The Potts model and the symmetric group

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

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.


The symmetric group Sk acts on a vector space V of dimension k by permuting the basis elements v1,v2,vk. The groups Sn acts on nV by permuting the tensor product factors. We show that the algebra of all matrices on nV commuting with Sk is generated by Sn and the operators e1 and e2 where e1 ( vp1 vp2 vpn ) =1k i=1k vivp2 vpn e2 ( vp1 vp2 vpn ) =δp1,p2 vp1vp2 vpn. The matrices e1 and e2 give the vertical and horizontal transfer matrices adding one site in the square lattice 2-dimensional Potts model.

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