Celebratio Mathematica — Jones

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V. F. R. Jones: “The Potts model and the symmetric group,” pp. 259–267 in Subfactors: Proceedings of the Taniguchi symposium on operator algebras (Kyuzeso, Japan, 6–10 July 1993). Edited by H. Araki, H. Kosaki, and Y. Kawahigashi. World Scientific (Singapore), 1994. MR 1317365 Zbl 0938.20505 incollection

The symmetric group \( S_k \) acts on a vector space \( V \) of dimension \( k \) by permuting the basis elements \( v_1,v_2 \), \( \dots,v_k \). The groups \( S_n \) acts on \( \bigotimes^n V \) by the tensor product factors. We show that the algebra of all matrices on \( \bigotimes^n V \) commuting with \( S_k \) is generated by \( S_n \) and the operators \( e_1 \) and \( e_2 \) where \begin{align*} 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} \end{align*} The matrices \( e_1 \) and \( e_2 \) give the vertical and horizontal transfer matrices adding one site in the square lattice Potts model.

Yasuyuki Kawahigashi	Related
Huzihiro Araki	Related
Hideki Kosaki	Related