## The Potts model and the symmetric group

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 $e1 ( vp1⊗ vp2⊗ ⋯⊗ vpn ) =1k ∑i=1k vi⊗vp2 ⊗⋯⊗vpn e2 ( vp1⊗ vp2⊗ ⋯⊗ vpn ) =δp1,p2 vp1⊗vp2 ⊗⋯⊗vpn.$ 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.