## 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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