Last update: 18 April 2014
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.
Let be a vector space of dimension over a field of characteristic zero, with basis indexed by a set of size The symmetric group is linearly represented on by Thus for each is an As a vector space, is just the fixed points for the representation of on by conjugation. But this is a permutation representation for if has matrix with respect to the basis then has matrix Hence a basis for is given by where each sum is taken over all in an orbit for the action of on the Cartesian product Each orbit for this action is given by a partition of into subsets (obviously at most of them). If denotes the equivalence relation defined by such a partition then the corresponding orbit on consists of these (setting for which iff The number of equivalence relations with classes is the Stirling number so that the dimension of is For this is the Bell number (see [Sta1986] p.33). Compare the first few Bell numbers with the dimensions of the Bratteli diagram.
If is an equivalence relation on we will call the element of defined by the corresponding It clearly has the following matrix: Note that is zero if the number of classes for is more than From our discussion is a basis for with elements (forgetting any that may be zero).
In fact we will use another set which linearly spans and will also be a basis if For each as above define by Where the sum is over all partitions coarser than By Möbius inversion ([Sta1986] p.116) the can be expressed in terms of the so they also span Note though that the are always nonzero so they will not form a basis for
We leave the verification of the following lemma to the reader as it should make clear what is going on.
|1)||If is defined by then|
|2)||If is defined by for then with as in the introduction.|
|3)||If is defined by and for then with as in the introduction.|