$\lambda=(12,11,10,9,3,2,1,0)$ is shown in Fig. (11). One can check easily
that $\Lambda(4,2,4)$ and $\lambda$ are incomparable with dominance partial
order. That being said, one can also check that $\lambda$ is a vanishing
orientation type of $G$.

##  Appendix E Proofs

### Proposition III.1

The elementary symmetric polynomials in $N$ variables $z_{1},\cdots,z_{N}$ can
be found from their generating function:

$E(t):=\prod_{i=1}^{N}(1+z_{i}t)=\sum_{r=0}^{N}e_{r}(z_{1},\cdots,z_{n})\>t^{n}$

If the projective roots are away from infinity, i.e. $[1:0]$, with the
definition $z_{i}=x_{i}/y_{i}$, the binary form $\beta_{N}(X,Y)$,

$\displaystyle\beta_{N}(X,Y)$
$\displaystyle=\left(\prod_{j=1}^{N}y_{j}\right)\prod_{i=1}^{N}(X-Yz_{i})$
$\displaystyle=\left(\prod_{j=1}^{N}y_{j}\right)\sum_{r=0}^{N}(-1)^{r}e_{r}(z_{1},\cdots,z_{N})X^{N-r}Y^{r}$

Since all roots are away from infinity, then $\prod_{j=1}^{N}y_{j}\neq 0$. By
comparing to the definition of coefficients of binary form
$a_{0}=\prod_{j=1}^{N}y_{j}\neq 0$. Since the coefficients should be treated
as points $\mathbb{P}^{N}$, one can then take $a_{0}=1$. With that choice,
then

${N\choose r}\>a_{r}=(-1)^{r}e_{r}(z_{1},\cdots,z_{N})$

∎

### Theorem III.2

The proof of this theorem is mostly calculation. We will shorten the
calculations and leave the details to the reader.

The liner Lie algebra $\mathfrak{sl}_{2}(\mathbb{C})$ is the three (complex)
dimensional vector space with the Pauli matrices
$\sigma^{+},\sigma^{-},\frac{1}{2}\sigma^{z}$ as its basis. These three are
the infinitesimal generators of $\mathrm{SL}_{2}(\mathbb{C})$. Take elements
in $\mathrm{SL}_{2}(\mathbb{C})$ infinitesimally close to identity:

$\begin{cases}(X,Y)\xrightarrow{1+\epsilon\sigma^{+}}(X+\epsilon Y,Y)\\\
(X,Y)\xrightarrow{1+\epsilon\frac{\sigma^{z}}{2}}((1+\frac{\epsilon}{2})X,(1-\frac{\epsilon}{2})Y)\\\
(X,Y)\xrightarrow{1+\epsilon\sigma^{-}}(X,Y+\epsilon X)\end{cases}$

Remember that to find the transformations $g\star a_{r}$, we need to satisfy
$\beta_{N}(g(X,Y)^{t};\\{g\star a_{r}\\})=\beta_{N}(X,Y;\\{a_{r}\\})$, or
better yet, $\beta_{N}(g^{-1}(X,Y)^{t};\\{a_{r}\\})=\beta_{N}(X,Y;\\{g\star
a_{r}\\})$. Now

$\begin{cases}\beta_{N}(X,Y)\xrightarrow{1-\epsilon\sigma^{+}}\sum_{k=0}^{N}{N\choose
k}[a_{k}-\epsilon ka_{k-1}]{\scriptstyle X^{N-k}Y^{k}}\\\
\beta_{N}(X,Y)\xrightarrow{1-\epsilon\frac{\sigma^{z}}{2}}\sum_{k=0}^{N}{N\choose
k}[a_{k}-\epsilon(\frac{N}{2}-k)a_{k}]{\scriptstyle X^{N-k}Y^{k}}\\\
\beta_{N}(X,Y)\xrightarrow{1-\epsilon\sigma^{-}}\sum_{k=0}^{N}{N\choose
k}[a_{k}-\epsilon(N-k)a_{k+1}]{\scriptstyle X^{N-k}Y^{k}}\end{cases}$

This means that via the induces action of $\mathrm{SL}_{2}(\mathbb{C})$ on the
space of coefficients $\mathbb{P}^{N}$, one has

$\begin{cases}(1+\epsilon\sigma^{+})\star
a_{k}=\widetilde{a}^{+}_{k}:=a_{k}-\epsilon ka_{k-1}\\\
(1+\epsilon\frac{\sigma^{z}}{2})\star
a_{k}=a_{k}-\epsilon(\frac{N}{2}-k)a_{k}\\\ (1+\epsilon\sigma^{-})\star
a_{k}\xrightarrow{1+\epsilon\sigma^{-}}\widetilde{a}^{-}_{k}:=a_{k}-\epsilon(N-k)a_{k+1}\end{cases}$

Note that, although this seemingly depends on $a_{N+1}$ and $a_{-1}$, these
two auxiliary entities always are multiplied by a zero. Define the following
operators

$\displaystyle\Delta$
$\displaystyle=\sum_{k=0}^{N}a_{k}\frac{\partial}{\partial a_{k}},$
$\displaystyle\quad\ell^{+}$
$\displaystyle=\sum_{k=1}^{N}ka_{k-1}\>\frac{\partial}{\partial a_{k}}$
$\displaystyle\ell^{z}$
$\displaystyle=\frac{N}{2}\Delta-\sum_{k=1}^{N}ka_{k}\frac{\partial}{\partial
a_{k}},$ $\displaystyle\quad\ell^{-}$
$\displaystyle=\sum_{k=0}^{N-1}(N-k)a_{k+1}\frac{\partial}{\partial a_{k}}$

Operator $\Delta$ is just the Euler operator, measuring the degree of a
homogeneous polynomial $Q$. One easily checks that the following are
equivalent:

  1. 1.

$Q$ is a binary invariant.

  2. 2.

$Q$ is invariant under induced action of $1+\epsilon\frac{\sigma^{\pm}}{2}$
and $1+\epsilon\frac{\sigma^{z}}{2}$ (since $\mathrm{SL}_{2}(\mathbb{C})$ is
the exponential of $\mathfrak{sl}_{2}(\mathbb{C})$).

  3. 3.

$\ell^{+}Q=\ell^{z}Q=\ell^{-}Q=0$.

We want to find $\ell^{\pm},\ell^{z}$ in terms of roots. To do so, define the
operator
$D_{m}:\mathbb{C}((z_{1},\cdots,z_{N}))\to\mathbb{C}((z_{1},\cdots,z_{N}))$
(where $\mathbb{C}((z_{1},\cdots,z_{N}))$ is the ring of formal power series)
as

$D_{m}=-\sum_{j=1}^{N}z_{j}^{m+1}\frac{\partial}{\partial z_{j}}$

for all $m\in\mathbb{Z}$. One can check that $[D_{m},D_{n}]=(m-n)D_{m+n}$.
This is a differential representation of Wick algebra on
$\mathbb{C}((z_{1},\cdots,z_{N}))$.

######  Claim.

Let $e_{k}(z_{1},\cdots,z_{N})$ be the $k$th elementary symmetric polynomial.
Then for $1\leq k\leq N$, one has

$\begin{cases}D_{-1}e_{k}=-(N+1-k)e_{k-1}\\\ D_{0}e_{k}=-ke_{k}\\\
D_{+1}e_{k}=-e_{1}e_{k}+(k+1)e_{k+1}\end{cases}$

with the convention $e_{0}=1$.

###### Proof.

First of all note that $D_{0}$ is the just negative of the Euler operator.
Since $e_{k}$ is homogeneous of order $k$, the identity $D_{0}e_{k}=-ke_{k}$
is trivial. We therefore need to show the $\pm 1$ case.

Define
$p_{k}(z_{1},\cdots,z_{N})=\sum_{j=1}^{N}z_{j}^{k}\in\mathbb{C}((z_{1},\cdots,z_{N}))$.
Clearly, $D_{m}p_{k}=-kp_{m+k}$ for all $m$. In particular if $k\geq 1$ then
$p_{k}$ is a symmetric polynomial, and $D_{\pm 1}p_{k}$ is also a symmetric
polynomial (with $p_{0}=N$). Newton’s identities (see (Macdonald, 1998, 2.11))
state that

$ke_{k}=\sum_{r=1}^{k}(-1)^{r-1}e_{k-r}p_{r}$

We therefore prove the our assertions by induction on $k$. For the base of
induction note that $p_{1}=e_{1}$. We leave the details of the calculation to
the reader. ∎

Now using prop. III.1, for the case $a_{0}\neq 0$ but other than that an
arbitrary constant, ${N\choose k}a_{k}=(-1)^{k}a_{0}e_{k}$, for $1\leq k\leq
N$ one finds

$\begin{cases}D_{-1}a_{k}=ka_{k-1}\\\ D_{0}a_{k}=-ka_{k}\\\
D_{-1}a_{k}=-e_{1}a_{k}-(N-k)a_{k+1}\end{cases}$

Note that $e_{1}=z_{1}+\cdots+z_{N}=Z=-Na_{1}/a_{0}$. Combining these
together, we find that in the coefficient space,

$\displaystyle D_{-1}=\sum_{k=1}^{N}ka_{k-1}\frac{\partial}{\partial
a_{k}},\qquad D_{0}=-\sum_{k=1}^{N}ka_{k}\frac{\partial}{\partial a_{k}}$
$\displaystyle D_{+1}=-Z\sum_{k=1}^{N}a_{k}\frac{\partial}{\partial
a_{k}}-\sum_{k=1}^{N-1}(N-k)a_{k+1}\frac{\partial}{\partial a_{k}}$

Both sums in $D_{+1}$ start from $k=1$. However, if we were to add their $k=0$
terms, it would become $C:=(-Za_{0}-Na_{1})\partial/\partial a_{0}$. But
$Z=-Na_{1}/a_{0}$, so $C=0$. We will therefore lift the sum index in $D_{+1}$
to $k=0$. Now suppose we restrict the action of all of these operator to
symmetric polynomials $P$ in $z_{1},\cdots,z_{N}$ with local degree $\delta$
(which means $\Delta P^{\flat}=\delta P^{\flat}$). Then

$\displaystyle\ell^{+}$
$\displaystyle=D_{-1}=-\sum_{j=1}^{N}\frac{\partial}{\partial z_{j}}=-L^{+}$
$\displaystyle\ell^{z}$
$\displaystyle=\frac{N\delta}{2}+D_{0}=\frac{N\delta}{2}-\sum_{j=1}^{N}z_{i}\frac{\partial}{\partial
z_{i}}=L^{z}$ $\displaystyle\ell^{-}$ $\displaystyle=-Z\delta-
D_{+1}=-Z\delta+\sum_{j=1}^{N}z_{i}^{2}\frac{\partial}{\partial z_{i}}=-L^{-}$

This shows that $P$ is a $(N,\delta)$ FQH-like polynomial if and only if
$P^{\flat}$ is a binary invariant of order $N$ and degree $\delta$.

##### Discussion:

Let $g\in\mathrm{SL}_{2}(\mathbb{C})$. The action of $g$ on coefficients is
translated to an action on the projective root. In fact, under $g$

$\begin{pmatrix}x_{i}\\\ y_{i}\end{pmatrix}\mapsto
g\cdot\begin{pmatrix}x_{i}\\\ y_{i}\end{pmatrix}$

Exactly like $(X,Y)^{t}$ transforms. This shows given $P$, the binary dual
$P^{\flat}$ is a binary inavriant if and only if by uniformly transforming all
of the projective roots by any $g\in\mathrm{SL}_{2}(\mathbb{C})$, the
polynomial $P$ (or more precisely the section corresponding to $P$) stays
invariant. At the same time, thinking of Riemann sphere as the complex
projective line, since the automorphism group of $\mathbb{C}^{2}$ are
$\mathrm{GL}_{2}(\mathbb{C})$, the automorphism group of $\mathbb{P}^{1}$ is
simply $\mathrm{PGL}_{2}(\mathbb{C})$. The action of this automorphism is by
matrix multiplication. Moreover
$\mathrm{PGL}_{2}(\mathbb{C})=\mathrm{SL}_{2}(\mathbb{C})/{\pm 1}$. In other
words, binary invariance is basically synonymous with invariance under the
natural action of the automorphism group of the Riemann sphere. The special
$\mathrm{SL}_{2}(\mathbb{C})$ transformations of interest are:

  1. 1.

$\exp(t\sigma^{+})=\begin{pmatrix}1&t\\\ 0&1\end{pmatrix}$
which is a translation by amount $t$ for the points in
$\mathbb{P}^{1}-\\{\infty\\}$ (while fixing the infinity).

  2. 2.

$\exp(t\sigma^{-})=\begin{pmatrix}1&0\\\ t&1\end{pmatrix}$
which is a translation by amount $t$ for points in $\mathbb{P}^{1}-\\{0\\}$
(while fixing zero).

  3. 3.

$\exp((\ln r+i\theta)\sigma^{z})=\begin{pmatrix}re^{i\theta}&0\\\
0&[re^{i\theta}]^{-1}\end{pmatrix}$
which is a scaling by amount $r$ followed by a rotation by amount $\theta$ in
$\mathbb{P}^{1}-\\{\infty\\}$ (since the transition map between the two chart
of $\mathbb{P}^{1}$ is $z\mapsto 1/z$, the transformation in
$\mathbb{P}^{1}-\\{0\\}$ is automatic).

Note that, since the two translations generate the whole automorphism group,
$P$ is a FQH-like polynomial, if and only if it is translational invariant
over the Riemann sphere. ∎

### Theorem III.3(Cayley)

See Elliott (Elliott, 1913, §72-74 & §88-89).

### Proposition IV.1

We will prove here that any bipartite (multiple) graph has a perfect matching.
Then using that perfect matching one constructs a perfect display. But first,
we need to define a few concepts:

Given a graph $G=(V,E)$, a matching $M$ in $G$ is a set of pairwise non-
adjacent edges; that is, no two edges share a common vertex. A matching $M$ is
maximum of there is no other matching $M^{\prime}$ with $|M^{\prime}|>|M|$. A
matching $M$ is _perfect_ if every vertex of $G$ appears as the end point of
some (and therefore only one) edge in $M$. Any perfect matching is maximum. If
$X\subset V$, the _neighborhood_ of $X$, denoted by $N(X)$ is the set of all
vertexes $x\in V-X$ that are adjacent to some vertex in $X$. We say a set
$S\subset E$ _covers_ a set $X\subset V$ if every $x\in X$ is the end point of
some $e\in S$. With all of those definitions, we can now state the famous
Hall’s marriage theorem

######  Theorem (Hall’s Marriage Theorem).

Let $G$ be a simple bipartite graph with bipartition $(A,B)$. Then there is a
matching $M$ which covers $A$ if and only if $|N(X)|\geq|X|$ for every
$X\subseteq A$.

For three different proves of this, see (Diestel, 2000, thm. 2.1.2). As a
corollary of Hall’s marriage theorem we have:

######  Corollary.

Let $G$ be a $d$-regular biparite multi-graph. Then $G$ has a perfect
matching.

###### Proof.

We will actually prove that if $G$ is the $d$-regular bipartite multigraph,
then the _underlying_ simple graph of $G$, which we denote by $G_{s}$ has a
perfect matching. By underlying simple graph we mean when all parallel edges
are treated as a single edge.

Let the partitions of $G$ be $A,B$. Take any subset $X\subset A$. The number
of edges with one end in $X$ is $\epsilon_{X}=|X|d$ due to $d$-regularity. Let
$N(X)\subset B$ be the neighborhood of $X$. The number of edges with one end
in $N(X)$ is $\epsilon_{N(X)}=|N(X)|d$. But any edge incident to $X$ is also
incident to $N(X)$ by definition of neighborhood. So
$\epsilon_{X}\leq\epsilon_{N(X)}$, or $|X|\leq|N(X)|$. Note that the
neighborhood of $X$ in $G$ is the same as neighborhood of $X$ in $G_{s}$. So
now by Hall’s marriage theorem we find that $G_{s}$ has a perfect matching.
But one can also treat $G_{s}$ as a subgraph of $G$. So $G$ has a perfect
matching. ∎

Now we prove that any bipartite $d$-regular multi-graph $G$ of order $2v$
admits a perfect display. Fix some perfect matching $M$ of $G$. Label the
vertexes in $A$ and $B$ as $(-1,j)$ and $(+1,j^{\prime})$ respectively (with
$j,j^{\prime}\in\mathbb{Z}_{v}$) such that each edge in $M$ is of the form
$(-,j)(+,j)$; i.e. it connects $(-,j)$ to $(+,j)$.∎

### Theorem V.1

The proof of this theorem is completely analogous to thm. V.2 (This theorem is
just the (ee) case of thm. V.2).

### Theorem V.2

We will use the notation used in thm. VI.1 for this proof. Although this
theorem was shown first in the paper, thm. VI.1 uses nothing discussed in thm.
V.2.

##### Elements of Dihedral Group:

$D_{m}=\langle q,x\mid q^{m}=x^{2}=(xq)^{2}=1\rangle$. All elements of $D_{m}$
can be written as $q^{i}x^{k}$ with $0\leq i<m$ and $k=0,1$. One has
$q^{i}xq^{j}=q^{i-j}x$.

##### Face Matrix

: Let us recall the face matrix $F$ since we will use it often: Here $v=2k$
and $F$ is a $v\times v$ matrix. Let $F^{\pm}=\pi^{\pm}_{k}(F)$. Suppose

  1. 1.

$F^{+}=F^{-}:=\widetilde{F}$.

  2. 2.

For all $s\in\mathbb{Z}_{v}$ one has $F_{s,s-1}=\rho\neq 0$.

where $\tilde{F}$ is a $k\times k$ matrix given by
$\widetilde{F}_{rs}=\sum_{j=0}^{k-1}\mu_{j}\delta_{s-r,j}$ such that if
$r+s=-1\pmod{k}$ then $\mu_{r}=\mu_{s}$. Also $\mu_{0}\neq 0$. With these
conditions clearly $F$ is a face matrix.

Let $(a,r)$ be the labeling we used in proof of thm. VI.1. We label the
vertexes of $G_{n}=\overline{TT_{n}\otimes\Phi}$ by the elements of $D_{nk}$
as follows:

$\begin{cases}(a,2i)&\mapsto q^{ni+a}\\\ (a,2i+1)&\mapsto
q^{ni+a}q^{n-1}x\end{cases}$

where $0\leq i<k-1$. Suppose $0\leq a<b<n$. Fix some $0\leq\ell\leq\lceil
k/2\rceil$. Define $\ell^{\prime}$ such that $\ell+\ell^{\prime}=-1\pmod{k}$.
According to the data in the face matrix

  * (ee)

$q^{ni+a}$ connects to $q^{n(i+\ell)+b}$ and $q^{n(i+\ell^{\prime})+b}$ with
$\widetilde{F}_{i,i+\ell}=\widetilde{F}_{i,i+\ell^{\prime}}=\mu_{\ell}$ edges.

  * (oo)

$q^{ni+a}q^{n-1}x$ connects to $q^{n(i+\ell)+b}q^{n-1}x$ and $q^{n(i+\ell)+b}$
with $\widetilde{F}_{i,i+\ell}=\widetilde{F}_{i,i+\ell^{\prime}}=\mu_{\ell}$
edges.

These edges (all of which are of multiplicity $\mu_{\ell}$) are all of the
form $(g,gs)$ with $g\in D_{2k}$ and $s\in S_{n,\ell}$ with

$S_{n,\ell}=\left\\{\left.q^{\pm(n\ell+j)}\right|j\in\mathbb{Z}_{k}\setminus\\{0\\}\right\\}$

And any pair of elements of the form $(g,gs)$ with $g\in D_{2k}$ and $s\in
S_{n,\ell}$ has already been accounted for in even-by-even (ee) and odd-by-odd
(oo) cases. This brings us to even-by-odd and odd-by-even cases.

  * (eo)

$(a,2i)$ is connected to $(b,2i-1)$, i.e. $q^{ni+a}$ is connected to
$q^{n(i-1)+b}q^{n-1}x$ with $\rho$ edges.

  * (oe)

$(a,2i+1)$ is connected to $(b,2i)$, i.e. $q^{ni+a}q^{n-1}x$ is connected to
$q^{ni+b}$ with $\rho$ edges.

These edges (all of which are of multiplicity $\rho$) are all of the form
$(g,gs)$ with $g\in D_{2k}$ and $s\in S_{n,\ell}$ with

$R_{n}=\left\\{x,qx,\cdots,q^{n-2}x\right\\}$

And any pair of elements of the form $(g,gs)$ with $g\in D_{2k}$ and $s\in
R_{n}$ has already been accounted for. So with $T_{n}$ and $\mu$ as defined in
the statement of the theorem,
$\overline{TT_{n}\otimes\Phi}=\mathrm{Cay}(D_{nk},T_{n},\mu)$.∎

### Theorem VI.1

Let us first address why $\Phi$ is strongly connected.

######  Claim.

Any connected finite regular digraph is strongly connected.

###### Proof.

The proof uses the concept of a “cut” and two lemmas related to the concept.
Let $D=(V,A)$ be digraph. A _cut_ $C=(S,T^{+},T^{-})$, is a triple of sets: A
subset of the vertex set $S\subset V$, and two so-called _cut-sets_
$T^{\pm}\subset A$. $T^{+}$ is the set of arcs going from $S$ to $V-S$ while
$T^{-}$ is the set of arc going from $V-S$ to $S$. By abuse of language, we
will refer to $S$ as the cut. Also, in $D$, we say $y$ is reachable from $x$
if there exists a directed path from $x$ to $y$.

######  Lemma.

Let $D=(V,A)$ be a connected digraph. Suppose $y\in V$ is not reachable from
$x\in V$ and $x\neq y$. Then there exists a cut $S$ such that

  1. 1.

$x\in S$ and $y\in V-S$.

  2. 2.

The cut-set $T^{+}$ is empty.

  3. 3.

The cut-set $T^{-}$ is non-empty.

###### Proof.

Define $S^{\prime}\subset V$ as the set of all vertexes that are reachable
from $x$. Then let $S=\\{x\\}\cup S^{\prime}$. Clearly $y\notin S$ and
$T^{+}=\emptyset$. Moreover, if $D$ is connected, $T^{-}$ cannot be empty. ∎

######  Lemma.

Suppose $D=(V,A)$ is a digraph such that the in-degree of each vertex is equal
to its out-degree. Then for any cut $S$, the cut-sets are of equal size, i.e.
$|T^{+}|=|T^{-}|$.

###### Proof.

For each $x\in S$ let $d_{\mathrm{ext}}^{+}(x)$ be the number arcs going from
$x$ to $S-V$, and $d_{\mathrm{ext}}^{-}(x)$ the number of arcs coming from
$S-V$ into $x$. Let $d(x)=d^{+}(x)=d^{-}(x)$ (since the out-degree and in-
degree are equal). Define

$d_{\mathrm{int}}^{\pm}(x)=d(x)-d^{\pm}_{\mathrm{ext}}(x)$

Here, $d_{\mathrm{int}}^{+}(x)$ (resp. $d_{\mathrm{int}}^{-}(x)$) is the
number of arcs of the form $S\to S$ that start at $x$ (resp. end at $x$). Note
that $\sum_{x\in S}d^{\pm}_{\mathrm{ext}}(x)=|T^{\pm}|$. Define

$\Delta=\sum_{x\in
S}d_{\mathrm{int}}^{-}(x)-d_{\mathrm{int}}^{+}(x)=|T^{+}|-|T^{-}|$

Consider an internal arc, i.e. $x\to y$ with both $x,y\in S$. Note that this
arc is contributing one to $d^{+}_{\mathrm{int}}(x)$ and one to
$d^{-}_{\mathrm{int}}(y)$. As such, $\Delta=0$, resulting in
$|T^{+}|=|T^{-}|$. ∎

Now let $D$ be a connected $d$-regular digraph. Suppose, however, that $D$ is
not strongly connected, i.e. there exists $x,y\in D$ with no directed path
from $x$ to $y$. Then by lemma there exists a cut with $T^{+}=\emptyset$ but
$T^{-}\neq\emptyset$. But, the second lemma we must have $|T^{+}|=|T^{-}|$.
This is a contradiction, showing that $D$ is strongly connected. ∎

Instead of proving the properties quoted in the statement for $G_{n}$, we will
prove a collection of facts, numbered as $(\\#n)$ about accordion graphs.
Combining these facts together gives the theorem. We will start by assuming
$\Phi$ is just fully looped and nothing else. As we move further into the
discussion, we impose more conditions on $\Phi$ and discuss the implications
of those new conditions. In particular, in this whole theorem we will never
need the (E) postulate.

As mentioned, we start by assuming $\Phi$ is fully looped digraph of some
order $v$. Let $G_{n}=\overline{TT_{n}\otimes\Phi}$. Label the vertexes of
$TT_{n}$ with elements of $\mathbb{Z}_{n}$ such that $i\to j$ iff $i<j$. Also
label the vertexes of $\Phi$ by elements of $\mathbb{Z}_{v}$ arbitrarily. Let
$F$ be the adjacency matrix of $\Phi$ in this ordering. Then the vertexes of
$G_{n}$ are labeled by $(a,r)\in\mathbb{Z}_{n}\times\mathbb{Z}_{v}$. Define

$A_{a}=\\{(a,r)\mid r\in\mathbb{Z}_{v}\\},\quad R_{r}=\\{(a,r)\mid
a\in\mathbb{Z}_{n}\\}$

We call $A_{a}$ the $a$th _axis_ and $R_{r}$ the $r$th _rung_. By definition
of tensor product of digraphs, we know the following facts:

  1. (#1)

_The order of $G_{n}$ is $nv$._

  2. (#2)

Since $TT_{n}$ has no loops, no two vertexes belonging to the same axis are
adjacent. In other words, _each axis is an independent set of size $v$.
Moreover, obviously $A_{a}\cap A_{b}=\emptyset$ if $a\neq b$_.

  3. (#3)

The diagonal entry, $F_{r,r}$ is counting the number loops over vertex $r$ in
$\Phi$. Since $\Phi$ is fully looped, $F_{r,r}\neq 0$. Moreover, due to tensor
product definition, given some $r\in\mathbb{Z}_{v}$ and $a\neq
b\in\mathbb{Z}_{n}$ the vertexes $(a,r)$ and $(b,r)$ of $G_{n}$ are adjacent
by $F_{r,r}\neq 0$ parallel edges (or an edge with weight $F_{r,r}$). In other
words, _each rung is a clique of size $n$._

  4. (#4)

Take any two vertexes $(a,r)$ and $(b,s)$, with $a<b$, then the edge between
them has a weight $F_{rs}$ (with the understanding that a zero weight means no
edge). This independence from the numbers $a,b$ shows that _if $T_{k}$ is the
union of any $2\leq k\leq n$ distinct axes, then the induced subgraph of
$T_{k}$ is isomorphic to $G_{k}$._

These facts are basically the unraveling of the definition of tensor product.
We now turn our attention away from tensor product.

######  Claim (#5).

The independence number of $G_{n}$ is $v$; i.e. $\alpha(G_{n})=v$.

###### Proof.

Let $S$ be any independent set of $G_{n}$ and $r\in\mathbb{Z}_{v}$. Then since
$R_{r}$ is a clique, the intersection $S\cap R_{r}$ is either empty, or a
singleton. Since there are only $v$ rungs, $|S|\leq v$ necessarily. Since
$A_{0}$ is an independent set of size of $v$, the independence number is $v$.
∎

######  Claim (#6).

The clique number of $G_{n}$ is $n$; i.e. $\omega(G_{n})=n$.

###### Proof.

Let $C$ be any clique. Since each axis is an independent set, $C\cap A_{a}$ is
either empty or a singleton for each $a\in\mathbb{Z}_{n}$. The rest is similar
to (#5). ∎

######  Claim (#7).

The chromatic number of $G_{n}$ is $n$; i.e. $\chi(G_{n})=n$.

###### Proof.

We do know that $G_{n}$ is $n$ colorable by painting each axis with a
different color, so $\chi(G_{n})\leq n$. Suppose there exists a proper
$(n-1)$-coloring $c$ with color classes $[1],[2],\cdots,[n-1]$. Since $R_{0}$
has $n$ elements, by pigeonhole principle, there exists a color class of $c$
containing two or more vertexes of $R_{0}$. But all vertexes in $R_{0}$ are
adjacent, and therefore, in a proper coloring cannot have the same color. This
is a contradiction, which proves $\chi(G_{n})=n$. ∎

######  Claim (#8).

The core of $G_{n}$ is $K_{n}$; i.e. $G_{n}^{\bullet}=K_{n}$.

###### Proof.

First of all it is very well-known fact that complete graphs are cores (and
easy to show). We also know that $K_{n}\subset G_{n}$, since each rung is a
clique. At the same time tensor product of digraphs already comes with a
natural projection homomorphism of digraphs $\pi:TT_{n}\otimes\Phi\to TT_{n}$.
Upon applying the forgetful functor, this descends to a homomorphism of
undirected graphs
$\overline{\pi}:\overline{TT_{n}\otimes\Phi}\to\overline{TT_{n}}=K_{n}$.
Therefore $K_{n}$ is the core of $G_{n}$.

∎

Note that, up until this point, none of the above facts required neither
connectivity of $\Phi$, not its regularity. We will now discuss connectivity
related properties. For the next claim, although CRLE digraphs are strongly
connected, assuming $\Phi$ is connected suffices.

######  Claim (#9).

Let $\Phi$ be a fully looped digraph. Then the following are equivalent

  1. (a)

$\Phi$ has $c$ connected components.

  2. (b)

$H_{2}=\overline{TT_{n}\otimes\Phi}$ has $c$ connected components.

  3. (c)

$H_{n}=\overline{TT_{n}\otimes\Phi}$ has $c$ connected components for all
$n\geq 2$.

###### Proof.

First of all, $(c)\to(b)$ is trivial. For the converse $(b)\to(c)$, take two
vertexes $(a,r)$ and $(b,s)$ of $H_{n}$ with $a<b$. If $(0,r)$ and $(1,s)$ are
connected in $H_{2}$, then $(a,r)$ and $(a+1,s)$ are connected by a path
$\gamma$ in the induced subgraph of $A_{a}\cup A_{a+1}$ in $H_{n}$. By
augmenting $\gamma$ path, if necessary, with $(a+1,s)-(a+2,s)-\cdots-(b,s)$
(these vertexes all belong to $R_{s}$, so such path exists), one finds a path
from $(a,r)$ to $(b,s)$. On the other hand, suppose $(0,r)$ and $(1,s)$ are
not connected in $H_{2}$. Then $(0,s)$ and $(1,r)$ are not connected in
$H_{2}$ either. Now suppose $(a,r)$ and $(b,s)$ are connected in $H_{n}$ by
some path

$(a,r)=(a_{1},r_{1})-(a_{2},r_{2})-\cdots-(a_{n},r_{n})=(b,s)$

By going into the induced subgraph of $A_{b}\cup A_{a_{n-1}}$, due to
existence of the edge $(a_{n-1},r_{n-1})-(b,s)$, we conclude that $(0,s)$,
$(1,s)$, $(0,r_{n-1})$ and $(1,r_{n-1})$ all belong to the same connected
component of $H_{2}$. Repeating the same process for each edge, inductively we
will find that $(0,r)$ and $(1,s)$ will belong to the same connected
component. This is a contradiction, which will result in the proof of
$(b)\to(c)$.

Before we start, recall that connectedness properties of a digraph $D$ are
defined as the connectedness properties of $\overline{D}$.

We first prove that $\Phi$ is connected if and only if $H_{2}$ is connected.
Let the labeling of $\Phi$ be same way we have done before.

Suppose $H_{2}$ is connected. Take vertexes $i,j$ in $\Phi$. One can find a
path from $(0,i)$ to $(1,j)$ in $H_{2}$. Say this path is a finite sequence of
the form $(\epsilon_{k},m_{k})$ with $\epsilon_{k}=0$ if $k=$odd and
$\epsilon_{k}=1$ for $k=$odd. But then simply consider the sequence $(m_{k})$
as a sequence of vertexes of $\Phi$. This is a path connecting $i$ to $j$ in
$\overline{\Phi}$.

Conversely suppose $\Phi$ is connected and take two vertexes $(0,i)$ and
$(0,j)$ in $H_{2}$. Let $i=i_{1}-i_{2}-\cdots-i_{m+1}=j$ be a path connecting
$i$ to $j$ in $\overline{\Phi}$. We now make a path between $(0,i)$ and
$(0,j)$ in $H_{2}$. Define

$\gamma_{k}=\begin{cases}(0,i_{k})-(1,i_{k+1})-(0,i_{k+1})&\text{if }i_{k}\to
i_{k+1}\\\ (0,i_{k})-(1,i_{k})-(0,i_{k+1})&\text{if }i_{k}\leftarrow
i_{k+1}\end{cases}$

where by “if $i_{k}\to i_{k+1}$” we mean if the direction of the edge
$i_{k}-i_{k+1}$ in $\overline{\Phi}$ is in fact $i_{k}\to i_{k+1}$ in $\Phi$.
Clearly $\gamma_{k}$ is a path from $(0,i_{k})$ to $(0,i_{k+1})$ in $H_{2}$.
By augmenting all these paths together we find a path from $(0,i)$ to $(0,j)$.
Finding paths in the other cases, e.g. for $(0,i)$ and $(1,j)$, is now
immediate from this construction.

Finally suppose $\Phi=\Phi^{1}\cup\cdots\cup\Phi^{c}$ with each $\Phi$ being a
connected component of $\Phi$. Then
$H_{2}^{k}=\overline{TT_{2}\otimes\Phi^{k}}$ is connected, and by construction
of tensor product, it is obvious that $H_{2}=H_{2}^{1}\cup\cdots\cup
H_{2}^{c}$ with $H_{2}^{k}$ being connected components of $H_{2}$. The
converse is also trivial. ∎

This brings us to the purpose of strong connectivity. Recall that since
strongly connectivity of $\Phi$ is a consequence of (R) and (C) together, we
are technically using the (R) condition too. But, for the next few claims, it
suffices to assume $\Phi$ is strongly connected, but not necessarily regular.

######  Claim (#10).

If $\Phi$ is fully looped and strongly connected, then the only maximum
independent sets of $G_{n}$ are the axes of $G_{n}$.

###### Proof.

The maximum independent sets of $G_{n}$ are of size $\alpha(G_{n})=v$. Let $S$
be such set, then by necessity $S\cap R_{r}$ is a singleton for every $r$.
Therefore

$S=\\{(a_{0},0),(a_{1},1),\cdots,(a_{v-1},v-1)\\}$

for some $a_{0},\cdots,a_{v-1}\in\mathbb{Z}_{n}$. let
$a=\min\\{a_{0},\cdots,a_{v-1}\\}$ and define $s\in\mathbb{Z}_{v}$ such that
$(a,s)\in S$ (i.e. $a=a_{s}$). Suppose there exists an arc $s\to r$ in $\Phi$.
Then for all $b>a$ there exists an edge between $(a,s)$ and $(b,r)$.
Therefore, by definition of independent set, $(b,r)\notin S$ for all $b>a$.
Then by minimality of $a$, and since $S\cap R_{r}$ is a singleton, one must
necessarily have $a_{r}=a$ too, i.e. $(a,r)\in S$. Now note that since $\Phi$
is strongly connected, for each $r\in\Phi$, with $r\neq s$, there is a
directed path $s\to r_{1}\to r_{2}\to\cdots\to r_{m}=r$. A simple induction
argument then shows that $a_{r}=a$ for all $r\neq s$. Therefore $S=A_{a}$. ∎

######  Claim (#11).

If $\Phi$ is fully looped and strongly connected, then $G_{n}$ is uniquely
$n$-colorable. If $u_{G_{n}}$ is this unique $n$-coloring, the color classes
of $u_{G_{n}}$ are exactly the axes of $G_{n}$.

###### Proof.

We have already proved that $\chi(G_{n})=n$. Since $G_{n}$ has $nv$ and
$\alpha(G_{n})=v$, if $c$ is any proper $n$-coloring, each color class of $c$
is necessarily of size $v$. Since the axes of $G_{n}$ are the only possible
independent sets of size $v$, $G_{n}$ is uniquely colorable. ∎

As mentioned in the main body, it is possible to discard the (C) condition by
changing the (R,E) condition to: $\Phi$ is $d$-regular and if $\Phi^{\prime}$
is a connected component of $\Phi$, then $|\Phi^{\prime}|d=$even. If one
insists on doing this, then all the results we have shown in this paper should
be applied to connected components first. Nothing of any meaningful
significance is gained by doing so.

Finally, we will also add the (R) condition. Suppose $\Phi$ is $d$-regular
digraph (no other condition is necessary). Then one immediately finds that

  * (#12)

$G_{n}$ is a regular graph of degree $(n-1)d$.

Finally, we the combination of CRL gives us:

######  Claim (#13).

If $\Phi$ is fully looped, connected and $d$-regular, then each maximum
independent set $S$ of $G_{n}$ is dominating. Furthermore, if $x\notin S$,
then $x$ connects to $S$ with exactly $d$ edges.

###### Proof.

Fix some axis $A_{a}$ and some vertex $(b,r)$ with $a\neq b$. Let $H_{ab}$
denote the induced subgraph of $A_{a}\cup A_{b}$. We know that $H_{ab}$ is
bipartite, with partitions being $A_{a}$ and $A_{b}$, and regular of degree
$d$. So $(b,r)$ connects to $A_{a}$ with exactly $d$-edges. So in particular,
$A_{a}$ is dominating. ∎

This concludes the theorem.∎

### Proposition VI.2

Consider $G=(V,E)$ be a loopless multi-graph with $V=\\{1,2,\cdots,N\\}$
already ordered. Let $I=\\{(i,j)\in V^{2}\mid i<j\\}$ and also let
$W=(w_{ij})$ be the adjacency matrix of $G$. Then the graph monomial of $G$
with these conventions is

$\widetilde{P}_{G}(z_{1},\cdots,z_{N})=\prod_{(i,j)\in
I}(z_{i}-z_{j})^{w_{ij}}$

This is a multiplication of $|E|$ linear factors of the form $(z_{i}-z_{j})$.
More precisely, given any edge $e=ij\in E$ (there are actually $w_{ij}$ copies
of $ij$ in $E$ since $G$ is multiple), choosing either $z_{i}$ or $-z_{j}$ is
equivalent to putting the direction on $e$. Doing the same for all edges, i.e.
making $|E|$ choices, will give an orientation $\omega$ in $G$. Let
$\omega_{0}$ be the orientation induced by natural ordering of the labels,
i.e. $i\to j$ iff $i<j$ and $ij\in E$. We associate the orientation
$\omega_{0}$ to when out of every linear factor $z_{i}-z_{j}$ we choose
$z_{i}$ (i.e. the variable with smaller index). Define
$S(\omega,\omega_{0})\subset E$ as the set of all edges $e$ such that
$\omega(e)\neq\omega_{0}(e)$. Define
$\mathrm{sgn}(\omega)=(-1)^{|S(\omega,\omega_{0})|}$. Then note that if
$\omega$ involves the choice of $-z_{j}$ out of $s$ linear forms, then
$|S(\omega,\omega_{0})|=s$. Given an orientation $\omega$, let $G_{\omega}$ be
the corresponding digraph. Make the sequence

$\Delta^{+}(\omega)=(\delta^{+}_{1},\delta^{+}_{2},\cdots,\delta^{+}_{N})$

where $\delta^{+}_{i}$ is the number of arcs going out $i$ in $G_{\omega}$.
Note that $\delta^{+}_{i}$ is exactly the number of times variable $z_{i}$ has
been chosen overall. Make the convention

$\mathbf{z}^{\Delta^{+}(\omega)}:=z_{1}^{\delta^{+}_{1}}z_{2}^{\delta^{+}_{2}}\cdots
z_{N}^{\delta^{+}_{N}}$

Combining all of this, we find that

$\widetilde{P}_{G}(z_{1},\cdots,z_{N})=\sum_{\omega}\mathrm{sgn}(\omega)\>\mathbf{z}^{\Delta^{+}(\omega)}$

where the sum is done over all orientations of $G$. Given any orientation
$\omega$, the sequence $\Delta^{+}(\omega)$ can be reordered into a partition
of $|E|$. We call this partition, $\lambda(\omega)$, the orientation type of
$\omega$. Let $\Omega_{\lambda}$ be the set of all orientations of type
$\omega$, and let $\mathcal{T}_{G}$ be the set of all orientation types of
$G$. Clearly if $\omega\in\Omega_{\lambda}$, then by definition of orientation
type,

$\mathscr{S}\left(\mathbf{z}^{\Delta^{+}(\omega)}\right)=m_{\lambda}(z_{1},\cdots,z_{N})$

with $\widetilde{m}_{\lambda}$ the free bosonic state corresponding to
$\lambda$. Therefore, we immediately obtain

$P_{G}=\sum_{\lambda\in\mathcal{T}_{G}}\widetilde{c}_{\lambda}\widetilde{m}_{\lambda},\quad\widetilde{c}_{\lambda}=\sum_{\omega\in\Omega_{\lambda}}\mathrm{sgn}(\omega)$

∎

### Theorem VI.3

Let $\bar{\omega}$ be an orientation of
$G_{n}=\overline{TT_{n}\otimes\Phi}=(V_{n},E_{n})$ that makes it into the
digraph $TT_{n}\otimes\Phi$. In other words, in the construction one can
already read this orientation $\omega$ before applying the forgetful functor.
We want to first show that $\bar{\omega}$ is of type $\Lambda(n,v,d)$. Using
the notation we used in the proof of thm. VI.1, if there is an edge between
$(a,r)$ and $(b,s)$ with $a<b$, then $(a,r)\to(b,s)$ according to $\omega$. As
a result, $\delta^{+}(0,r)=\delta_{n}=(n-1)d$ for all $r\in\mathbb{Z}_{r}$. In
other words, any edge with one end in $A_{0}$ will go out of $A_{0}$.
Furthermore, since by deleting the zeroth axis the resulting graph is
$G_{n-1}$, by induction on $n$ one finds that the type of $\bar{\omega}$ is

$(\underbrace{\delta_{n}\cdots\delta_{n}}_{\times
v}\underbrace{\delta_{n-1}\cdots\delta_{n-1}}_{\times
v}\cdots\underbrace{\delta_{2}\cdots\delta_{2}}_{\times v})$

which is exactly $\Lambda(n,v,d)$.

Let us find all orientations of type $\Lambda(n,v,d)$. Suppose $\omega$ is of
type $\Lambda(n,v,d)$. Let us define $\omega_{n}=\omega$ for further
convenience. Note that the set $X_{0}(\omega)\subset V$ which consists of all
vertexes of out-degree $\delta_{n}=(n-1)d$ is an independent set of size $v$.
As such $X_{0}(\omega)$ is a color class of $u_{G_{n}}$, the unique
$n$-coloring of $G_{n}$. Now the induced subgraph of $V-X_{0}$ is $G_{n-1}$.
Hence the orientation $\omega_{n}$ induces an orientation $\omega_{n-1}$ on
$G_{n-1}$, which is clearly of type $\Lambda(n-1,v,d)$. By induction on $n$,
one finds that for all $a\in\mathbb{Z}_{n}-\\{n-1\\}$, the set $X_{a}(\omega)$
consisting of all vertexes of out-degree $\delta_{n-a}=(n-1-a)d$ is an
independent set of size $v$. At the same time, the set $X_{n-1}(\omega)\subset
V_{n}$ of all vertexes of $V_{n}$ with out-degree zero (in $G_{n}$ with
orientation $\omega_{n}$) is an independent set of size $v$ too. In other
words, $\omega$ decomposes $V_{n}$ as

$V_{n}=\bigcup_{a\in\mathbb{Z}_{n}}X_{a}(\omega)$

with each $X_{j}$ a color class of $u_{G_{n}}$. Note that
$X_{a}(\bar{\omega})=A_{a}$. For any $\omega\in\Lambda(n,v,d)$, let
$\sigma(a)$ be such that $X_{a}(\omega)=A_{\sigma(a)}$. This $\sigma$ is a
permutation of elements of $\mathbb{Z}_{n}$. Let $H_{a,b}$ be the induced
subgraph of $A_{a}\cup A_{b}$ in $G_{n}$. If $a<b$, then in $\omega$ all edges
in subgraph $H_{\sigma(a),\sigma(b)}$ will go out of $A_{\sigma(a)}$ into
$A_{\sigma(b)}$. Therefore there are exactly $n!$ such orientations.

Let $\omega\in\Lambda(n,v,d)$. By what we just shows, over each $H_{a,b}$,
either $\omega$ and $\bar{\omega}$ agree on orientation of every edge of
$H_{a,b}$, or they disagree on every last one of them. Let
$S(\bar{\omega},\omega)$ be the number of edges of $G_{n}$ over which
$\omega,\bar{\omega}$ disagree. By what we just showed, and since
$H_{a,b}\simeq G_{2}$ has $vd$ edges, $|S(\bar{\omega},\omega)|$ is a multiple
of $vd$. But $vd=$even, so $\omega$ and $\bar{\omega}$ have the same sign.
(This is exactly why postulate (E) is put in place). Since by construction
$\mathrm{sgn}(\bar{\omega})=1$, we find that

$c_{\Lambda(n,v,d)}=\sum_{\omega\in\Lambda(n,v,d)}\mathrm{sgn}(\omega)=n!\neq
0$

In particular, $P_{G_{n}}$ does not identically vanish.∎

### Proposition VI.4

Given a partition $\lambda$, define
$S_{k}(\lambda)=\sum_{i=1}^{k}\lambda_{i}$, i.e. the $k$th partial sum. As
usual, we prove by induction. First off, $\Lambda(2,v,d)$ is trivially a
maximal in $G_{2}$. So we need to take care of induction step.

Suppose now that $\Lambda(n-1,v,d)$ is a maximal orientation type of
$G_{n-1}$, but there exists an orientation $\omega$ of $G_{n}$ with a type
$\lambda\neq\Lambda(n,v,d)$ that dominates $\Lambda(n,v,d)$; i.e. assume
$S_{k}(\lambda)\geq S_{k}(\Lambda(n,v,d))$ for all $k$. Then, in particular,
for $k=1$ we have $\lambda_{1}\geq\delta_{n}=(n-1)d$. The only way this can
happen is if $\lambda_{1}=\delta_{n}$. Similarly one shows that
$\lambda_{1}=\lambda_{2}=\cdots=\lambda_{v}=\delta_{n}$. Define
$\lambda^{\prime}=(\lambda_{v+1},\lambda_{v+2},\cdots)$, i.e. remove the first
$v$ parts of $\lambda$. Fr $k>v$, the inequality $S_{k}(\lambda)\geq
S_{k}(\Lambda(n,v,d))$ implies $\Lambda(n-1,v,d)\preceq\lambda^{\prime}$. At
the same time, the set of all vertexes that have out-degree $\delta_{n}$ in
$\omega$ is an independent set of size $v$. Therefore this set is a color
class of the unique coloring. Denote this set by $X$. The induced subgraph of
$V-X$ is isomorphic to $G_{n-1}$. Let $\omega^{\prime}$ be the orientation on
$G_{n-1}$ induced by $\omega$. The type of $\omega^{\prime}$ is exactly
$\lambda^{\prime}$. Note that if $\lambda\neq\Lambda(n,v,d)$, then
$\lambda^{\prime}\neq\Lambda(n-1,v,d)$. Therefore we have found an orientation
type in $G_{n-1}$ which dominates $\Lambda(n,v,d)$. This contradicts
maximality of $\Lambda(n-1,v,d)$ in $G_{n-1}$.∎

### Theorem VI.5

#####  Set-Partitions:

For this proof we will need the concept of a set-partition and quite a few
terminologies related to it. Let $X$ be a finite set. A _set-partition_ $\pi$
of $X$ is the determined by the following data:

  1. 1.

$\pi$ is a set of subsets of $X$; i.e. if $U\in\pi$, then $B\subset B$. An
element $B\in\pi$ is called a _block_ of $\pi$.

  2. 2.

If $B,B^{\prime}$ are two blocks, then $B\cap B^{\prime}=\emptyset$.

  3. 3.

If $\pi=\\{B_{1},\cdots,B_{l}\\}$, then $\bigcup_{j=1}^{l}B_{j}=X$.

without loss of generality, suppose the blocks are labeled such that
$|B_{1}|\geq|B_{2}|\geq\cdots\geq|B_{l}|$. If $\lambda_{i}=|B_{i}|$, then
$\lambda=(\lambda_{1}\lambda_{2}\cdots\lambda_{l})$ is a partition of $|X|$.
One says $\pi$ is of _shape_ $\lambda$. We denote the set of all set-
partitions of shape $\lambda$ by $\mathfrak{p}_{\lambda}$. With
$\lambda_{1}=b$, in the multiplicity picture, one writes $\lambda$ as
$\lambda=(1^{m_{1}}2^{m_{2}}\cdots b^{m_{b}})$. Then the number of set-
partitions of shape $\lambda$ is

$C_{\lambda}:=|\mathfrak{p}_{\lambda}|=\frac{N!}{\prod_{r=1}^{b}m_{r}!(r!)^{m_{r}}}$

##### Tiers of a Set-Partition:

For purposes of fusion, it is convenient to do one further refinement. Let $X$
be a set and $\pi$ a set-partition of $X$. Let $f:\pi\to\mathbb{N}$ be the
cardinality function, i.e. $B\mapsto|B|$. We call the $f^{-1}(r)\subset\pi$
the $r$th _tier_ of $\pi$, denoted by $\tau^{(r)}(\pi)$. Given a block
$B\in\pi$, we say the _tier number_ of $B$ is $r$ iff $B\in\tau_{r}(\pi)$.

##### Internally Ordered Set-Partitions:

We call the collection of a set-partition $\pi$, together with bijections
$L^{(r)}:\tau^{(r)}(\pi)\to\\{1,2,\cdots,\nu_{r}\\}$ and _internally ordered
set-partition_ and we denote it by $\vec{\pi}$. If $\nu_{r}=0$, the datum is
$L^{(r)}$ is not necessary. The maps $L^{(r)}$ put a total order on the tiers
of $\pi$. We will never write these $L^{(r)}$ maps explicitly. Implicitly,
however, for an internally ordered $\vec{\pi}$, when we write

$\tau^{(r)}=\\{B_{1}^{(r)},\cdots,B^{(r)}_{\nu_{g}}\\}$

the index $j$ of $B^{(r)}_{j}$ should be understood as the total order put on
$\tau^{(r)}$. We denote the set of internally ordered set-partitions of shape
$\lambda$ by $\vec{\mathfrak{p}}_{\lambda}$. There are
$\epsilon_{\lambda}:=|\vec{\mathfrak{p}}_{\lambda}|=N!/\prod_{r=1}^{b}(r!)^{m_{r}}$
internally ordered set-partitions of shape $\lambda$.

##### Conventions:

We reserve a few symbols before we start the proof. Throughout
$\kappa=(1^{\nu_{1}}2^{\nu_{2}}\cdots
g^{\nu_{g}})=(\kappa_{1}\kappa_{2}\cdots\kappa_{s})$; i.e. these symbols are
reserved:

  * •

$\kappa$ is a partition of the number $N$, the number of
variables/particles/vertexes.

  * •

$\kappa_{i}$ is $i$th part of $\kappa$.

  * •

$s$ is the length of $\kappa$ is $s$; meaning $\kappa_{s}\neq 0$.

  * •

$g=\kappa_{1}$, i.e. $g$ is the largest part of $\kappa$.

  * •

Given any number $1\leq r\leq g$, the multiplicity of $r$ in $\kappa$ is
$\nu_{r}\geq 0$. Therefore $s=\sum_{r=1}^{g}\nu_{r}$ and
$N=\sum_{r=1}^{g}r\nu_{r}$.

#####  $\kappa$-fusion:

The first real step in our journey down the road of this proof, is to
translate the definition $\kappa$-fusion into algebraic language. Let
$S=\\{z_{1},\cdots,z_{N})$ denote our set of variables. Let
$\vec{\pi}\in\vec{\mathfrak{p}}_{\kappa}(S)$. Then the $r$th tier is

$\tau^{(r)}=\\{B^{(r)}_{1},B^{(r)}_{2},\cdots,B^{(r)}_{\nu_{r}}\\}$

Also recall that for $1\leq r\leq g$ we defined

$\mathbf{Z}^{(r)}=\\{Z^{(r)}_{1},Z^{(r)}_{2},\cdots,Z^{(r)}_{\nu_{j}}\\}$

Associated to an internally ordered set-partition $\vec{\pi}$, we define a
$\mathbb{C}$-algbera homomorphism
$\phi_{\vec{\pi}}:\mathbb{C}[S]\to\mathbb{C}[\mathbf{Z}^{(1)},\cdots,\mathbf{Z}^{(g)}]$
by sending variables in block $B_{j}^{(r)}$ to the variable $Z^{(r)}_{j}$ (for
all $r$ and $j$). This $\phi_{\kappa}$ fuses the variables of a polynomial
$f\in\mathbb{C}[S]$ according to the ordered set-partition $\vec{\pi}$.

##### Operator $\mathscr{Q}_{\kappa}$:

Given any permutation of $N$, i.e. $\sigma\in\mathfrak{S}_{N}$, by abuse of
notation, let us denote the isomorphism $\sigma:\mathbb{C}[S]\to\mathbb{C}[S]$
via $f(z_{1},\cdots,z_{N})\mapsto f(z_{\sigma(1)},\cdots,z_{\sigma(N)})$.
Given any $\mathbb{C}$-algebra homomorphism $\psi:\mathbb{C}[S]\to R$, with
$R$ some $\mathbb{C}$-algebra, we define $\sigma^{*}\psi=\psi\circ\sigma$
(pullback).

Now define
$\mathcal{M}_{\kappa}=\\{\phi_{\vec{\pi}}\mid\vec{\pi}\in\vec{\mathfrak{p}}_{\kappa}\\}$.
Note that if $\phi\in\mathcal{M}_{\kappa}$, then
$\sigma^{*}\phi\in\mathcal{M}_{\kappa}$ and moreover
$\sigma^{*}:\mathcal{M}_{\kappa}\to\mathcal{M}_{\kappa}$ is a bijection. Using
all of that, now define the $\mathbb{C}$-linear map
$\mathscr{Q}_{\kappa}:\mathbb{C}[S]\to\mathbb{C}[\mathbf{Z}^{(1)},\cdots,\mathbf{Z}^{(g)}]$
as

$\mathscr{Q}_{\kappa}=\epsilon_{\kappa}^{-1}\sum_{\phi\in\mathcal{M}_{\kappa}}\phi$
(41)

i.e. as the “average” of $\mathcal{M}_{\kappa}$. Note that for any
$\sigma\in\mathfrak{S}_{\kappa}$, one has
$\sigma^{*}\mathscr{Q}_{\kappa}=\mathscr{Q}_{\kappa}$. Let us make a two
remarks about $\mathscr{Q}_{\kappa}$:

  1. 1.

If $P$ is a symmetric polynomial in variables $S$, then for any two
$\phi,\phi^{\prime}\in\mathcal{M}_{\kappa}$ one has
$\phi(P)=\phi^{\prime}(P)=P|_{\kappa}$.

  2. 2.

Due to $\sigma^{*}:\mathcal{M}_{\kappa}\to\mathcal{M}_{\kappa}$ being a
bijection, one finds that
$\sigma^{*}\mathscr{Q}_{\kappa}=\mathscr{Q}_{\kappa}$ for any
$\sigma\in\mathfrak{S}_{N}$.

Now combining the two, and recalling that the normalized symmetrization was
$\widehat{\mathscr{S}}=N!^{-1}\sum_{\sigma\in\mathfrak{S}_{N}}\sigma$, we find

$\mathscr{Q}_{\kappa}\circ\widehat{\mathscr{S}}=N!^{-1}\sum_{\sigma}\sigma^{*}\mathscr{Q}_{\kappa}=\mathscr{Q}_{\kappa}$

So in particular, if $\widetilde{P}\in\mathbb{C}[S]$, then

$\widehat{P}:=\widehat{\mathscr{S}}[\widetilde{P}]\Longrightarrow\widehat{P}|_{\kappa}=\mathscr{Q}_{\kappa}[\widetilde{P}]$
(42)

This is specially nice, since we do not have to deal with the pesky
symmetrization operator anymore. At the same time, this brings out right
outside of the door of graph theory and SGM construction. All we need to do is
to understand what $\phi_{\vec{\pi}}$ does to a graph monomial
$\widetilde{P}_{G}$.

##### Effect of $\phi_{\vec{\pi}}$ on graph monomials:

Let $G=(V,E)$ be a multigraph of order $N$, which we have already labeled with
$S=\\{z_{1},z_{2},\cdots,z_{N}\\}$. Let $\widetilde{P}_{G}$ be the graph
monomial with respect to this order. Let $\vec{\pi}$ be an internally ordered
partition as before. Note that if some block $B_{j}^{(r)}$ contains a pair of
adjacent vertexes, then $\phi_{\vec{\pi}}\widetilde{P}_{G}=0$. However, if
there are no such blocks in $\vec{\pi}$, then each block is an independent
set. Therefore $\vec{\pi}$ induces a $\kappa$-coloring on $G$. This induced
$\kappa$-coloring, $\gamma_{\vec{\pi}}$, is explicitly defined as follows:
Define the color set as

$\mathrm{Col}=\\{(r,j)\mid 1\leq r\leq g,\>1\leq j\leq\nu_{r}\\}$

with the understanding that if $\nu_{r}=0$ for some $r$, then there is no
element of the form $(r,\bullet)$ in $\mathrm{Col}$. This set is obviously of
size $N$. We define the coloring $\gamma_{\vec{\pi}}$ as a mapping
$\gamma_{\vec{\pi}}:V\to\mathrm{Col}$. Take vertex $z$ of $G$, if it belongs
to the block $B^{(r)}_{j}$ of $\vec{\pi}$, then $\gamma_{\vec{\pi}}(z)=(r,j)$.

The $\kappa$-coloring $\gamma_{\vec{\pi}}$, in turn, leads to to construction
of a colored mulit-graph $H_{\vec{\pi}}$ which is already labeled with
elements $\mathbf{Z}^{(1)}\cup\cdots\cup\mathbf{Z}^{(g)}$. The construction of
$H_{\vec{\pi}}$ is as follows:

  * $-$

The vertex set is simply the set-partition $\pi$ (we forget the order of
$\vec{\pi}$). In other words, a vertex of $H_{\vec{\pi}}$ is a block of $\pi$.

  * $-$

The vertex $B^{(r)}_{j}$ is labeled by $Z^{(r)}_{j}$.

  * $-$

The coloring $\gamma_{\vec{\pi}}^{*}$ is found by assigning to each block of
$\pi$, its tier number.

  * $-$

Finally for edges:

$\displaystyle\\#\text{edges between blocks }B\text{ and }B^{\prime}\text{ in
}H_{\vec{\pi}}=$ $\displaystyle\sum_{z\in B}\sum_{z^{\prime}\in
B^{\prime}}\text{edges between vertexes }z\text{ and }z^{\prime}\text{ in }G$

Once we forget the extra labels $Z^{(r)}_{j}$, which the construction will
automatically do later on, $H_{\vec{\pi}}$ becomes the compression
$G\downarrow\gamma_{\vec{\pi}}$. Now, by construction of $H_{\vec{\pi}}$, one
finds that

$(\phi_{\vec{\pi}}\widetilde{P}_{G})(\mathbf{Z}^{(1)},\cdots,\mathbf{Z}^{(g)})=\widetilde{P}_{H_{\vec{\pi}}}(\mathbf{Z}^{(1)},\cdots,\mathbf{Z}^{(g)})$

where the graph monomial of $H_{\vec{\pi}}$ is calculated with respect to the
labels already on it.

##### A Step in the Reverse Direction:

Again let $G=(V,E)$ be a multigraph of order $N$, with predetermined labels in
$S$. Take a $\kappa$-coloring $\gamma:V\to\\{1,2,\cdots,s\\}$ b. For $1\leq
k\leq s$, let $[k]=\gamma^{-1}(k)$ be the color class for color $k$. Then the
set

$\pi_{\gamma}=\\{[k]\mid 1\leq k\leq s\\}$

is clearly an element of $\mathfrak{p}_{\kappa}$. We can turn $\pi_{\gamma}$
into an internally ordered set-partition in $\prod_{r=1}^{g}(\nu_{r})!$
different ways. If $\vec{\pi}\in\vec{\mathfrak{p}_{\kappa}}$ is equal to
$\pi_{\gamma}$ for some $\kappa$-coloring $\gamma$ together with an internal
order, then we say $\vec{\pi}$ is a “good” element of
$\vec{\mathfrak{p}_{\kappa}}$. Define

$\mathcal{M}^{\mathrm{good}}_{\kappa}=\\{\phi_{\vec{\pi}}\mid\vec{\pi}\text{
is a good element of }\vec{\mathfrak{p}}_{\kappa}\\}$

We have already seen that if
$\phi_{\vec{\pi}}\notin\mathcal{M}^{\mathrm{good}}_{\kappa}$, then
$\phi_{\vec{\pi}}(\widetilde{P}_{G})=0$. So it enough to work with
$\mathcal{M}^{\mathrm{good}}_{\kappa}$ only. Let us understand
$\mathcal{M}^{\mathrm{good}}_{\kappa}$ a bit better:

  1. 1.

Recall that two $s$-colorings $\gamma,\gamma^{\prime}$ said to be equivalent,
$\gamma\sim\gamma^{\prime}$ if $\gamma^{\prime}=\gamma\circ\sigma$ for some
$\sigma\in\mathfrak{S}_{s}$. If that happens, then the sets $\pi_{\gamma}$ and
$\pi_{\gamma^{\prime}}$ are exactly the same. Therefore if
$\gamma\in\mathcal{C}_{\kappa}$ we denote its equivalence class by
$\overline{\gamma}\in\mathcal{C}_{\kappa}^{*}=\mathcal{C}_{\kappa}/\sim$, then
the construction of $\mathcal{M}_{\kappa}^{\mathrm{good}}$ really depends on
$\overline{\gamma}$ and not $\gamma$. In fact if $\eta,\eta^{\prime}$ are not
equivalent, then they will lead to distinct set-partitions $\pi_{\eta}$ and
$\pi_{\eta}$

  2. 2.

Now let $\eta\in\mathcal{C}^{*}_{\kappa}$, which by abuse of language, we
still call a “$\kappa$-coloring” and $\pi_{\eta}$ the corresponding set-
partition. Construct $\vec{\pi}^{0}=\vec{\pi}_{\eta}^{0}$ with putting some
internal order on $\pi^{0}_{\eta}$. For any $1\leq r\leq g$ and
$\sigma^{(r)}\in\mathfrak{S}_{\nu_{r}}$ and any $\mathbb{C}$-algebra
homomorphism $\psi:R\to\mathbb{C}[\mathbf{Z}^{(r)}]$, define
$\sigma^{(r)}_{*}\psi=\sigma^{(r)}\circ\psi$ (pushforward). Now given any
$\vec{\pi}:=\vec{\pi}_{\eta}$, which is $\pi_{\eta}$ together with some
internal ordering, there exists $\sigma^{(r)}\in\mathfrak{S}_{\nu_{r}}$ for
each $1\leq r\leq g$, such that

$\phi_{\vec{\pi}}=\sigma^{(1)}_{*}\otimes\sigma^{(2)}_{*}\otimes\cdots\otimes\sigma^{(g)}_{*}\phi_{\vec{\pi}^{0}}$

This is pretty much the definition of internal ordering. In the above, we are
using the isomorphism
$\mathbb{C}[\mathbf{Z}^{(1)},\cdots,\mathbf{Z}^{(g)}]\simeq\mathbb{C}[\mathbf{Z}^{(1)}]\otimes\cdots\otimes\mathbb{C}[\mathbf{Z}^{(g)}]$.

##### Finishing Touch:

Note that $\epsilon_{\kappa}=c_{\kappa}\prod_{r=1}^{g}\nu_{r}!$

By what what we have shown so far,

$\displaystyle\widehat{P}_{G}|_{\kappa}=\frac{c_{\kappa}^{-1}}{\prod_{r=1}^{g}\nu_{r}!}\sum_{\phi\in\mathcal{M}^{\mathrm{good}}_{\kappa}}\phi\widetilde{P}_{G}(\mathbf{Z}^{(1)},\cdots,\mathbf{Z}^{(g)})=$
$\displaystyle
c_{\kappa}^{-1}\sum_{\eta\in\mathcal{C}_{\kappa}^{*}}\left(\prod_{r=1}^{g}\frac{1}{\nu_{r}!}\sum_{\sigma^{(r)}\in\mathfrak{S}_{\nu_{r}}}\right)\bigotimes_{r=1}^{g}\sigma^{(r)}_{*}\widetilde{P}_{H_{\vec{\pi}^{0}_{\eta}}}({\scriptstyle\mathbf{Z}^{(1)},\cdots,\mathbf{Z}^{(g)}})$
$\displaystyle=c_{\kappa}^{-1}\prod_{r=1}^{g}\widehat{\mathscr{S}}^{(r)}\sum_{\eta\in\mathcal{C}_{\kappa}^{*}}\widetilde{P}_{H_{\vec{\pi}^{0}_{\eta}}}(\mathbf{Z}^{(1)},\cdots,\mathbf{Z}^{(g)})$

Where the normalized symmetrization operator $\mathscr{S}^{(r)}$ only acts on
the variables $\mathbf{Z}^{(r)}$. Now note that, due this symmetrizations, we
no longer need to be careful about how to label the vertexes of the colored
graph $H_{\vec{\pi}^{0}_{\eta}}$ which have the same color. In other words,
the vertexes of $r$th color ($r$th tier) can be labeled by $\mathbf{Z}^{(r)}$
in any fashion one desires. This allows us to get rid of labeling of
$H_{\vec{\pi}^{0}_{\eta}}$ completely. This forgetful process turns the data
in $H_{\vec{\pi}^{0}_{\eta}}$ simply to the compression $G\downarrow\eta$ (by
$G\downarrow\eta$ we mean $G\downarrow\gamma$ for some $\gamma\in\eta$). Now
by definition of chromatic SGM, which we was given in the main body of the
paper, one concludes that:

$\widehat{P}_{G}|_{\kappa}(\mathbf{Z}^{(1)},\cdots,\mathbf{Z}^{(g)})=c_{\kappa}^{-1}\sum_{\eta\in\mathcal{C}_{\kappa}^{*}}\widehat{P}^{\chi}_{G\downarrow\eta}(\mathbf{Z}^{(1)},\cdots,\mathbf{Z}^{(g)})$

∎

### Corollary VI.6

Let $G_{n}=\overline{TT_{n}\otimes\Phi}=(V_{n},E_{n})$ with $\Phi$ a
$(v,d)$-CRLE digraph. Also let $P_{n}$ denote the SGM of $G_{n}$. To calculate
$P_{n}|_{v}$ we need to find proper colorings of type $(v1^{(n-1)v})$. This is
a $s=(n-1)v+1$ coloring problem. Up to equivalence of colorings, finding a
$\gamma:V\to\\{1,2,\cdots,(n-1)v+1\\}$ of this pattern is equivalent to
finding an independent set of size $v$, which we denote by
$S_{\overline{\gamma}}$ (recall that $\overline{\gamma}$ is equivalence class
of $\gamma$). There are exactly $n$ such independent sets, which are color
classes of unique coloring $u_{G_{n}}$.

Now the induced subgraph of $V_{n}-S_{\overline{\gamma}}$ in $G_{n}$ is
$G_{n-1}$. Moreover, we know that $S_{\overline{\gamma}}$ is a dominating set
of $G_{n}$ and every vertex in $x\in G_{n-1}$ connects to
$S_{\overline{\gamma}}$ with $d$ edges. So upon compression
$G_{n}\downarrow\gamma$ one ends up with a vertex set
$V^{\prime}=V_{n-1}\cup\\{X\\}$, an edge set $E^{\prime}=E_{n-1}\cup E_{X}$,
and a coloring $\gamma^{*}$, where

  * $-$

$X$ is the so called “fat” vertex; This is the vertex $S_{\overline{\gamma}}$
compresses to.

  * $-$

$(V_{n-1},E_{n-1})$ is the graph $G_{n-1}$, which compression does not touch.

  * $-$

For every vertex $x\in G_{n-1}$, the edge subset $E_{X}$ contains $d$ parallel
edges connecting $x$ to $X$. Those are the only edges in $E_{X}$.

  * $-$

The coloring $\gamma^{*}$ colors $X$ red and all other vertexes blue.

The normalized chromatic SGM of $G_{n}\downarrow\overline{\gamma}$ is now:

$\displaystyle\widehat{P}^{\chi}_{G_{n}\downarrow\overline{\gamma}}(Z;z_{1},\cdots,z_{(n-1)v})=$
$\displaystyle\widehat{\mathscr{S}}\left[\prod_{i=1}^{(n-1)v}(Z-z_{i})^{d}\widetilde{P}_{G_{n-1}}(z_{1},\cdots,z_{(n-1)v})\right]=$
$\displaystyle\prod_{i=1}^{(n-1)v}(Z-z_{i})^{d}\widehat{P}_{n-1}(z_{1},\cdots,z_{(n-1)v})$

where the symmetrization is over the usual variables
$z_{1},\cdots,z_{(n-1)v}$. Now that $c_{\kappa}^{-1}=v!((n-1)v)!/(nv)!$. Since
this is completely independent from $\overline{\gamma}$, and since there are
$n$ such colorings (equivalence classes), we conclude that (in the
unnormalized version)

$\displaystyle P_{n}|_{v}(Z,z_{1},\cdots,z_{(n-1)v})=$ $\displaystyle
nv!\prod_{i=1}^{(n-1)v}(Z-z_{i})^{d}P_{n-1}(z_{1},\cdots,z_{(n-1)v})$

Meaning the aggregation sequence $SGM(\overline{TT\otimes\Phi)}$ satisfies a
$(v,d)$-clustering property. ∎

### Theorem VI.7

Let $P,Q$ be two FQH-like polynomials of order $N:=nv$, local degree
$\delta=(n-1)d$ and degree $M=N\delta/2$. Define $R=P-Q$.

######  Claim.

$R$ is either zero or of local degree $\delta$.

###### Proof.

Assume $R\neq 0$, i.e. $P\neq Q$. By going into binary dual picture,
$R^{\flat}=P^{\flat}-Q^{\flat}\neq 0$. Since both $P^{\flat},Q^{\flat}$ are
homogeneous of degree $\delta$, and $R^{\flat}\neq 0$, the degree of
$R^{\flat}$ is also $\delta$. Therefore, the local degree of $R$ is also
$\delta$. ∎

Let us give a few definitions before we proceed. Let the notation
$J^{(\alpha)}_{\lambda}$ stand for a (specialized) Jack polynomial with
partition $\lambda$ and parameter $\alpha$ 111 Depending on the source, the
parameter $\alpha$ might be defined differently. In Ref. Feigin  _et al._
(2002), $J^{(\alpha)}_{\lambda}$ in $n$ variables is an eigenvector of
$H_{\alpha}=\sum_{i=1}^{n}(z_{i}\partial_{i})^{2}+\alpha\sum_{i<j}\frac{z_{i}+z_{j}}{z_{i}-z_{j}}(z_{i}\partial_{j}-z_{j}\partial_{i})$
while, for example, in the definition Bernevig and Haldane (2008a) (and most
physics sources involving Jacks) $J^{(\alpha)}_{\lambda}$ is defined as an
eigenvector of $H_{1/\alpha}$. The two notions are obviously equivalent, but
result in a discrepancy of notation in the literature. We are using the
definition in Ref. Bernevig and Haldane (2008a).. For a reference on Jack
polynomials see Stanley (1989).

######  Claim.

Let $f=J^{(\alpha)}_{\lambda}$ be over $N$ variables, of degree $M$ and local
degree $\delta$. If $\lambda=(\lambda_{1}\lambda_{2}\cdots)$, under these
conditions, $\lambda$ is a partition of $M$, with $\ell(\lambda)\leq N$ and
$\lambda_{1}=\delta$.

###### Proof.

A Jack $J^{(\alpha)}_{\lambda}$ is homogeneous, symmetric and has $\lambda$ as
the predominant root partition (see Stanley (1989)). So if $f$ is over $N$
variables and of degree $M$, then $\lambda$ is a partition of $M$ of length at
most $N$. Furthermore, by predominance, the local degree of $f$ is the local
degree of $\widetilde{m}_{\lambda}$. Therefore if local degree of $f$ is
$\delta$, then $\lambda_{1}=\delta$. ∎

Let $F_{N}(k)$ denote the space of all symmetric polynomials $f$ in $N$
variables such that $f|_{k+1}=0$. Also suppose $k,r$ are such that $k+1$ and
$r-1$ are relatively prime and let $\alpha(k,r)=-(k+1)/(r-1)$. For those
$k,r$, define the space $I_{N}(k,r)$ as the $\mathbb{C}$-span of specialized
Jack polynomials $J^{(\alpha(k,r))}_{\lambda}$ over $N$ variables in which
$\lambda$ is a $(k,r,N)$-admissible partition. The main tool we now need is
Theorem 4.2 of Feigin  _et al._ (2002). This theorem states that
$F_{N}(k)=I_{N}(k,2)$.

######  Claim.

Let $P,Q$ be as before. If $P|_{v}=Q|_{v}$ and $W_{n,v,d}=\emptyset$, then
$P=Q$.

###### Proof.

Suppose not! Then $R=P-Q\neq 0$ but one has $R|_{v}=0$. In other words, $R\in
F_{N}(v-1)=I_{N}(v-1,2)$. So $R$ is a $\mathbb{C}$-linear combination of Jacks
in $I(v-1,2)$, say

$R=c_{1}J^{(\alpha)}_{\lambda_{1}}+\cdots+c_{p}J^{(\alpha)}_{\lambda_{p}}$

let $L=\\{\lambda_{1},\cdots,\lambda_{p}\\}$. Since $\deg R=M$ and $R$ is
homogeneous, all elements of $L$ are partitions of $M$ which are
$(v-1,2,N)$-admissible. Moreover, since local degree of $R$ is $\delta$, the
largest part of all $\lambda\in L$ are $\leq\delta$ and there exist at least
one $\rho=(\rho_{1},\rho_{2},\cdots)$ in $L$ with $\rho_{1}=\delta$. By
definition of $W_{n,v,d}$, we conclude that $\rho\in W_{n,v,d}$. But by
hypothesis of the theorem, $W_{n,v,d}=\emptyset$ for all $n\geq 2$. This is a
contradiction, proving $P=Q$. ∎

With $i=1,2$, let
$\Pi^{(i)}=(P_{2}^{(i)},P_{3}^{(i)},\cdots,P^{(i)}_{n},\cdots)$ be two
sequences of FQH-like polynomials such that

  1. 1.

$P_{n}^{(i)}$ is over $nv$ variables and of local degree $(n-1)d$.

  2. 2.

$\Pi^{(i)}$ has the $(v,d)$-clustering property.

By what we have proved, if $k\geq 2$ and $q_{2}\in\mathbb{C}$ is such that
$P_{k}^{(1)}=q_{2}P_{k}^{(2)}$, then there is $q_{k}\in\mathbb{C}$ for all $k$
such that $P_{k+1}^{(1)}=q_{k}P_{k+1}^{(2)}$. At the same time, if $\dim
B_{2v,d}=1$, then one necessarily has $P_{2}^{(1)}=q_{2}P_{2}^{(2)}$ for some
$q_{2}\in\mathbb{C}$. So up to constant factors, the sequences $\Pi^{(1)}$ and
$\Pi^{(2)}$ are the same. ∎

### Proposition VI.8

##### Case $(v,2)$:

A partition $\lambda$ in $W_{n,v,2}$ is characterized by the following
conditions:

  1. (1)

$\lambda$ satisfies the $(v-1,2)$-generalized Pauli exclusion.

  2. (2)

$\widetilde{m}_{\lambda}$ lives in a Landau level of size $\delta_{n}=2(n-1)$.

  3. (3)

$\widetilde{m}_{\lambda}$ has angular momentum $M_{n}=n(n-1)v$.

  4. (4)

$\lambda$ has length at most $nv$ (or $\widetilde{m}_{\lambda}$ is for $nv$
particles).

Let us find the highest angular momentum a partition satisfying (1) and (2)
can have. This is achieved by putting $v-1$ particles in the last orbital,
then skip an orbital, then another $v-1$ bosons in orbital $2(n-2)$, etc. This
partition (which is actually $\Lambda(v-1,2,n)$) has angular momentum
$M_{\max}=n(n-1)(v-1)$. Since $M_{\max}<M_{n}$, conditions (1), (2) and (3)
cannot be satisfied simultaneously. Therefore $W_{n,v,2}=\emptyset$ for all
$n\geq 2$.

##### Case $(2,d)$ with $d=2,3,4$:

For this it is actually better to backtrack to why we defined $W_{n,v,d}$ as
we did. In proof of thm. VI.7, we defined $F_{N}(k)$ as the space of
polynomials in $N$ variables that vanish when $k+1$ particles are fused
together. The condition $W_{n,2,d}=\emptyset$ is equivalent to: “There are no
symmetric polynomial $P$ such that

  1. (1)

$P\in F_{2n}(1)$.

  2. (2)

The local degree of $P$ is $\delta_{n}=(n-1)d$.

  3. (3)

$\deg P=n(n-1)d$.”

Let us find the minimum local degree a $P$ that satisfies (1) can have. If a
symmetric polynomial $P$ is such that $P|_{z_{1}=z_{2}}=0$, then $P$ is
necessarily of the form $P=\prod_{i<j}^{2n}(z_{i}-z_{j})^{2}P^{\prime}$, with
$P^{\prime}$ sone symmetric polynomial. The minimum local degree such a
polynomial can have is $\delta_{\min}=4(n-1)+2$ which is achieved by
$P^{\prime}$ being constant. But for $d=2,3,4$ one has
$\delta_{\min}>\delta_{n}$. So conditions (1) and (2) are at odds. Hence
$W_{n,2,d}=\emptyset$ for all $n\geq 2$.∎

### Theorem VI.9 (Hermite’s Reciprocity Theorem)

See Elliot (Elliott, 1913, §131) or Kung-Rota (Kung and Rota, 1984, thm. 4.3).

## References

  * Laughlin (1983) R. B. Laughlin, Physical Review Letters 50, 1395 (1983).
  * Prange and Girvin (1987) R. E. Prange and S. M. Girvin, “The quantum hall effect, graduate texts in contemporary physics,” (1987).
  * Haldane (1983) F. D. M. Haldane, Physical Review Letters 51, 605 (1983).
  * Halperin (1984) B. I. Halperin, Physical Review Letters 52, 1583 (1984).
  * Jain (1989) J. K. Jain, Physical review letters 63, 199 (1989).
  * Olver (1999) P. J. Olver,  _Classical invariant theory_ , Vol. 44 (Cambridge University Press, 1999).
  * Sylvester (1878) J. J. Sylvester, American Journal of Mathematics 1, 64 (1878).
  * Petersen (1891) J. Petersen, Acta Mathematica 15, 193 (1891).
  * Moore and Read (1991) G. Moore and N. Read, Nuclear Physics B 360, 362 (1991).
  * Read and Rezayi (1999) N. Read and E. Rezayi, Physical Review B 59, 8084 (1999).
  * Simon  _et al._ (2007) S. H. Simon, E. Rezayi, N. Cooper, and I. Berdnikov, Physical Review B 75, 075317 (2007).
  * Green (2002) D. Green, arXiv preprint cond-mat/0202455 (2002).
  * Bernevig and Haldane (2008a) B. A. Bernevig and F. Haldane, Physical Review B 77, 184502 (2008a).
  * Bernevig and Haldane (2008b) B. A. Bernevig and F. Haldane, Physical review letters 100, 246802 (2008b).
  * Feigin  _et al._ (2002) B. Feigin, M. Jimbo, T. Miwa, and E. Mukhin, International Mathematics Research Notices 2002, 1223 (2002).
  * Estienne and Santachiara (2009) B. Estienne and R. Santachiara, Journal of Physics A: Mathematical and Theoretical 42, 445209 (2009).
  * Bernevig  _et al._ (2009) B. A. Bernevig, V. Gurarie, and S. H. Simon, Journal of Physics A: Mathematical and Theoretical 42, 245206 (2009).
  * Estienne  _et al._ (2010a) B. Estienne, N. Regnault, and R. Santachiara, Nuclear physics B 824, 539 (2010a).
  * Estienne  _et al._ (2010b) B. Estienne, B. A. Bernevig, and R. Santachiara, Physical Review B 82, 205307 (2010b).
  * Zamaere  _et al._ (2014) C. B. Zamaere, S. Griffeth, and S. V. Sam, Communications in Mathematical Physics 330, 415 (2014).
  * Rezayi and Haldane (1994) E. Rezayi and F. Haldane, Physical Review B 50, 17199 (1994).
  * Wen and Wang (2008) X.-G. Wen and Z. Wang, Physical Review B 77, 235108 (2008).
  * Sabidussi (1992) G. Sabidussi, Discrete mathematics 101, 251 (1992).
  * Macdonald (1998) I. G. Macdonald,  _Symmetric functions and Hall polynomials_ (Oxford university press, 1998).
  * Godsil and Royle (2013) C. Godsil and G. F. Royle,  _Algebraic graph theory_ , Vol. 207 (Springer Science & Business Media, 2013).
  * Elliott (1913) E. B. Elliott,  _An introduction to the algebra of quantics_ (Clarendon Press, 1913).
  * Diestel (2000) R. Diestel,  _Graph theory $\\{$graduate texts in mathematics; 173$\\}$_ (Springer-Verlag Berlin and Heidelberg GmbH & amp, 2000).
  * Note (1) Depending on the source, the parameter $\alpha$ might be defined differently. In Ref. Feigin  _et al._ (2002), $J^{(\alpha)}_{\lambda}$ in $n$ variables is an eigenvector of  $H_{\alpha}=\sum@\slimits@_{i=1}^{n}(z_{i}\partial_{i})^{2}+\alpha\sum@\slimits@_{i<j}\frac{z_{i}+z_{j}}{z_{i}-z_{j}}(z_{i}\partial_{j}-z_{j}\partial_{i})$ while, for example, in the definition Bernevig and Haldane (2008a) (and most physics sources involving Jacks) $J^{(\alpha)}_{\lambda}$ is defined as an eigenvector of $H_{1/\alpha}$. The two notions are obviously equivalent, but result in a discrepancy of notation in the literature. We are using the definition in Ref. Bernevig and Haldane (2008a).
  * Stanley (1989) R. P. Stanley, Advances in Mathematics 77, 76 (1989).
  * Kung and Rota (1984) J. P. Kung and G.-C. Rota, Bulletin of the American Mathematical Society 10, 27 (1984).