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 $V$ be a vector space of dimension $k$ over a field $K$ of characteristic zero, with basis $\left\{{v}_{a}\right\}$ indexed by a set $A$ of size $k\text{.}$ The symmetric group ${S}_{k}$ is linearly represented on $V$ by $\sigma \left({v}_{a}\right)={v}_{\sigma \left(a\right)}\text{.}$ Thus for each $n,$ ${\otimes}^{n}V$ is an ${S}_{k}\text{-module.}$ As a vector space, ${\text{End}}_{{S}_{k}}\left({\otimes}^{n}V\right)$ is just the fixed points for the representation of ${S}_{k}$ on $\text{End}\left({\otimes}^{n}V\right)$ by conjugation. But this is a permutation representation for if $X\in \text{End}\left({\otimes}^{n}V\right)$ has matrix ${X}_{{a}_{1}{a}_{2}\cdots {a}_{n}}^{{b}_{1}{b}_{2}\cdots {b}_{n}}$ with respect to the basis $\{{v}_{{a}_{1}}\otimes {v}_{{a}_{2}}\otimes \cdots \otimes {v}_{{a}_{n}}\}$ then $\sigma \left(X\right)$ has matrix ${X}_{\sigma \left({a}_{1}\right)\sigma \left({a}_{2}\right)\cdots \sigma \left({a}_{n}\right)}^{\sigma \left({b}_{1}\right)\sigma \left({b}_{2}\right)\cdots \sigma \left({b}_{n}\right)}\text{.}$ Hence a basis for ${\text{End}}_{{S}_{k}}\left({\otimes}^{n}V\right)$ is given by $\left\{\sum _{\text{orbit}}{X}_{{a}_{1}\cdots {a}_{n}}^{{b}_{1}\cdots {b}_{n}}\right\}$ where each sum is taken over all $\{{a}_{1},{a}_{2},\cdots {a}_{n},{b}_{1},{b}_{2},\cdots {b}_{n}\}$ in an orbit for the action of ${S}_{k}$ on the $2n\text{-fold}$ Cartesian product ${A}^{2n}\text{.}$ Each orbit for this ${S}_{k}$ action is given by a partition of $\{1,2,\dots ,2n\}$ into subsets (obviously at most $k$ of them). If $\sim $ denotes the equivalence relation defined by such a partition then the corresponding orbit on ${A}^{2n}$ consists of these $2n\text{-tuples}$ $({a}_{1},\cdots ,{a}_{n},\cdots ,{a}_{2n})$ (setting ${b}_{i}={a}_{n+i}\text{)}$ for which ${a}_{i}={a}_{j}$ iff $i\sim j\text{.}$ The number of equivalence relations with $r$ classes is the Stirling number $S(2n,r)$ so that the dimension of ${\text{End}}_{{S}_{k}}\left({\otimes}^{n}V\right)$ is $\sum _{i=1}^{k}S(2n,i)\text{.}$ For $2n\le k$ this is the Bell number $B\left(2n\right)=\sum _{i=1}^{2n}S(2n,i)$ (see [Sta1986] p.33). Compare the first few Bell numbers with the dimensions of the Bratteli diagram.
If $\sim $ is an equivalence relation on $\{1,\dots ,2n\}$ we will call ${T}_{\sim}$ the element of $\text{End}\left({V}^{\otimes n}\right)$ defined by the corresponding ${S}_{k}\text{-orbit.}$ It clearly has the following matrix: $${\left({T}_{\sim}\right)}_{{a}_{1}\cdots {a}_{n}}^{{a}_{n+1}\cdots {a}_{n}}=\{\begin{array}{cc}1& \text{if}\phantom{\rule{1em}{0ex}}({a}_{i}={a}_{j}\iff i\sim j)\\ 0& \text{otherwise.}\end{array}$$ Note that ${T}_{\sim}$ is zero if the number of classes for $\sim $ is more than $k\text{.}$ From our discussion $\left\{{T}_{\sim}\hspace{0.17em}\right|\hspace{0.17em}\sim \hspace{0.17em}\text{an equivalence relations}\}$ is a basis for ${\text{End}}_{{S}_{k}}\left({\otimes}^{n}V\right)$ with $\sum _{i=1}^{k}S(2n,i)$ elements (forgetting any ${T}_{\sim}$ that may be zero).
In fact we will use another set which linearly spans ${\text{End}}_{{S}_{k}}\left({\otimes}^{n}V\right)$ and will also be a basis if $k\ge 2n\text{.}$ For each $\sim $ as above define ${L}_{\sim}\in {\text{End}}_{{S}_{k}}\left({\otimes}^{n}V\right)$ by $${L}_{\sim}=\sum _{\sim \prime \le \sim}{T}_{\sim}\text{.}$$ Where the sum is over all partitions $\sim \prime $ coarser than $\sim \text{.}$ By Möbius inversion ([Sta1986] p.116) the ${T}_{\sim}\text{'s}$ can be expressed in terms of the ${L}_{\sim}\text{'s}$ so they also span ${\text{End}}_{{S}_{k}}\left({\otimes}^{n}V\right)\text{.}$ Note though that the ${L}_{\sim}\text{'s}$ are always nonzero so they will not form a basis for $k<2n\text{.}$
We leave the verification of the following lemma to the reader as it should make clear what is going on.
Lemma 1.
1) | If $\sim $ is defined by $i\sim i+n$ then ${L}_{\sim}=\text{id.}$ |
2) | If $\sim $ is defined by $i\sim i+n$ for $i>1$ then ${L}_{\sim}=k{e}_{1}$ with ${e}_{1}$ as in the introduction. |
3) | If $\sim $ is defined by $1\sim 2\sim (n+1)\sim (n+2)$ and $i\sim i+n$ for $i>2$ then ${L}_{\sim}={e}_{2}$ with ${e}_{2}$ as in the introduction. |