intro.workingwiki.tex
In \cite{LP} we describe rank 1 and rank 2 central functions and determine their ring structure.  It is the purpose of this paper to generalize both the tensorial and graphical definitions of central functions to arbitrary rank and to further the study of the ring structure to the rank 3 case.  In doing this, we provide software to compute these functions both using the tensorial description and also using a purely combinatorial description in terms of spin networks.

%Beyond rank 3 it seems that only the combinatorial method will be tractible.  From a ``tensorial'' point-of-view central functions (up to a choice of normalization and association) are uniquely determined.  However, there are many different ways to impose relations on ``spaces of graphs'' and only one (up to normalization and association) to make spin networks correspond to central functions.  Perhaps for this reason, we call our version {\it trace diagrams}.  All such spaces of graphs (which correspond to some set of invariant functions) are based on intertwiners acting on irreducible representations; in this way we do not differ.  However, all subsequent choices are made to correspond exactly to central functions (as we define them).  Once the ``correct'' choices are made, it is quite amazing that the graphical representation of central functions distills most of their computational complexity down to mere combinatorics.

Let $\F_r=\langle \xt_1,...,\xt_r\rangle$ be a rank $r$ free group and let $G=\SL$.  The representations $\R_r=\hm(\F_r,G)$ are an affine variety isomorphic to $G^{\times r}$.  $G$ acts on $\R_r$ by conjugation.  However, the orbit space $\R_r/G$ is not a variety.  There is a categorical quotient in the space of affine schemes $\X_r=\R_r\aq G$ which has a geometric quotient (an honest orbit space) that is comprised of orbits of representations that may be written as finite direct sums of irreducible subrepresentations.

As we will see, central functions are defined to be special elements in the coordinate ring $\C[\X_r]=\C[\R_r]^G$.  In fact, they form a natural additive basis for the invariant ring as an infinite-dimensional trivial representation of $G$.  Moreover, these functions take natural diagramatic form, shown in Figure \ref{f:centralfunction}, a fact first mentioned by Baez \cite{Ba96}. One of the most important points of this paper is that their tensorial nature can be forgotten and they can be understood entirely in terms of combinatorics given by a graphical calculus.
    \begin{figure}[htb]
    $$
    \chi_{\vi,\vec m,\vec p} \equiv
    \tikz[scale=1.2]{
        \draw[trivalent]
            (0,0)to[bend left=80]node[small matrix]{$X_1$}(0,1)node[leftlabel,pos=.8]{$i_1$}
            (0,0)to[bend right=80]node[small matrix]{$X_2$}(0,1)node[rightlabel,pos=.8]{$i_2$}
            (0,0)to[bend right=20](.5,-.2)node[bottomlabel,pos=.5]{$m_1$}
            to[bend right=80]node[small matrix]{$X_3$}(.5,1.2)node[rightlabel,pos=.8]{$i_3$}
            to[bend right=20](0,1)node[toplabel,pos=.5]{$p_1$}
            (.5,-.2)to[bend right=20](1,-.4)node[bottomlabel,pos=.5]{$m_2$}
            (1,1.4)to[bend right=20](.5,1.2)node[toplabel,pos=.5]{$p_2$};
        \draw[draw=none](1.25,0)--(2.25,-.2)node[pos=.2]{.}node[pos=.5]{.}node[pos=.8]{.};
        \draw[draw=none](1.25,1)--(2.25,1.2)node[pos=.2]{.}node[pos=.5]{.}node[pos=.8]{.};
        \draw[trivalent,shift={(.5,0)}]
            (1.5,-.6)to[bend right=20](2,-.8)node[bottomlabel,pos=.4]{$m_{r-2}$}
            to[bend right=80]node[small matrix]{$X_r$}(2,1.8)node[leftlabel,pos=.8]{$i_r$}
            to[bend right=20](1.5,1.6)node[toplabel,pos=.6]{$p_{r-2}$}
            (2,-.8)to[bend right=20](2.5,-1)
            to[bend right=80](2.5,2)node[rightlabel,pos=.4]{$m_{r-1}$}
            to[bend right=20](2,1.8);
    }$$
    \caption{Diagrammatic Form of Central Functions}\label{f:centralfunction}
    \end{figure}

For $r > 2$, the ring structure of $\C[\X_r]$ in terms of the additive basis of central functions is not known. We explore the $r = 3$ case in this paper using computations made with Mathematica. In fact, we use the tensorial contraction method which reflects the definition of the central functions, and we also use a purely combinatorial method which takes advantage of the graphical nature of these functions.

\section{Background on $\SL$ Character Varieties and Representation Theory}\label{s:background}

\subsection{$\SL$-character varieties}
Let $\C[x^k_{ij}]/\Delta$ be the complex polynomial ring in $4r$ variables ($1\leq k \leq r$ and $1\leq i,j \leq 2$), where $\Delta$ is the ideal generated by
the $r$ irreducible polynomials $$x^k_{11}x^k_{22}-x^k_{12}x^k_{21}-1.$$  It is not hard to see that $\C[\R_r]=\C[x^k_{ij}]/\Delta$.  Let $\widehat{x}^k_{ij}$ be the image of $x^k_{ij}$ under $\C[x^k_{ij}] \to \C[x^k_{ij}]/\Delta$.
Define $$\xb_k=\left(
\begin{array}{cc}
\widehat{x}^k_{11} & \widehat{x}^k_{12} \\
\widehat{x}^k_{21} & \widehat{x}^k_{22}  \\
\end{array}\right).$$  Such elements are called {\it generic unimodular matrices}.  We note that $$\left(
\begin{array}{cc}
x^k_{11} & x^k_{12} \\
x^k_{21} & x^k_{22}  \\
\end{array}\right)$$ are simply called {\it generic matrices}.
Let $G=\SL$.  $G$ acts on $\C[\R_r]$ as follows:  $$g\cdot \widehat{x}^k_{ij}=y^k_{ij} \text{ where } \yb_k:=g\xb_k g^{-1},\text{ and }g\in G.$$
The ring of invariants $\C[\R_r]^{G}$ is a finitely-generated domain (see \cite{Na}), which implies $$\X_r:=\mathrm{Spec}_{max}\left(\C[\R_r]^{G}\right)$$ is an affine variety over $\C$, called the
$G$-{\it character variety} of $\F_r$.  It is the variety whose coordinate ring is the ring of invariants, that is, $$\C[\X_r]=\C[\R_r]^{G}.$$

Closely related to $\C[\X_r]$ is the ring of invariants $$\C[\mathfrak{Y}_r]:=\C[\mathfrak{gl}(2,\C)^{\times r}]^{\SL}=\C[x^k_{ij}]^{\SL}.$$
In fact one can show (see \cite{L3})
$$\C[\mathfrak{Y}_r]/\Delta\approx \C[\X_r]$$
Otherwise stated, $$\C[x^k_{ij}]^{\SL}/\Delta\approx \left( \C[x^k_{ij}]/\Delta \right)^{\SL},$$ which is true because $\SL$ is {\it linearly} reductive.
In 1976 Procesi proved (in the context of $n\times n$ generic matrices)
\begin{theorem}[Procesi]
$\C[\mathfrak{Y}_r]$ is generated by the invariants $$\{\Tr{\xb_{i_1}\xb_{i_2}\cdots\xb_{i_k}}\},$$ where $\xb_j$ are generic matrices.
\end{theorem}
Evidently, this ring is multigraded.
Finding minimal generators amounts to finding all linear relations among generators of the same multidegree in the vector space
$$\mathcal{V}_r=\C[\mathfrak{Y}_r]^+/\left(\C[\mathfrak{Y}_r]^+\right)^2,$$ where $\C[\mathfrak{Y}]^+$ is the ideal of positive terms.
It is worth observing that there can be no relation among generators of differing multidegree.  Any such linear relation in $\mathcal{V}_r$ pulls back to a reduction relation in $\C[\Y_r]$ which in turns projects to a ``reduction relation'' in $\C[\X_r]$.  The Cayley-Hamilton equation gives $$\xb^2-\Tr{\xb}\xb+\mathrm{det}(\xb)\id=0.$$  And if we assume $\mathrm{det}(\xb)=1$, as is the case in $\C[\X_r]$, we easily derive $\Tr{\xb^{-1}}=\Tr{\xb}$ and $\Tr{\xb^2}=\Tr{\xb}^2-2$.  Hence the generators $\Tr{\xb^2}$ in $\C[\Y_r]$ project to $\Tr{\xb}^2-2$ in $\C[\X_r]$ and so are freely eliminated.

We work a couple examples before moving on:
$\C[\Y_1]$ is algebraically generated by $\{\Tr{\xb^n}\}$.  But if $n\geq 3$ the  Cayley-Hamilton equation provides relations which
express the generator in terms of the generators $\Tr{\xb}$ and $\Tr{\xb^2}$.  However, the dimension of this variety is computed to be $2$.  Thus there can be no further relations.

One can prove the following theorem relating $\C[\Y_r]$ to $\C[\X_r]$:

Let $\mathcal{M}(R)$ be a set of minimal generators of a ring $R$.  Then in \cite{L3} it is shown, as with a generalization to $\SLm{n}$, that $$\mathcal{M}(\C[\Y_r])-\{\Tr{\xb_1^2},...,\Tr{\xb_r^2}\}=\mathcal{M}(\C[\X_r]),$$ as long as the generators of $\C[\Y_r]$ are taken to be of the form in Procesi's theorem.

Multiplying the Cayley-Hamilton equation on both sides by words $\ub$ and $\vb$ allows us to freely eliminate the generators of type: $\Tr{\ub\xb^n\vb}$ as long as $n\geq 2$ and at least one of $\ub$ or $\vb$ is not the identity.
So for the case $\C[\Y_2]$, we are left with the generators $$\{\Tr{\xb_1},\Tr{\xb_2},\Tr{\xb_1^2},\Tr{\xb_2^2},\Tr{\xb_1\xb_2}\}$$ since
any other expression in two letters would result in a sub-expression with an exponent greater than one, which we just showed was impossible.
Since in this case the dimension of the variety is $5$, there can be no futher relations and thus these generators are minimal and $\Y_r\approx \C^5$.  We can conclude that $\X_1\approx \C$ and $\X_2\approx \C^3$.

More generally, it can be shown that there are no generators necessary that have word length 4 or more (see \cite{G9} for an exposition).  In particular,
\begin{align}
\Tr{\wb_1\wb_2\wb_3\wb_4}=&\Tr{\wb_1}\Tr{\wb_2}\Tr{\wb_3}\Tr{\wb_4}+\Tr{\wb_1}\Tr{\wb_2\wb_3\wb_4}\nonumber\\
&+\Tr{\wb_2}\Tr{\wb_3\wb_4\wb_1}+\Tr{\wb_3}\Tr{\wb_4\wb_1\wb_2}\nonumber\\
&+\Tr{\wb_4}\Tr{\wb_1\wb_2\wb_3}+\Tr{\wb_1\wb_2}\Tr{\wb_3\wb_4}\nonumber\\
&+\Tr{\wb_1\wb_4}\Tr{\wb_2\wb_3}+\Tr{\wb_1\wb_2}\Tr{\wb_3\wb_4}\nonumber\\
&-\Tr{\wb_1\wb_3}\Tr{\wb_2\wb_4}-\Tr{\wb_1}\Tr{\wb_2}\Tr{\wb_3\wb_4}\nonumber\\
&-\Tr{\wb_1}\Tr{\wb_4}\Tr{\wb_2\wb_3}-\Tr{\wb_2}\Tr{\wb_3}\Tr{\wb_1\wb_4}\nonumber\\
&-\Tr{\wb_3}\Tr{\wb_4}\Tr{\wb_1\wb_2}\label{length4reduction}.
\end{align}
and
\begin{align}
\Tr{\xb\yb\zb}+\Tr{\xb\zb\yb}&=\Tr{\xb\yb}\Tr{\zb}+\Tr{\xb\zb}\Tr{\yb}\nonumber\\
&+\Tr{\zb\yb}\Tr{\xb}-\Tr{\xb}\Tr{\yb}\Tr{\zb}.\label{sumformula}\end{align}

Equations \eqref{length4reduction} and \eqref{sumformula} together imply there are $N_r=\frac{r(r^2+5)}{6}$ minimal generators of $\C[\X_r]$.  They are:
$\mathcal{G}_1=\{\Tr{\xb_1},...,\Tr{\xb_r}\}$ of order $r$,
$\mathcal{G}_2=\{\Tr{\xb_i\xb_j}\ |\ 1\leq i<j \leq r\}$ of order $\frac{r(r-1)}{2}$, and
$\mathcal{G}_3=\{\Tr{\xb_i\xb_j\xb_k}\ | \ 1\leq i<j<k \leq r \}$ of order $\frac{r(r-1)(r-2)}{3}$.

Enumerating these $N_r$ generators, $\{t_1,...,t_{N_r}\}$ defines a polynomial mapping
$$t=(t_1,...,t_{N_r}):\X_r\longrightarrow \C^{N_r}.$$  It is not hard to show that $t$ is a proper injection and hence defines a homeomorphism (with respect to the induced topology from $\C^{N_r}$) onto its image.  We conclude with the following global geometric result:  the smallest affine embedding $\X_r\longrightarrow \C^{N}$ is when $N=\frac{r(r^2+5)}{6}$, which follows from the fact that $\{t_1,...,t_{N_r}\}$ is a {\it minimal} generating set.

We will describe the $r=3$ case in more detail in Section \ref{s:rankthree}.  We note now that $\X_3$ is a branched double cover of $\C^6$.  In general, $$\X_r=\mathrm{Spec}_{max}\left(\C[t_1,...,t_{N_r}]/\mathfrak{I}_r\right)$$ where $\mathfrak{I}_r$ is an ideal generated by $\frac{1}{2}\left(\binom{r}{3}^2+\binom{r}{3}\right)+r\binom{r}{4}$ polynomials (see \cite{Dr}).

\subsection{Some $\SL$ Representation Theory}\label{ss:reptheory}

We now review some basic $\SL$ representation theory, and reintroduce some notation from our first paper \cite{LP}.

Let $V_0=\C =V_0^*$ be the trivial representation of $\SL$.  Denote the standard basis for $\C^2$ by $e_1=\tmxt10$ and $e_2=\tmxt01$, and the dual basis by its transpose:
$e_1^*$ and $e_2^*.$

Then the standard representation and its dual are
$$V=V_1 = \C e_1\oplus \C e_2 \quad\text{and}\quad V^*=V_1^*=\C e_1^*\oplus \C e_2^*.$$
Denote the symmetric powers of these representations by
$$V_n=\mathrm{Sym}^n(V) \textrm{ and } V_n^*=\mathrm{Sym}^n(V^*).$$
One can show $V_n\approx (V_n)^*\approx V_n^*$ as $\SL$-modules, which is particular to $\SL$ and not obvious.

\begin{proposition}The symmetric powers of the standard representation of $\SL$ are all irreducible representations and
moreover they comprise a complete list.\end{proposition}

For proof see \cite{FH}.

The tensor product $V_a\otimes V_b,$ where $a,b \in \N $, is also a representation of $\SL$ and
decomposes into irreducible representations as follows:
\begin{proposition}[Clebsch-Gordan formula]\label{clebshgordan}
  \begin{equation}\label{eq:clebsch-gordan}
  V_a\otimes V_b\approx\bigoplus^{\mathrm{min}(a,b)}_{j=0} V_{a+b-2j}.
  \end{equation}
\end{proposition}

The particular $V_{a+b-2j}$ summands in this formula are described by the following:
\begin{definition}\label{def:admissible}
    Given $a,b\in\N$, we write $c\in\iadm{a,b}$ and say that $\{a,b,c\}$ is an \emph{admissible triple} for all $c=a+b-2j$, $0\le j\le \min(a,b)$.
\end{definition}

We remind ourselves of Schur's Lemma for later use:
\begin{proposition}[Schur's Lemma]
Let $G$ be a group, $V$ and $W$ irreducible representations of $G$, and $f\in\hm_G(V,W)$ with $f\neq0$.
If $V\approx W$, then $\dim_\C\mathrm{Hom}_G(V,W)=1$; and
if $V\not\approx W$, then $\dim_\C\mathrm{Hom}_G(V,W)=0$.
\end{proposition}

A tensor product $v_1\otimes v_2 \otimes \cdots \otimes v_n \in V^{\otimes n}$ projects to $V_n$ by \emph{symmetrizing}. We define its image under this operation by
    $$v_1\circ v_2 \circ \cdots \circ v_n \equiv \frac{1}{n!} \sum_{\sigma\in \Sigma_n} v_{\sigma(1)}\otimes v_{\sigma(2)} \otimes \cdots \otimes v_{\sigma(n)},$$
where the sum is over all permutations on $n$ elements. There exist bases for $V_n$ and $V_n^*$, given by
the elements%
\begin{align*}
    {\sf n}_{n-k}&=e_1^{n-k}e_2^k=\underbrace{e_1\circ e_1\circ \cdots \circ e_1}_{n-k}\circ
        \underbrace{e_2\circ e_2\circ \cdots \circ e_2}_{k}\quad\text{and}\\
    {\sf n}_{n-k}^*&=(e_1^*)^{n-k}(e_2^*)^k=\underbrace{e_1^*\circ e_1^*\circ \cdots \circ e_1^*}_{n-k}\circ
        \underbrace{e_2^*\circ e_2^*\circ \cdots \circ e_2^*}_{k}\:,
\end{align*}
respectively, where $0\le k\le n$. These elements will be described in diagrammatic form in section \ref{ss:diagrammatic-symmetrization}.

The ``dual'' pairing between $V_n$ and $V_n^*$ is given by
$${\sf n}^*_{n-k}(v_1\circ v_2 \circ \cdots \circ v_n)=
\frac{1}{n!} \sum_{\sigma \in \Sigma_n}({\sf n}_{n-k})^*(v_{\sigma(1)}\otimes v_{\sigma(2)} \otimes
\cdots \otimes v_{\sigma(n)}),$$ where $\Sigma_n$ is the symmetric group on $n$ elements. In particular,
    $${\sf n}^*_{n-k}({\sf n}_{n-l})
    =\frac{(n-k)!k!}{n!}\delta_{kl}=\raisebox{2pt}{$\delta_{kl}$}\!\Big/\!\raisebox{-2pt}{$\tbinom{n}{k}$}.$$

Let $g=\tmx{g_{11} & g_{12}}{g_{21} & g_{22}}\in \SL$. The $\SL$-action on $V_n$ is given by
\begin{align*}
  g\cdot {\sf n}_{n-k}
    &=(g_{11}e_1+g_{21}e_2)^{n-k}(g_{12}e_1+g_{22}e_2)^k\\%
    &=\sum_{\substack{0\le j\le n-k\\0\le i \le k}}\tbinom{n-k}{j}\tbinom{k}{i}
        \left(g_{11}^{n-k-j}g_{12}^{k-i}g_{21}^{j}g_{22}^{i}\right){\sf n}_{n-(i+j)}.
\end{align*}

For the dual, $\SL$ acts on $V_n^*$ in the usual way:
\begin{displaymath}
    (g \cdot {\sf n}^*_{n-k})(v)= {\sf n}^*_{n-k}(g^{-1}\cdot v )
    \quad\textrm{ for } v\in V_n.
\end{displaymath}

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