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.

§3. Proof of the theorem.

The diagrams joining the 2n bottom ×'s to the 2n top ×'s form the basis of an algebra sometimes called the Temperley Lieb algebra. It can be thought of as a subalgebra of the Brauer algebra defined in the introduction as each diagram gives a partition of the 4n ×'s into subsets of size 2. The multiplication is defined by a 2-step procedure.

Step 1. Stack the two rectangles on top of each other, lining up the ×'s.

Step 2. Remove the middle edges and middle ×'s. One then has a new diagram possibly containing some closed loops. If there are p closed loops, the product is then the resulting diagram, with the closed loops removed, times the scalar δp where δ is some fixed element of the field. One thus obtains a family of algebras K(2n,δ) of dimension 12n+1(4n2n). We illustrate multiplication in K(4,δ) below. α= β= so αβ=δ=

We now define special elements E1,E2,E2n-1 of K(2n,δ): E1 E2 En One easily has { Ei2=δEi EiEj=Ej Ei,f|i-j| 2 EiEi±1Ei =Ei and it is easy to see that the Ei's, together with 1, generate K(2n,δ). Counting reduced words on the Ei's one obtains the Catalan numbers (see [Jon1983]) so the above relations present K(2n,δ).

Lemma 3. If d is a diagram giving a basis element of K(2n,δ) as above, let d be the equivalence relation it generates. Then the map φ(Ei)= { LEi iodd kLEi ieven extends to an algebra homomorphism from K(2n,1k) to End(nV) so that for a diagram d, φ(d) is a non zero multiple of Ld.

Proof. Since the above relations present K(2n,1k), the existence of φ follows from the fact that the φ(Ei) satisfy the same relations. We leave it to the reader to check that if d and d are diagrams then LdLd is a multiple of Ldd.


Remark. It seems suggestive that the mapping φ defined above is injective as soon as k4 whereas we have seen that for non-planar partitions there are linear relations between the L as soon as n>k.

The same thing happens with the K(2n,δ) as a subalgebra of the Brauer algebra. This linear independence is easily proved using the Markov trace technique of [Jon1983].

Theorem. The commutant of Sk on nV is generated by Sn, e1 and e2 where V, e1 and e2 are as in Lemma 1 and Sn acts by permuting the tensor factors.

Proof. It suffices to show that any L is in the algebra generated by Sn and e1,e2. We have seen that by multiplying L on the left and right by elements of Sn, we can assume that is planar. The elements φ(E3),φ(E4),φ(En) are just conjugates of φ(E1) and φ(E2) under the action of Sn so the theorem follows from Lemma 3.


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