## The Potts model and the symmetric group.(English)Zbl 0938.20505

Araki, Huzihiro (ed.) et al., Subfactors. Proceedings of the Taniguchi symposium on operator algebras, Kyuzeso, Japan, July 6-10, 1993. Singapore: World Scientific. 259-267 (1994).
Summary: The symmetric group $$S_k$$ acts on a vector space $$V$$ of dimension $$k$$ by permuting the basis elements $$v_1,v_2,\cdots,v_k$$. The group $$S_n$$ acts on $$\bigotimes^nV$$ by permuting the tensor product factors. We show that the algebra of all matrices on $$\bigotimes^nV$$ commuting with $$S_k$$ is generated by $$S_n$$ and the operators $$e_1$$ and $$e_2$$, where \begin{aligned} e_1(v_{p_1}\otimes v_{p_2}\otimes\cdots\otimes v_{p_n})&=\frac 1k\sum^k_{i=1}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}.\end{aligned} 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.
For the entire collection see [Zbl 0914.00068].

### MSC:

 20C30 Representations of finite symmetric groups 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics

### Keywords:

symmetric groups; transfer matrices; Potts model