{c}{e}_1(v_{p_1}\otimes v_{p_2}\otimes \cdots \otimes v_{p_n})=\frac{1}{k}\sum _{i=1}^k{v}_i\otimes v_{p_2}\otimes \cdots \otimes v_{p_n}\\ e_2(v_{p_1}\otimes v_{p_2}\otimes \cdots \otimes v_{p_n})={\delta }_{p_1,p_2}{v}_{p_1}\otimes v_{p_2}\otimes \cdots \otimes v_{p_n}\text{.}\end{array}$$ The matrices $e_1$ and $e_2$ give the vertical and horizontal transfer matrices adding one site in the square lattice 2-dimensional Potts model.

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