# Higher-Rank Dimer Models

Richard Kenyon111Department of Mathematics, Yale University, New Haven;
richard.kenyon at yale.edu Nicholas Ovenhouse222Department of Mathematics,
Yale University, New Haven

###### Abstract

Let ${\mathcal{G}}$ be a bipartite planar graph with edges directed from black
to white. For each vertex $v$ let $n_{v}$ be a positive integer. A _multiweb_
in ${\mathcal{G}}$ is a multigraph with multiplicity $n_{v}$ at vertex $v$. A
_connection_ is a choice of linear maps on edges $\Phi=\\{\phi_{bw}\\}_{bw\in
E}$ where
$\phi_{bw}\in\mathrm{Hom}({\mathbb{R}}^{n_{b}},{\mathbb{R}}^{n_{w}})$.
Associated to $\Phi$ is a function on multiwebs, the trace
$\mathrm{Tr}_{\Phi}$. We define an associated Kasteleyn matrix $K=K(\Phi)$ in
this setting and write $\det K$ as the sum of traces of all multiwebs. This
generalizes Kasteleyn’s theorem and the result of [DKS22].

We study connections with positive traces, and define the associated
probability measure on multiwebs. By careful choice of connection we can thus
encode the “free fermionic” subvarieties for vertex models such as the
$6$-vertex model and $20$-vertex models, and in particular give determinantal
solutions.

We also find for each multiweb system an equivalent scalar system, that is, a
planar bipartite graph $H$ and a local measure-preserving mapping from dimer
covers of $H$ to multiwebs on ${\mathcal{G}}$. We identify a family of
positive connections as those whose scalar versions have positive face
weights.

###### Contents

  1. 1 Introduction
  2. 2 Webs and Tensor Networks
    1. 2.1 $\mathbf{n}$-Multiwebs
    2. 2.2 Graphs with boundary
    3. 2.3 Multiweb traces
    4. 2.4 Traces and colorings
    5. 2.5 Kasteleyn matrix
    6. 2.6 Trace theorem
  3. 3 Grassmannians and Boundary Measurements
  4. 4 Multiweb probability models
    1. 4.1 Free-fermionic six-vertex model
    2. 4.2 20-vertex model
  5. 5 Realization of Higher-Rank Connections by Scalar Networks
    1. 5.1 Worked example
    2. 5.2 Gadgets
    3. 5.3 Gauge Transformations
    4. 5.4 Measure
  6. 6 $2$-multiwebs on honeycomb graphs
    1. 6.1 The parameter space
    2. 6.2 Scalarization
    3. 6.3 Positive weights

##  1 Introduction

The _dimer model_ is the study of random dimer covers (also known as perfect
matchings) of graphs. Kasteleyn showed in [Kas63] that one can enumerate
(weighted) dimer covers of a planar graph with the determinant of an
associated matrix, now called the _Kasteleyn matrix_. The dimer model,
originally defined as a stat mech model, has turned out to have connections
with many other parts of mathematics, including cluster algebras [Spe07,
MS10], representation theory [DKS22, KS23], geometry [KOS06, KLRR18],
integrable systems [GK13], and string theory [FHV+06].

The recent works [DKS22] and [KS23] discuss relations between the dimer model
and $\mathrm{GL}_{n}$-connections on graphs. In particular associated to a
bipartite planar graph ${\mathcal{G}}$ with a $\mathrm{GL}_{n}$-connection is
a certain matrix, the $\mathrm{GL}_{n}$-Kasteleyn matrix, whose determinant
counts traces of multiwebs in ${\mathcal{G}}$ (see definitions below). We
generalize here the result of [DKS22] from the case of
$\mathrm{GL}_{n}$-connections to the case of a “quiver representation”: a
graph of vector spaces and linear maps between them. See Theorem 3 below.

One application of this result is for planar, biperiodic graphs, with bi-
periodic connections. We give conditions (see Section 5) under which all
traces of multiwebs are nonnegative, so that one can discuss Gibbs measures on
random multiwebs. See for example Theorem 6 for a result regarding random
$2$-webs on the honeycomb with a periodic $\mathrm{GL}_{2}$-connection. Using
the Postnikov/Talaska parameterization of the totally nonnegative Grassmannian
$\mathrm{Gr}^{\geq}_{k,n}$, see [Pos06, Tal08], we also construct an explicit,
local, measure-preserving mapping between multiwebs on ${\mathcal{G}}$ and
dimer covers of a related planar graph. See Theorem 5 and the example in
Figures 16 and 17.

As another application we are able to parameterize and solve (in the sense of
computing the free energy, local statistics, limit shapes, etc.) the so-called
free-fermionic subvarieties of vertex models such as the $6$-vertex and
$20$-vertex models, in a conceptually simple manner. See Sections 4.1 and 4.2.

Figure 1: Left: orientation version of $6$-vertex model. Right: square ice
version. The bijection is given by the proximity of the arrowheads to the
vertices.

Recall that the six-vertex model is a probability measure on orientations of
${\mathbb{Z}}^{2}$ (or subgraphs of ${\mathbb{Z}}^{2}$) with the constraint
that at each vertex two edges point in and two point out; see Figure 1, left.
Equivalently, one can consider “square ice” configurations in which water
molecules are arranged so that an oxygen atom sits at each point of
${\mathbb{Z}}^{2}$ and a hydrogen atom on each edge, so that each oxygen is
bonded to two of the neighboring hydrogens (Figure 1, right). In either
version there are $6$ possible local configurations at a vertex. By
definition, the probability of a configuration is proportional to the product
of local weights $c_{1},c_{2},a_{1},a_{2},b_{1},b_{2}$ at each vertex which
depend on the structure there; see Figure 2.

Figure 2: Local weights for the $6$-vertex model.

Computing the free energy in the thermodynamic limit (that is, on
${\mathbb{Z}}^{2}$) is a long standing open problem. Lieb [Lie67] computed the
free energy in the symmetric case $a_{1}=a_{2},b_{1}=b_{2},c_{1}=c_{2}$.
Another subvariety of the space of vertex weights, the $5$-vertex model (where
one of $a_{1},a_{2},b_{1},b_{2}$ is set to $0$) was recently solved in
[dGKW21]. A third subcase, the free-fermionic subvariety, is defined by the
constraint that $a_{1}a_{2}+b_{1}b_{2}-c_{1}c_{2}=0$. This case is solvable by
determinantal methods, see e.g. [Bax16]. This last case, and the more general
free fermionic 6V models with staggered weights (that is, invariant under a
coarser sublattice), can be solved with our methods.

In a similar manner the $20$-vertex model (Figure 3)

Figure 3: A configuration in the $20$-vertex model.

is a model of orientations of the triangular grid (i.e. the square grid with
diagonals) with indegree $3$ at each vertex; it is defined by $20$ local
vertex weights. It also has a free-fermionic subvariety, which is a certain
subvariety of dimension $9$, discussed below. Again in this case we show that
the model restricted to this subvariety is solvable by determinantal methods.
Vertex models of other multiplicities and on other planar graphs can likewise
be solved by similar means.

As a final application, the dimer model on a bipartite graph on a torus has an
associated Liouville integrable system [GK13]. Pulling this system back from
the scalarized version to the quiver representation gives a novel integrable
structure on such quiver representations. We intend to explore this connection
in a follow-up paper [KO23].

Acknowledgments. This research was supported by NSF grant DMS-1940932 and the
Simons Foundation grant 327929. We thank Daniel Douglas, Sam Panitch and Sri
Tata for discussions.

##  2 Webs and Tensor Networks

In this section, we review and generalize the definitions of tensor networks,
webs, multiwebs and multiweb traces. Background can be found in [FP16, Kup96,
DKS22].

For a positive integer $n$, a _planar $n$-web_ is a planar bipartite
$n$-regular graph. Such webs are used in the study of the representation
theory of $\mathrm{SL}_{n}$ [Sik01]. The definition of web can be generalized
to work inside a given graph [DKS22], where it is called a _multiweb_. We can
also allow $n$ to vary from vertex to vertex (Postnikov [Pos18] also discusses
this generalization from another viewpoint).

###  2.1 $\mathbf{n}$-Multiwebs

Let ${\mathcal{G}}=(V=B\cup W,E)$ be a bipartite graph. Let
$\mathbf{n}:V\to{\mathbb{Z}}_{>0}$ be a positive integer function on vertices,
called the vertex multiplicity. An _$\mathbf{n}$ -multiweb in ${\mathcal{G}}$_
is a function $m:E\to{\mathbb{Z}}_{\geq 0}$ that sums to $n_{v}=\mathbf{n}(v)$
at each vertex $v$, that is, such that $\sum_{u\sim v}m_{uv}=n_{v}$. We call
$m_{e}$ the _multiplicity_ of the edge $e$. The multiplicity of an edge may be
zero. Let $\Omega_{\mathbf{n}}=\Omega_{\mathbf{n}}({\mathcal{G}})$ be the set
of $\mathbf{n}$-multiwebs. When $n_{v}\equiv 1$, a $1$-multiweb is also known
as a _dimer cover_ , or _perfect matching_ , of ${\mathcal{G}}$. When
$n_{v}\equiv 2$, a $2$-multiweb is also known as a _double-dimer cover_ , see
for example [KW11].

The set $\Omega_{\mathbf{n}}$ is nonempty only if $\mathbf{n}$ satisfies
$\sum_{b\in B}n_{b}=\sum_{w\in W}n_{w}$ (by the handshake lemma) and also
certain linear inequalities. These inequalities are given by Hall’s matching
theorem:

######  Theorem 1 ([Hal35]).

$\Omega_{\mathbf{n}}$ is nonempty if and only if $\sum_{b\in
B}n_{b}=\sum_{w\in W}n_{w}$ and for every set $X$ of black vertices, there are
sufficiently many white neighbors, counted with multiplicity: for any set
$X\subset B$, we have $\sum_{b\in X}n_{b}\leq\sum_{w\in Y}n_{w}$ where $Y$ is
the set of neighbors of $X$.

###### Proof.

Replace each vertex $v$ with $n_{v}$ vertices and each edge $bw$ with a
complete bipartite graph $K_{n_{b},n_{w}}$. Then the statement follows
immediately from Hall’s matching theorem [Hal35] regarding the existence of a
perfect matching of a bipartite graph. ∎

###  2.2 Graphs with boundary

Even though our main theorem, Theorem 3, deals with graphs without boundary,
in principle there is a generalization to graphs with boundary (with a
slightly more technical statement and proof which we do not give here); see
[KS23] which deals with the case of graphs with boundary in the case
$\mathbf{n}\equiv 3$.

Suppose ${\mathcal{G}}$ is planar, embedded in the disk, with a distinguished
set of vertices $V_{\partial}=W_{\partial}\cup B_{\partial}$ on its outer
face, called boundary vertices. Fix $\mathbf{n}:V\to{\mathbb{Z}}_{>0}$ as
before. Also fix a “boundary multiplicity function”
$d:V_{\partial}\to{\mathbb{Z}}_{>0}$ satisfying $d_{v}\leq n_{v}$. We define
an $(\mathbf{n},d)$-multiweb as above except that at a boundary vertex $v$ we
require $\sum_{u\sim v}m_{uv}=d_{v}$ instead of $\sum_{u\sim v}m_{uv}=n_{v}$.
The connection $\Phi$ is defined as above using $n_{v}$, however.

We let $\Omega_{\mathbf{n},d}$ be the resulting set of multiwebs. Note that
the handshake lemma in this case implies that

$\sum_{b\in B_{int}}n_{b}+\sum_{b\in B_{\partial}}d_{b}=\sum_{w\in
W_{int}}n_{w}+\sum_{w\in W_{\partial}}d_{w}$

in order for $\Omega_{\mathbf{n},d}$ to be nonempty.

###  2.3 Multiweb traces

Material in this section is a generalization of material from [DKS22]. For
constant $\mathbf{n}\equiv n$, $n$-webs are used to graphically represent
$\mathrm{SL}_{n}$-invariants in tensor products of representations [Sik01].
For general $\mathbf{n}$, $\mathbf{n}$-multiweb traces have no such (global)
invariance, but there is a local $\mathrm{SL}_{n_{v}}$-invariance at each
vertex $v$; see below.

With boundary vertices of multiplicities $(n_{1},d_{1}),\dots,(n_{k},d_{k})$,
and a multiweb $m\in\Omega_{\mathbf{n},d}$, we will define the trace
$\mathrm{Tr}_{\Phi}(m)$ as an element of

$(V_{1}^{\otimes d_{1}}\otimes\dots\otimes V_{k}^{\otimes d_{k}})^{*}$

where $V_{i}={\mathbb{R}}^{n_{i}}$ if the $i$th boundary vertex $v_{i}$ is
black and $V_{i}=({\mathbb{R}}^{n_{i}})^{*}$ if $v_{i}$ is white. For a graph
without boundary, the trace $\mathrm{Tr}_{\Phi}(m)$ is a number. The notation
$\mathrm{Tr}_{\Phi}$ emphasizes the dependence on the connection $\Phi$, but
we sometimes just write $\mathrm{Tr}(m)$ to simplify the notation.

To define the multiweb trace we require ${\mathcal{G}}$ to have the additional
structure of a total ordering of the edges out of each vertex. Planarity of
${\mathcal{G}}$ gives a circular ordering to these edges (by convention,
counterclockwise at black vertices and clockwise at white vertices) which we
can augment to a total ordering by specifying a starting half-edge at each
vertex. Graphically we put a mark, called a _cilium_ , in the wedge between
the starting and ending edge of the ordering. These cilia then determine the
total order at each vertex. The choice of cilia only affects the sign of the
multiweb trace, and the choice of cilium at a vertex with $n_{v}$ odd does not
affect the multiweb trace at all, so we can and will only define cilia at
vertices of even $n_{v}$.

Let ${\mathcal{G}}$ be a planar bipartite graph with choice of cilia and a
connection $\Phi$ as above. Let $m\in\Omega_{\mathbf{n},d}$ be an
$(\mathbf{n},d)$-multiweb. To define the multiweb trace
$\mathrm{Tr}_{\Phi}(m)$ it is convenient to first assume that $m$ is _simple_
, that is, all edge multiplicities $m_{e}$ are $1$ or $0$; we’ll discuss
higher multiplicity edges below.

Fix an internal black vertex $b$, let $n=n_{b}$ and $V={\mathbb{R}}^{n}$. Let
$e_{1},\dots,e_{n}$ be its adjacent half-edges in order, with multiplicities
$m_{e_{i}}=1$ (we ignore edges of multiplicity $0$). We place a copy $V^{i}$
of $V$ on each half-edge $e_{i}$. There is a distinguished element $v_{b}\in
V^{1}\otimes\dots\otimes V^{n}$, called the _codeterminant_. In a fixed basis
$u_{1},\dots,u_{n}$ of $V$, it is defined as

$v_{b}=\sum_{\sigma\in S_{n}}(-1)^{\sigma}u^{1}_{\sigma(1)}\otimes\dots\otimes
u^{n}_{\sigma(n)}$ (1)

where $u^{i}_{j}$ is the $j$th basis element of $V^{i}$. Note that $v_{b}$ is
independent of $\mathrm{SL}_{n}$-change of basis of $V$. Also note that if $n$
is even, changing the location of the cilium by one “notch” changes the sign
of $v_{b}$, since the signature of the cyclic permutation of length $n$ is
$-1$.

Similarly, for a white interior vertex $w$, we take a copy of
$V^{*}=({\mathbb{R}}^{n_{w}})^{*}$, the dual space to ${\mathbb{R}}^{n_{w}}$,
on each half-edge incident to $w$, and define a codeterminant
$v_{w}\in(V^{*})^{1}\otimes\dots\otimes(V^{*})^{n_{w}}$ in a similar manner.

The trace $\mathrm{Tr}_{\Phi}(m)$ is a multilinear function whose inputs are
vectors and covectors at the boundary vertices: at a black boundary vertex $b$
the input is a vector in $v_{b}\in({\mathbb{R}}^{n_{b}})^{\otimes d_{b}}$ and
at a white boundary vertex $w$ the input is a vector
$v_{w}\in(({\mathbb{R}}^{n_{w}})^{*})^{\otimes d_{w}}$. For each vertex of
${\mathcal{G}}$, we thus have a distinguished element of the corresponding
vector space at that vertex: for internal vertices it is the appropriate
codeterminant, and for boundary vertices it is the input vector or covector.
We take the tensor product of all these elements to get an element

$x\in\bigotimes_{b\in B_{\mathrm{int}}}({\mathbb{R}}^{n_{b}})^{\otimes
n_{b}}\otimes\bigotimes_{w\in
W_{\mathrm{int}}}(({\mathbb{R}}^{n_{w}})^{*})^{\otimes
n_{w}}\otimes\bigotimes_{b\in B_{\partial}}({\mathbb{R}}^{n_{b}})^{\otimes
d_{b}}\otimes\bigotimes_{w\in
W_{\partial}}(({\mathbb{R}}^{n_{w}})^{*})^{\otimes d_{w}}.$

Next, we use the connection matrices $\phi_{bw}$, and apply the linear map
$\pi=\bigotimes_{bw\in E}\phi_{bw}$ to the tensor factors corresponding to the
black vertices, resulting in $\pi(x)\in\bigotimes_{w\in W}(V^{\otimes
n_{w}}\otimes(V^{*})^{\otimes n_{w}})$. Lastly, we take the trace of this
tensor by contracting at all indices.

This defines the trace for simple multiwebs. Now suppose $m$ has edges of
higher multiplicity. Replace each edge $e=bw$ of multiplicity $m_{e}>1$ with
$m_{e}$ parallel edges (connecting the same endpoints), extending the linear
order at each vertex in the natural way, so that the new edges are consecutive
in the new order. On each new edge use the same mapping $\phi_{bw}$. This
defines a simple multiweb $\hat{m}$ on this new graph; define

$\mathrm{Tr}_{\Phi}(m)=\frac{\mathrm{Tr}_{\Phi}(\hat{m})}{\prod_{e}m_{e}!}.$
(2)

######  Example 1.

The simplest example is depicted in Figure 4. In this example, the graph is
simply one edge connecting two boundary vertices $b,w$ with multiplicities
$n_{b},n_{w}$ and $d_{b}=d_{w}=1$. The unique multiweb $m$ has $m_{e}=1$. Its
trace is $v_{w}\phi_{bw}v_{b}$. More generally, suppose $d_{b}=d_{w}=d$; the
unique web has multiplicity $m_{e}=d$. Let
$v_{b}\in({\mathbb{R}}^{n_{b}})^{\otimes d}$ and
$v_{w}\in(({\mathbb{R}}^{n_{w}})^{*})^{\otimes d}$. Then the trace is
$v_{w}((\phi_{bw})^{\otimes d})v_{b}$. In other words,
$\mathrm{Tr}_{\Phi}(m)\in((\mathbb{R}^{n_{b}})^{\otimes
d}\otimes((\mathbb{R}^{n_{w}})^{*})^{\otimes d})^{*}$ is the tensor whose
coefficients with respect to the standard basis are the matrix entries of
$\phi_{bw}^{\otimes d}$.

Figure 4:  A simple web.

######  Example 2.

Consider the graph $G$ with a single internal 3-valent white vertex and 3
black boundary vertices, shown in Figure 5.

Figure 5:  Two multiwebs.

In this example, we take $n_{b}=n_{w}=2$ at all vertices, and the $2\times 2$
matrices on the edges are $A,B,C$. The input vectors on the boundary are
$u,x,y\in({\mathbb{R}}^{2})^{\otimes d}$, where $d$ is the corresponding
boundary multiplicity. The left part of the figure shows an element
$m\in\Omega_{2,d}(\mathcal{G})$ where $d_{u}=2$, and $d_{x}=d_{y}=0$. The
right part of the figure shows an example where $d_{u}=d_{x}=1$ and $d_{y}=0$.
All other $(2,d)$-multiwebs (for all possible choices of $d$) are obtained
from these examples by rotation. The blue edges indicate the multiplicities.
For the left web $u\in{\mathbb{R}}^{2}\otimes{\mathbb{R}}^{2}$, and web trace
is $c_{a}\det A,$ where $c_{a}$ is the coefficient of the projection of $u$
onto ${\mathbb{R}}^{2}\wedge{\mathbb{R}}^{2}$. (That is, letting $\sigma$
denote the exchange of tensor factors of
${\mathbb{R}}^{2}\otimes{\mathbb{R}}^{2}$, we have
$u_{s}:=\frac{1}{2}(u+\sigma u)$ and $u_{a}:=\frac{1}{2}(u-\sigma(u))$, and
$u_{a}$ is a multiple $c_{a}$ of $e_{1}\wedge e_{2}:=e_{1}\otimes
e_{2}-e_{2}\otimes e_{1}$.) We can see this by splitting the edge into two
edges of multiplicity $1$ and applying (2). Specifically, letting
$e_{1},e_{2}$ be the standard basis of ${\mathbb{R}}^{2}$, and $f_{1},f_{2}$
the dual basis, if $u=e_{1}\otimes e_{2}$, then $u_{a}=\frac{1}{2}e_{1}\wedge
e_{2}$ and the trace is

$\frac{1}{2!}\langle v_{w}|A|u\rangle=\frac{1}{2}\langle f_{1}\otimes
f_{2}-f_{2}\otimes f_{1}|Ae_{1}\otimes
Ae_{2}\rangle=\frac{1}{2}(A_{11}A_{22}-A_{12}A_{21})=\frac{1}{2}\det A.$

For the web on the right $u,x\in{\mathbb{R}}^{2}$ and the trace is

$\langle v_{w}|A\otimes B|u\otimes x\rangle=\langle f_{1}\otimes
f_{2}-f_{2}\otimes f_{1}|Au\otimes Bx\rangle=Au\wedge Bx,$

that is, the determinant of the matrix whose columns are $Au$ and $Bx$. Note
that $Au$ comes before $Bx$ due to the location of the cilium, and the fact
that the edges are ordered clockwise at a white vertex.

###  2.4 Traces and colorings

Let $m\in\Omega_{\mathbf{n}}$. The trace $\mathrm{Tr}(m)$ has an
interpretation in terms of edge colorings, as follows. At a vertex $v$ of
multiplicity $n_{v}$ we take a set ${\mathcal{C}}_{v}=\\{1,2,\dots,n_{v}\\}$
of colors. Given a multiweb $m$ and edge $e=bw$ we pick a set
$S_{e}\subset{\mathcal{C}}_{b}$ and $T_{e}\subset{\mathcal{C}}_{w}$, both of
size $m_{e}$. Around each internal black vertex $b$ we require that the sets
$S_{bw}$ for neighbors $w\sim b$ partition ${\mathcal{C}}_{b}$, and likewise
at internal white vertices the sets $T_{bw}$ partition ${\mathcal{C}}_{w}$. We
then associate to $b$ a sign $c_{b}$ depending on the sets
$\\{S_{bw}\\}_{w\sim b}$, as follows. Starting from the cilium at $b$, order
the sets $S_{bw}$ in counterclockwise order, and within each $S_{bw}$ order
the colors in natural order. This gives a permutation of ${\mathcal{C}}_{b}$
and $c_{b}$ is its signature. Likewise at a white vertex $w$, $c_{w}$ is the
signature of the permutation of the colors around $w$ starting from the cilium
in _clockwise_ order, with each $T_{bw}$ ordered in natural order.

Let $v$ be a boundary vertex of multiplicity $(n_{v},d_{v})$, with neighbors
$u_{1},\dots,u_{\ell}$. Assume for the moment that the cilium at $v$ is in the
external face. Let $u_{v}\in({\mathbb{R}}^{n_{v}})^{\otimes d_{v}}$ be a basis
vector. It corresponds to a map $U_{v}:[d_{v}]\to{\mathcal{C}}_{n_{v}}$, that
is, an ordered $d_{v}$-tuple of colors at $v$, not necessarily distinct. Given
a multiweb $m$, with multiplicities $m_{i}$ on edges $vu_{i}$, we assign to
the first edge, $vu_{1}$, the first $m_{1}$ colors
$U_{v}(\\{1,\dots,m_{1}\\})$, assign to the second edge the next $m_{2}$
colors $U_{v}(\\{m_{1}+1,\dots,m_{1}+m_{2}\\})$, and so on. If the cilium at
$v$ is not in the external face, we adjust these colors cyclically by the
appropriate amount.

The following proposition was proved in [DKS22], Prop. 3.2., for constant
$\mathbf{n}$, and no boundary. Their proof generalizes directly to the current
setting.

######  Proposition 2.

For a fixed choice $C_{\partial}=\\{U_{v}\\}_{v\in V_{\partial}}$ of color
sets at the boundary vertices, the trace of $m$ evaluated at the corresponding
basis vectors $\\{u_{v}\\}$ is

$\mathrm{Tr}(m)(\\{u_{v}\\})=\sum_{C}\prod_{v}c_{v}\prod_{e=bw}\det[(\phi_{bw})_{S_{e}}^{T_{e}}]$
(3)

where the sum is over colorings $C$ of the half-edges of $m$ having boundary
colors $C_{\partial}$, assigned as above, and $(\phi_{bw})_{S_{e}}^{T_{e}}$ is
the submatrix of $\phi_{bw}$ with rows $T_{e}$ and columns $S_{e}$.

As a special case of this proposition, suppose $m$ is simple, that is,
$m_{e}=0$ or $1$ for each edge. Then

$\mathrm{Tr}(m)(\\{u_{v}\\})=\sum_{C}\prod_{v}c_{v}\prod_{e=bw}(\phi_{bw})_{t,s}$
(4)

where the sum is over colorings $C$ of the half-edges of $m$ having boundary
colors $C_{\partial}$ in the correct order, and the second product is over
edges of $m$, with edge $bw$ getting colors $S_{e}=\\{s\\}$ and
$T_{e}=\\{t\\}$. This form in fact implies (3), using the definition (2) of
trace for edges with $m_{e}>1$.

###  2.5 Kasteleyn matrix

Given a planar bipartite graph ${\mathcal{G}}$ with no boundary, with
multiplicities $\mathbf{n}$ and connection $\Phi$, let $L$ be a choice of
cilia at vertices of even multiplicity. We choose a $\pm 1$-connection
$\epsilon=\\{\epsilon_{e}\\}_{e\in E}$ on ${\mathcal{G}}$, the _Kasteleyn
connection_ , or _Kasteleyn signs_ , with monodromy $(-1)^{\ell+1+k}$ around
each face, where $2\ell$ is the length of the face (its number of edges) and
$k$ is the number of cilia in that face. Recall that we do not specify cilia
at vertices with $n_{v}$ odd, so $k$ only counts cilia at vertices with
$n_{v}$ even.

We define a matrix $K=K(\Phi)$, the _Kasteleyn matrix_ , as follows. If $w\sim
b$, $K_{wb}=\epsilon_{wb}\phi_{bw}$ and otherwise $K_{wb}=0$. Note that this
generalizes the usual Kasteleyn rule of [Ken09] (which corresponds to the case
when all faces have an even number of cilia; in this case the sign only
depends on the face length mod $4$; see also [DKS22]).

The following lemma will be used below.

######  Lemma 1.

For any loop $\gamma$ in ${\mathcal{G}}$, the monodromy of $\epsilon$ around
$\gamma$ is $(-1)^{L/2+1+k+n_{\mathrm{int}}}$, where $L$ is the length of
$\gamma$, $k$ is the number of cilia of vertices along $\gamma$ which point
into $\gamma$, and $n_{\mathrm{int}}$ is the sum of multiplicities of vertices
strictly enclosed by $\gamma$.

###### Proof.

This is true by definition when $\gamma$ is a single face and follows by
induction on the number of faces enclosed by $\gamma$: suppose we subdivide a
face $f$ into two faces by adding in a path between two vertices $u,v$ of $f$,
so that $f$ becomes two faces $f_{1},f_{2}$ (see Figure 6).

Figure 6: Subdividing a face $f$ into two faces $f_{1},f_{2}$.

Let $L_{1}$ be the length (number of edges) of the part $\rho_{1}$ of $f$ in
common with $f_{1}$, and let $L_{2}$ be the length of the part $\rho_{2}$ of
$f$ in common with $f_{2}$. Let $L_{3}$ be the length of the new path. Let
$k_{1}$ be the number of cilia along $\rho_{1}$ (pointing into $f$), and
$k_{2}$ that for $\rho_{2}$. Let $k_{3}$ be the cilia along the new path
pointing into $f_{1}$, and $k_{4}$ the cilia along the new path pointing into
$f_{2}$. We have

$(-1)^{\frac{L_{1}+L_{3}}{2}+1+k_{1}+k_{3}}(-1)^{\frac{L_{2}+L_{3}}{2}+1+k_{2}+k_{4}}=(-1)^{\frac{L_{1}+L_{2}}{2}+1+k_{1}+k_{2}+(L_{3}+k_{3}+k_{4}+1)}.$

It remains to identify $L_{3}+k_{3}+k_{4}+1\bmod 2$ with $n_{int}\bmod 2$.
Each of the $L_{3}-1$ vertices along the new path either has a cilium (in
$f_{1}$ or $f_{2}$, and is thus of even multiplicity) or is of odd
multiplicity. The sum $k_{3}+k_{4}$ is the number of even-multiplicity
vertices along the new path, so $L_{3}-k_{3}-k_{4}-1=n_{int}\bmod 2$. ∎

We then let $\tilde{K}=\tilde{K}(\Phi)$ be obtained from $K$ be replacing each
entry $K_{wb}$, with the corresponding $n_{w}\times n_{b}$ block of scalars
(in particular $0$ entries of $K$ become blocks of $0$s).

######  Example 3.

Consider the graph $\mathcal{G}$ pictured in Figure 7 where the $\phi_{bw}$
are $A,B,\dots,G$. Note that $\epsilon_{bw}=+1$ everywhere except on the edge
$33$ (hence the sign in the label $-F$). We have

$K=\begin{pmatrix}A&D&0\\\ B&C&G\\\
E&0&-F\end{pmatrix},~{}~{}~{}~{}\widetilde{K}=\begin{pmatrix}a&d_{1}&d_{2}&0&0\\\
b_{1}&c_{11}&c_{12}&g_{11}&g_{12}\\\ b_{2}&c_{21}&c_{22}&g_{21}&g_{22}\\\
b_{3}&c_{31}&c_{32}&g_{31}&g_{32}\\\ e&0&0&-f_{1}&-f_{2}\end{pmatrix}.$ (5)

Figure 7: Example of ${\mathcal{G}}$ with multiplicities, signs and cilia. The
labels on the vertices are the indices, and blue numbers are the
multiplicities.

###  2.6 Trace theorem

When $\mathbf{n}\equiv 2$, a $2$-multiweb, also known as a double-dimer cover,
is a collection of loops and doubled edges. If the connection takes values in
$\mathrm{SL}_{2}$, the web trace is the product of the matrix traces of the
monodromies of the connection around these loops (with either orientation). In
[Ken14], it was shown that the web trace of a general bipartite graph with an
$\mathrm{SL}_{2}$-connection is given by a generalization of the Kasteleyn
determinant formula. This was generalized in [DKS22] both to $n>2$ and to the
case of connection matrices in $\mathrm{GL}_{n}$ and even
$M_{n}({\mathbb{R}})$. We further generalize this result (with a similar
proof) to the case of varying $n$ as follows.

######  Theorem 3.

Let ${\mathcal{G}}$ be a bipartite planar graph without boundary and with
choice of cilia $L$. Let $\mathbf{n}:V\to{\mathbb{Z}}_{>0}$, and choose a
connection $\Phi$. Let $K$ be an associated Kasteleyn matrix. Then

$\det(\widetilde{K})=\pm\sum_{m\in\Omega_{\mathbf{n}}}\mathrm{Tr}_{\Phi}(m).$

###### Proof.

Let ${\mathcal{G}}_{\mathbf{n}}$ be obtained from ${\mathcal{G}}$ by replacing
each vertex $v$ with $n_{v}$ vertices $v_{1},\dots,v_{n_{v}}$, and each edge
$bw$ by a complete bipartite graph $K_{n_{b},n_{w}}$. On edge $b_{j}w_{i}$
lying over edge $bw$ of ${\mathcal{G}}$, put (scalar) weight
$\epsilon_{bw}(\phi_{bw})_{i,j}$, where $\epsilon_{bw}$ is the Kasteleyn sign
from $bw$ in ${\mathcal{G}}$ and $(\phi_{bw})_{i,j}$ is the $i,j$-entry of
$\phi_{bw}$. Note that $\det\tilde{K}$ is a (signed, weighted) sum of dimer
covers of ${\mathcal{G}}_{\mathbf{n}}$.

There is a natural projection map
${\mathcal{G}}_{\mathbf{n}}\to{\mathcal{G}}$. Each dimer cover of
${\mathcal{G}}_{\mathbf{n}}$ projects to an $\mathbf{n}$-multiweb of
${\mathcal{G}}$, and we can group the terms in $\det\tilde{K}$ according to
their corresponding multiwebs:

$\det\tilde{K}=\sum_{m\in\Omega_{\mathbf{n}}}\sum_{\sigma\in
m}(-1)^{\sigma}\tilde{K}_{\sigma}.$

Here $\sigma\in m$ means the dimer cover $\sigma$ projects to $m$, and
$\tilde{K}_{\sigma}$ is the corresponding product of entries of $\tilde{K}$.

It remains to show that, up to a global sign $s$,

$s\mathrm{Tr}(m)=\sum_{\sigma\in
m}(-1)^{\sigma}\tilde{K}_{\sigma}=\sum_{\sigma\in
m}(-1)^{\sigma}\prod_{\tilde{w}}\tilde{K}_{\tilde{w},\sigma(\tilde{w})}.$ (6)

We assume first that $m$ is simple, that is, all edge multiplicities $m_{e}$
are $0$ or $1$. If $m$ is not simple we temporarily modify the graph by
replacing each edge on which $m_{e}>1$ with $m_{e}$ parallel edges; see below.

Let $w\in{\mathcal{G}}$ and $\tilde{w}\in{\mathcal{G}}_{\mathbf{n}}$ lying
over $w$. Then $\sigma(\tilde{w})=\tilde{b}$ is a black vertex of
${\mathcal{G}}_{\mathbf{n}}$ lying over some neighbor $b$ of $w$ in
${\mathcal{G}}$. Under the identification of vertices over $v$ with colors at
$v$, this $\sigma$ thus colors the edge $e=wb$, with one color from
${\mathcal{C}}_{w}$ and one color from ${\mathcal{C}}_{b}$. These colors on
edge $wb$ also correspond to a matrix entry $\phi_{ij}$. Thus (6) agrees with
(4), once we verify the agreement of signs. We need to check that

$(-1)^{\sigma}\prod_{bw\in m}\epsilon_{bw}\prod_{v}c_{v}=s,$ (7)

a constant sign.

Suppose we exchange colors at two circularly adjacent half-edges at $v$ (not
separated by the cilium at $v$). Then both $(-1)^{\sigma}$ and $c_{v}$ change
signs. Thus all colorings of a given simple multiweb $m$ have the same left-
hand side of (7).

It remains to check the sign change when we change multiwebs. But first let us
check what happens when we glue parallel edges together to make non-simple
multiwebs. If simple multiweb $m$ has $k$ parallel edges between $w$ and $b$,
consider all colorings of (the half edges of) these edges with colors from
size-$k$ subsets $S_{e}\subset{\mathcal{C}}_{w}$ and
$T_{e}\subset{\mathcal{C}}_{b}$. For fixed $S_{e},T_{e}$ there are $(k!)^{2}$
such colorings; summing over all $\sigma$ involving such colorings gives a
contribution of $k!\det((\phi_{wb})_{S_{e}}^{T_{e}})$. When we glue these
parallel edges of $m$ into a single edge of multiplicity $k$, and divide by
$k!$, the contribution from this edge from these colors is
$\det((\phi_{wb})_{S_{e}}^{T_{e}})$, which agrees with the corresponding term
in (3). So at this point we can assume $m$ is a general (nonsimple) multiweb.

Now we need to check the sign change when we change multiwebs. Let $\gamma$ be
a simple closed path in ${\mathcal{G}}$ with vertices
$b_{1},w_{1},\dots,b_{\ell},w_{\ell}$ in counterclockwise order. Suppose
multiweb $m\in\Omega_{\mathbf{n}}$ has positive multiplicities on edges
$b_{i}w_{i}$ of $\gamma$. Let $m^{\prime}$ be obtained from $m$ by decreasing
the multiplicities on edges $b_{i}w_{i}$ by $1$ and increasing the
multiplicities by $1$ on the other edges $b_{i}w_{i-1}$ (indices cyclic). We
show below that these operations preserve (7) and connect the set of all
$\mathbf{n}$-multiwebs; this will complete the proof.

By changing colors if necessary (we already dealt with color swaps above) we
can assume that both half edges of $b_{i}w_{i}$ contain color $1$, the first
color at each of $b,w$. We then move these colors $1$ to the edges
$b_{i}w_{i-1}$, increasing their multiplicity. Note that
$m^{\prime}\in\Omega_{\mathbf{n}}$ as well. Let us compare the sign (7) of $m$
with $m^{\prime}$. Suppose edge $b_{i}w_{i}$ has multiplicity $m_{i}$, and let
$q(b_{i})$ be the total multiplicity of edges $b_{i}w$ of $m$ with $w$
strictly enclosed by $\gamma$. If the cilium at $b_{i}$ is also in the region
enclosed by $\gamma$, then when moving color $1$ the sign change of
$c_{b_{i}}$ is $(-1)^{m_{i}+q(b_{i})}$: color $1$ is the first color of edge
$b_{i}w_{i}$, so moves past $m_{i}-1+q(b_{i})$ other colors and also past the
cilium, to become the first color of $b_{i}w_{i-1}$. Similarly if the cilium
at $b$ is not enclosed by $\gamma$ the sign change of $c_{b_{i}}$ is
$(-1)^{m_{i}+q(b_{i})-1}$. Likewise at $w_{i}$ the sign change is
$(-1)^{m_{i}+q(w_{i})}$ if the cilium at $w_{i}$ is enclosed, and otherwise is
$(-1)^{m_{i}+q(w_{i})-1}$. Therefore the total sign change in the $c_{v}$,
taking the product around all vertices of $\gamma$, is $(-1)^{k+Q}$, where $k$
is the number of cilia at the vertices of $\gamma$ which are enclosed by
$\gamma$, and $Q=\sum_{v\in\gamma}q(v)$. Note that, by the handshake lemma,
$(-1)^{Q}=(-1)^{n_{\mathrm{int}}}$ where $n_{\mathrm{int}}$ is the total
multiplicity of vertices strictly enclosed by $\gamma$.

This sign change $(-1)^{k+n_{\mathrm{int}}}$ is exactly compensated by the
monodromy of the Kasteleyn signs $\epsilon$ around the face, so that (7)
remains unchanged. More precisely, the change in the sign is due to the change
in the signature of $\sigma$, which is $(-1)^{\ell+1}$ (since $\sigma$ changes
by multiplication by an $\ell$-cycle) times the monodromy around the face of
the Kasteleyn signs, which is $(-1)^{\ell+1+k+n_{\mathrm{int}}}$ by Lemma 1,
times the contribution from the cilia of $(-1)^{k}$. The result is no net sign
change.

It remains to show that $\Omega_{\mathbf{n}}$ is connected by such “loop
moves”. We can interpret each $m\in\Omega_{\mathbf{n}}$ as a flow
$[m]\in{\mathbb{Z}}^{E}$, with value $m_{e}$ on edge $e$ directed from black
to white. Then given $m,m^{\prime}\in\Omega_{\mathbf{n}}$, the difference
$[m]-[m^{\prime}]$ is a divergence-free flow. It can be decomposed into a
collection of non-crossing oriented cycles, that is, disjoint cycles on the
associated ribbon surface. We can change $m$ to $m^{\prime}$ on each cycle
using the operation described above. ∎

######  Example 4.

Consider again the case from Example 3, pictured in Figure 7. It is easy to
see by inspection that there are only 3 elements of
$\Omega_{\mathbf{n}}(\mathcal{G})$, pictured below:

$m_{1}$$m_{2}$$m_{3}$

In this case, one can see that $\sum_{i=1}^{3}\mathrm{Tr}(m_{i})$ realizes the
expansion of $\det(\widetilde{K})$ along the first column. Indeed, if
$\widetilde{K}^{ij}$ is the matrix obtained by removing row $i$ and column
$j$, then $\mathrm{Tr}(m_{1})=a\det(\widetilde{K}^{11})$,
$\mathrm{Tr}(m_{2})=e\det(\widetilde{K}^{51})$, and
$\mathrm{Tr}(m_{3})=b_{1}\det(\widetilde{K}^{21})+b_{2}\det(\widetilde{K}^{31})+b_{3}\det(\widetilde{K}^{41})$.

##  3 Grassmannians and Boundary Measurements

In this section, we review Postnikov’s parameterization of the positive
Grassmannian using networks, which he calls _plabic graphs_ (the word plabic
is an abbreviation of “planar bi-colored”) [Pos06]. These are planar graphs,
drawn in a disk, with some vertices on the boundary. Vertices are drawn in two
colors (black and white).

The _Grassmannian_ $\mathrm{Gr}_{k,n}$ is the set of $k$-dimensional subspaces
of an $n$-dimensional vector space. It can be viewed concretely as the
quotient of the set of $k\times n$ matrices, of rank $k$, modulo row
operations (i.e. left multiplication by $\mathrm{GL}_{k}$). The
$k$-dimensional subspace represented by a matrix is simply its row span. The
_Plücker coordinates_ are the minors $\Delta_{I}$, where $I\in\binom{[n]}{k}$
is a $k$-element subset of $[n]=\\{1,2,\dots,n\\}$, and $\Delta_{I}$ is the
determinant of the $k\times k$ submatrix using the columns from $I$. The _non-
negative Grassmannian_ $\mathrm{Gr}^{\geq}_{k,n}$ is the set of points whose
Plücker coordinates are all non-negative.

Postnikov showed that $\mathrm{Gr}^{\geq}_{k,n}$ has a cell decomposition
whose cells, called _positroid cells_ , are indexed by equivalence classes of
plabic graphs. For each “reduced” plabic graph333A plabic graph is _reduced_
if it has the minimal number of faces required to parameterize the appropriate
cell; see [Pos06, GK13]., the assignment of weights (positive real numbers) to
edges (or faces) of the graph gives a parameterization of the corresponding
positroid cell. This parameterization was called the _boundary measurement
map_ in [Pos06], and we will briefly describe it now.

An orientation of a plabic graph is called a _perfect orientation_ if every
black vertex has in-degree 1, and every white vertex has out-degree 1. The
following construction uses a choice of perfect orientation, but the end
result will not depend on the choice. The chosen orientation will make some of
the boundary vertices sources, and others sinks. Suppose $k$ is the number of
sinks (and $n-k$ the number of sources), and $I\subset[n]$ the set of sinks.
The corresponding $k\times n$ matrix $X$, called the _boundary measurement
matrix_ , is defined as follows. The rows will correspond to the sinks in the
set $I$, and the columns will correspond to all boundary vertices. The
submatrix in columns $I$ will simply be the identity matrix. For all other
entries, in row $i$ and column corresponding to boundary vertex $j$, the entry
of the matrix will be the generating function for directed paths from source
$j$ to sink $i$:

$X_{ij}=(-1)^{N_{ij}}\sum_{p\colon j\to i}(-1)^{w(p)}\mathrm{wt}(p)$

Here, the sum is over all directed paths from $j$ to $i$, $N_{ij}$ is the
number of sinks on the boundary between $i$ and $j$, $w(p)$ is the winding
number of the path, and $\mathrm{wt}(p)$ is simply the product of the edge
weights along the path. If the graph is acyclic, then the factors of
$(-1)^{w(p)}$ are irrelevant. When there are cycles, these signs ensure that
the rational expressions obtained by summing the geometric series are
subtraction-free.

######  Example 5.

Figure 8 shows a network, and two choices of perfect orientations. As in the
previous section, we use the convention that the preferred orientation is from
black-to-white, and all our edge weights are with respect to this convention.
Therefore when a perfect orientation assigns some edges a white-to-black
direction, we invert the weight on that edge. The boundary measurement
matrices of the two orientations are

$X_{1}=\begin{pmatrix}1&0&-\frac{c}{d}&-\frac{1}{d}\\\
0&1&b+\frac{ac}{d}&\frac{a}{d}\end{pmatrix}\quad\text{ and }\quad
X_{2}=\begin{pmatrix}1&\frac{c}{ac+bd}&0&-\frac{b}{ac+bd}\\\
0&\frac{d}{ac+bd}&1&\frac{a}{ac+bd}\end{pmatrix}$

Note that the entries of $X_{1}$ are Laurent polynomials, while the entries of
$X_{2}$ are genuine rational expressions. This comes from the directed cycle
in the second perfect orientation. Note however that the two matrices are
equivalent up to left action by $\mathrm{GL}_{2}$, which can be seen by the
fact that they have the same $2\times 2$ minors (up to a common factor of
$\frac{d}{ac+bd}$).

$1$$2$$3$$4$$a$$b$$c$$d$$1$$2$$3$$4$$a$$b$$c$$d^{-1}$$1$$2$$3$$4$$\frac{1}{a}$$b$$\frac{1}{c}$$d$
Figure 8: (left) A plabic graph for $\mathrm{Gr}_{2,4}$, (middle) an acyclic
perfect orientation, (right) a cyclic perfect orientation.

In [PSW09], an interpretation of the boundary measurements in terms of dimer
covers was given. Suppose the graph is bipartite, and all boundary vertices
are black (this can be arranged by some local transformations which do not
change the boundary measurements, specifically by inserting some 2-valent
vertices). An _almost perfect matching_ is a subset of edges where every
internal vertex is incident to exactly one edge (but boundary vertices are
allowed to be unmatched). It can be shown that for a given graph, there is a
number $k$ (called the _excedence_) such that every almost perfect matching
uses exactly $k$ boundary vertices. This is the same $k$ as above, since there
is a simple bijection between almost perfect matchings and perfect
orientations: the edges of the matching are the edges directed white-to-black
in the perfect orientation. Then the sinks of the perfect orientation are
exactly the boundary vertices used in this matching.

If $M$ is an almost perfect matching, let $\partial(M)\subset[n]$ be the set
of boundary vertices used by $M$. The following result can be seen in [PSW09,
KW11, Lam15].

######  Theorem 4.

Let $X$ be the boundary measurement matrix corresponding to a chosen perfect
orientation. Then

$\Delta_{I}(X)=\sum_{M\colon\partial(M)=I}\mathrm{wt}(M)$

In other words, the Plücker coordinate $\Delta_{I}$ is the generating function
for almost perfect matchings with boundary set $I$.

The boundary measurement matrix can be given the following concrete
interpretation in terms of the Kasteleyn matrix. Suppose ${\mathcal{G}}$ is
bipartite, all boundary vertices are black, and let $K$ be a (typically not
square) Kasteleyn matrix for ${\mathcal{G}}$. Let $m$ be an almost perfect
matching, and $W^{\prime}$ the set of white vertices paired in $m$ to a
boundary vertex. Then after renumbering the vertices, $K$ has block form
$K=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ where $D$ has rows indexed by
$W\setminus W^{\prime}$ and columns indexed by $B_{\mathrm{int}}$, the
internal black vertices. Note that $D$ is a square matrix since its vertices
are paired by $m$. Then the boundary measurement matrix, up to a fixed choice
of gauge, is the Schur reduction $X=A-BD^{-1}C$; see [KW11, KS23]. The matrix
$X$ will be totally nonnegative after a gauge transformation; a choice of
Kasteleyn signs which yields totally nonnegative
$X\in\mathrm{Gr}^{\geq}_{k,n}$ is one where, for each $i=1,\dots,n-1$, the
product of signs along the boundary path between adjacent boundary vertices
$i$ and $i+1$ is $(-1)^{\ell/2+1}$ where $\ell$ is the length of the path.

######  Example 6.

Consider the situation from Example 5, pictured in Figure 8. Figure 9
illustrates Theorem 4 in this example, showing the various almost perfect
matchings. The Plücker coordinates are

$\Delta_{12}=d,\quad\quad\Delta_{13}=ac+bd,\quad\quad\Delta_{14}=a$
$\Delta_{23}=c,\quad\quad\Delta_{24}=1,\quad\quad\Delta_{34}=b$

These agree with the minors of $X_{1}$ up to a common factor of $d$, and they
agree with the minors of $X_{2}$ up to a common factor of $\frac{1}{ac+bd}$.

$a$$b$$c$$d$$a$$b$$c$$d$$a$$b$$c$$d$$a$$b$$c$$d$$a$$b$$c$$d$$a$$b$$c$$d$$a$$b$$c$$d$
Figure 9: The almost perfect matchings of the graph from Figure 8

Lastly, we will describe an inverse to the boundary measurement map, at least
in a special case. Among the positroid cells, there is one of maximal
dimension, which a dense open set in $\mathrm{Gr}^{\geq}_{k,n}$. It is called
the _totally positive_ Grassmannian, and denoted $\mathrm{Gr}^{+}_{k,n}$. The
graphs used to parameterize this top-dimensional cell correspond to the
standard cluster algebra structure on the coordinate ring of the Grassmannian.
They can be obtained as follows. Start by drawing a $k\times(n-k)$ rectangular
array of vertices. The vertices along the left and bottom edges will be
connected to the boundary. Direct all edges in the rectangular grid up and to
the left. This is pictured in the left part of Figure 10. To turn this into a
perfectly oriented plabic graph, we can split all 4-valent vertices into two
3-valent vertices. The resulting graph will be perfectly oriented, and we can
color the vertices black and white accordingly. Also, to make the graph more
uniform (and bipartite), we can insert some 2-valent vertices. This is
pictured in the right part of Figure 10.

Figure 10: The plabic graph representing the top-dimensional cell
$\mathrm{Gr}^{+}_{3,7}$ in $\mathrm{Gr}^{\geq}_{3,7}.$

##  4 Multiweb probability models

A _multiweb probability model_ is a certain probability measure on
$\Omega_{\mathbf{n}}$, the set of multiwebs on a given bipartite planar graph
(without boundary), with a connection having the property that the trace is a
positive function on $\Omega_{\mathbf{n}}$. In this case we define
$Pr(m)=\frac{\mathrm{Tr}(m)}{Z}$ where
$Z=\sum_{m\in\Omega_{\mathbf{n}}}\mathrm{Tr}(m)$ is the normalizing constant,
called the _partition function_. By Theorem 3, $Z$ is the determinant of an
associated Kasteleyn matrix.

We give some examples here. Some results of [KOS06] are used; we refer the
reader there or to [Ken09] for the relevant background.

###  4.1 Free-fermionic six-vertex model

Consider the $6$-vertex model discussed in the introduction. We work with the
square ice version. Let ${\mathcal{G}}$ be the graph obtained from
${\mathbb{Z}}^{2}$ by putting a vertex in the center of each edge. Color these
new vertices black and the original vertices white (Figure 11, left). The set
of square ice configurations is in bijection with the space
$\Omega_{\mathbf{n}}$ of ${\bf n}$-multiwebs on ${\mathcal{G}}$, where
$n_{w}=2$ at white vertices and $n_{b}=1$ at black vertices.

We put $2\times 1$ matrices $\phi_{bw}$ on the edges, and cilia northeast of
each white vertex as shown in Figure 11, right (we don’t need to specify cilia
at black vertices).

Figure 11: Graph for the $6V$ multiweb model, and connection matrices around a
white vertex.

Then for any configuration $m$ its trace is the product, over all white
vertices $w$, of the “molecule” $m_{w}$ there. The trace of the molecule
$m_{w}$ is a $2\times 2$ minor of the $2\times 4$ Grassmannian matrix

$M_{w}=\begin{pmatrix}c_{1}&c_{2}&c_{3}&c_{4}\\\
d_{1}&d_{2}&d_{3}&d_{4}\end{pmatrix}.$

Specifically the trace of $m_{w}$ is the minor $\Delta_{i,j}$ where $i,j$ are
the edges of the molecule attached to $w$, indexed clockwise starting from the
cilium. If we choose each $M_{w}$ to be in $\mathrm{Gr}^{+}_{2,4}$, then all
traces $\mathrm{Tr}(m_{w})$ are positive and $\mathrm{Tr}(m)=\prod_{w\in
W}\mathrm{Tr}(m_{w})$ is positive. These traces define a probability measure
on $\Omega_{\mathbf{n}}$, that is, on the space of square ice configurations,
with $Pr(m)=\frac{\mathrm{Tr}(m)}{Z}$, where $Z$, the partition function, is
$Z=\det K(\Phi)$.

The six vertex weights $a_{1},\dots,c_{2}$ defined by $M_{w}$ are the six
$2\times 2$ minors of $M_{w}$. As such they satisfy a relation, the Plücker
relation $a_{1}a_{2}+b_{1}b_{2}-c_{1}c_{2}=0$. This equation defines the so-
called _free-fermionic subvariety_ of the six vertex model. It is well-known
that the model can be solved by determinantal methods in this case, see e.g.
[Bax16].

A particularly symmetric choice is $a_{1}=a_{2}=b_{1}=b_{2}=1$ and
$c_{1}=c_{2}=\sqrt{2}$. In this case the square ice model on the $n\times n$
grid graph with “domain wall boundary conditions”, that is, for the graphs
shown in Figure 1 or Figure 11, left, is equivalent to domino tilings of the
Aztec diamond, see [EKLP92].

For the model on the whole plane defined by $M_{w}\in\mathrm{Gr}^{+}_{2,4}$,
we can compute the free energy per site $F$ as the Mahler measure of the
characteristic polynomial $P(z,w)=\det\tilde{K}(z,w)$ (see [KOS06]) where,
taking a fundamental domain consisting of a single white vertex and its east
and north black neighbors, $K(z,w)$ is the $1\times 2$ matrix

$K(z,w)=\begin{pmatrix}\begin{pmatrix}c_{1}\\\
d_{1}\end{pmatrix}-z\begin{pmatrix}c_{3}\\\
d_{3}\end{pmatrix}&\begin{pmatrix}c_{2}\\\
d_{2}\end{pmatrix}+w\begin{pmatrix}c_{4}\\\ d_{4}\end{pmatrix}\end{pmatrix}.$

Here the minus sign is a Kasteleyn sign. We have

$\tilde{K}(z,w)=\begin{pmatrix}c_{1}-c_{3}z&c_{2}+c_{4}w\\\
d_{1}-d_{3}z&d_{2}+d_{4}w\end{pmatrix}$

and

$P(z,w)=\det\tilde{K}=\Delta_{12}+z\Delta_{23}+w\Delta_{14}-wz\Delta_{34}.$

The free energy per site is

$F=\frac{1}{(2\pi)^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}\log
P(e^{i\theta},e^{i\phi})\,d\theta\,d\phi.$

###  4.2 20-vertex model

The $20$-vertex model is defined similarly to the $6$-vertex model, but on the
triangular lattice. A configuration consists in an orientation of the edges so
that at each vertex the outdegree and indegree are $3$. There is an equivalent
“triangular ice” version, for example ammonia ice, $\text{NH}_{3}$, with a
nitrogen atom at each vertex and three hydrogen atoms bonded to it along the
edges.

To represent the $20$-vertex model as a web model, let ${\mathcal{G}}$ be the
graph obtained from the triangular grid by putting a vertex in the center of
each edge. Color these new vertices black and the original vertices white. See
Figure 12.

Figure 12:

The set of ice configurations is in bijection with the space
$\Omega_{\mathbf{n}}$ of ${\bf n}$-multiwebs on ${\mathcal{G}}$, where
$n_{w}=3$ at white vertices and $n_{b}=1$ at black vertices.

We put $3\times 1$ matrices $\phi_{bw}$ on the edges, and periodic cilia, for
example east-southeast of each white vertex. The trace of the molecule $m_{w}$
is a $3\times 3$ minor of the $3\times 6$ Grassmannian matrix $M_{w}$: the
minor $\Delta_{i,j,k}$ where $i,j,k$ index the edges of the molecule attached
to $w$. As above we choose each $M_{w}$ to be in $\mathrm{Gr}^{+}_{3,6}$. then
$\mathrm{Tr}(m)=\prod_{w\in W}\mathrm{Tr}(m_{w})$ is positive. These traces
define a probability measure on $\Omega_{\mathbf{n}}$, that is, on the space
of triangular ice configurations, with $Pr(m)=\frac{\mathrm{Tr}(m)}{Z}$, where
$Z=\det K(\Phi)$.

The $20$ vertex weights $w_{1},\dots,w_{20}$ defined by $M_{w}$ are the
maximal minors of $M_{w}$. As such they satisfy certain Plücker relations.
These equation define a $9$-dimensional _free-fermionic subvariety_ of the
20-vertex model.

As in the square ice model, there is a circularly symmetric choice of weights,
arising from the unique circularly symmetric point in $\mathrm{Gr}^{+}_{3,6}$
described in [Kar18]. Up to a global scale factor this $6$-fold symmetric
version of the model has

$M=\begin{pmatrix}1&1&1&1&1&1\\\ \cos
0&\cos\frac{\pi}{3}&\cos\frac{2\pi}{3}&\cos\frac{3\pi}{3}&\cos\frac{4\pi}{3}&\cos\frac{5\pi}{3}\\\
\sin
0&\sin\frac{\pi}{3}&\sin\frac{2\pi}{3}&\sin\frac{3\pi}{3}&\sin\frac{4\pi}{3}&\sin\frac{5\pi}{3}\end{pmatrix}$

If we embed ${\mathcal{G}}$ symmetrically, so that faces are equilateral, then
each molecule gets a weight proportional to the product, over pairs of
hydrogen atoms, of sines of the half-angles between vectors connecting the
central atom to the arms. As such the weights of the $3$ possible types of
molecules in Figure 13 are in ratio $1:2:3$.

Figure 13: At the symmetric point, the three types of molecules have weights
in ratio $1:2:3$.

We can compute the free energy for the $20$-vertex model on the plane by
choosing a fundamental domain consisting of a single white vertex and the
three black neighbors east, north, and northwest from it. Then $K(z,w)$ is the
$1\times 3$ matrix

$K(z,w)=\begin{pmatrix}\begin{pmatrix}c_{1}\\\ d_{1}\\\
e_{1}\end{pmatrix}+z\begin{pmatrix}c_{4}\\\ d_{4}\\\
e_{4}\end{pmatrix}&\begin{pmatrix}c_{2}\\\ d_{2}\\\
e_{2}\end{pmatrix}+w\begin{pmatrix}c_{5}\\\ d_{5}\\\
e_{5}\end{pmatrix}&\begin{pmatrix}c_{3}\\\ d_{3}\\\
e_{3}\end{pmatrix}+\frac{w}{z}\begin{pmatrix}c_{6}\\\ d_{6}\\\
e_{6}\end{pmatrix}\end{pmatrix}.$

We have

$\tilde{K}(z,w)=\begin{pmatrix}c_{1}+c_{4}z&c_{2}+c_{5}w&c_{3}+c_{6}w/z\\\
d_{1}+d_{4}z&d_{2}+d_{5}w&d_{3}+d_{6}w/z\\\
e_{1}+e_{4}z&e_{2}+e_{5}w&e_{3}+e_{6}w/z\end{pmatrix},$

and

$P(z,w)=\det\tilde{K}=\Delta_{123}+z\Delta_{234}+zw\Delta_{345}-w(\Delta_{135}+\Delta_{246})+w^{2}\Delta_{456}+\frac{w^{2}}{z}\Delta_{156}+\frac{w}{z}\Delta_{126}.$

The free energy per site is

$F=\frac{1}{(2\pi)^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}\log
P(e^{i\theta},e^{i\phi})\,d\theta\,d\phi.$

As a specific example, take

$M=\begin{pmatrix}a&1&a&1&a&1\\\ a\cos
0&\cos\frac{\pi}{3}&a\cos\frac{2\pi}{3}&\cos\frac{3\pi}{3}&a\cos\frac{4\pi}{3}&\cos\frac{5\pi}{3}\\\
a\sin
0&\sin\frac{\pi}{3}&a\sin\frac{2\pi}{3}&\sin\frac{3\pi}{3}&a\sin\frac{4\pi}{3}&\sin\frac{5\pi}{3}\end{pmatrix}$

for a parameter $a>0$. This leads to a model with $3$-fold (but not $6$-fold)
rotational symmetry; edges from a white vertex in directions of cube roots of
unity have weight $a$ and edges in direction which are cube roots of $-1$ have
weight $1$. A molecule has a weight $1,2,$ or $3$ as before multiplied by its
edge weights. We have

$P(z,w)=a^{2}+az+a^{2}zw+aw^{2}+a^{2}\frac{w^{2}}{z}+a\frac{w}{z}-(3+3a^{2})w$

which is easier to understand in “Newton polygon” form

$P(z,w)=\begin{matrix}a^{2}\frac{w^{2}}{z}&aw^{2}\\\
a\frac{w}{z}&-(3+3a^{2})w&a^{2}wz\\\ &a^{2}&az\end{matrix}.$

This model has a “gas phase” when $a\neq 1$, see Figure 14: a bounded
complementary component of the amoeba of $P$. The maximal entropy Gibbs
measure on configurations on ${\mathcal{G}}$ is this gas phase; it has the
property that correlations between local arrangements decay exponentially
quickly in the distance. A typical configuration has molecules locked in a
periodic pattern except for localized perturbations, as in Figure 15.

Figure 14: The amoeba for $P$ in the $3$-fold symmetric $20$-vertex model.
Figure 15: Sample configuration from the maximal Gibbs measure for the
$20$-vertex model when $a=2$.

Interestingly, as $a\to 1$ the gaseous phase shrinks to a point; at $a=1$ we
are in the $6$-fold symmetric case and this is a so-called “liquid phase”: the
correlations decay only polynomially.

##  5 Realization of Higher-Rank Connections by Scalar Networks

We show here that to a bipartite planar graph ${\mathcal{G}}$, with the data
of multiplicities $\mathbf{n}$ and connection $\Phi$, we can associate a new
planar bipartite graph $\hat{\mathcal{G}}$ with an
${\mathbb{R}}^{*}$-connection for which there is a local, measure-preserving
surjective mapping
$\Psi:\Omega_{1}(\hat{\mathcal{G}})\to\Omega_{\mathbf{n}}({\mathcal{G}})$.

We begin by illustrating our construction with a worked example. We then
discuss the general construction below.

###  5.1 Worked example

We give an explicit realization of a $\mathrm{GL}_{2}$ connection on a
trivalent bipartite graph ${\mathcal{G}}$, by an ${\mathbb{R}}^{*}$-connection
on a new graph $\hat{\mathcal{G}}$.

Let ${\mathcal{G}}=(W\cup B,E)$ be a trivalent bipartite planar graph, and
$\mathbf{n}\equiv 2$. Let $\Phi=\\{\phi_{bw}\\}$ be a
$\mathrm{GL}_{2}$-connection. Let $\hat{\mathcal{G}}$ be the graph obtained
from ${\mathcal{G}}$ by replacing each edge with two parallel edges and each
vertex with a “gadget”: the $10$-vertex graph $H_{2,6}$ or $H_{2,6}^{*}$
(depending on whether the vertex is black or white) which parameterizes
$\mathrm{Gr}^{+}_{2,6}$, as shown in Figure 16. We put edge weights as shown
in Figure 18 on the gadgets $H_{2,6}$, and edge weights $1$ on the gadgets
$H_{2,6}^{*}$. The mapping from dimer covers ($1$-multiwebs) on
$\hat{\mathcal{G}}$ to $2$-multiwebs on ${\mathcal{G}}$ is illustrated in
Figure 17.

Figure 16: (Part of) a trivalent graph ${\mathcal{G}}$ and its “scalarization”
$\hat{\mathcal{G}}$.

Figure 17:  Mapping from dimer covers of $\hat{\mathcal{G}}$ (left) to
$2$-multiwebs on ${\mathcal{G}}$ (right). Edges of ${\mathcal{G}}$ of
multiplicity $m_{e}=0,1,2$ correspond to using respectively $2,1$ or $0$ edges
of $\hat{\mathcal{G}}$ between the vertex gadgets. If $m_{e}=2$ or $m_{e}=0$
note that there is a unique (local) preimage. If $m_{e}=1$ there are multiple
local preimages.

Figure 18: The “gadget” $H_{2,6}$ for a black vertex (left) and $H_{2,6}^{*}$
for a white vertex (right). On the right, all edge weights are $1$. Note also
that the index order of boundary vertices is counterclockwise for $H_{2,6}$,
and clockwise for $H_{2,6}^{*}$.

Note that $H_{2,6}$ is not exactly the graph one obtains from the construction
presented at the end of Section 3. There are, however, local transformations
of plabic graphs (see [Pos06]) preserving the boundary measurements, which
transform this graph into $H_{2,6}$.

Let us first focus on the gadget $H_{2,6}$ of Figure 18. With these edge
weights, and a particular choice of Kasteleyn signs, the Kasteleyn matrix for
the gadget is

$K=\begin{pmatrix}1&0&0\\\ b&c&0\\\ 0&1&0\\\ 0&d&e\\\ 0&0&1\\\ -a&0&f\\\
x&-y&u\end{pmatrix}.$

Here the last row corresponds to the white vertex internal to the gadget; the
other white vertices communicate with the rest of the graph outside the
gadget. We can define a boundary response matrix by taking the Schur reduction
of this $K$ onto its first six rows and first two columns. The reduced
Kasteleyn matrix onto columns $1,2$ and rows $1,\dots,6$ is given (up to a
scalar multiple of $z$ in the first column) by

$K_{\mathrm{red}}=\begin{pmatrix}u&bu&0&-ex&-x&-(au+fx)\\\
0&c&1&\frac{1}{u}(du+ey)&\frac{y}{u}&\frac{fy}{u}\end{pmatrix}^{t}.$ (8)

Choosing a different set of $2$ columns would result in an equivalent matrix:
one with the same maximal minors, up to a global scale. The matrix
$K_{\mathrm{red}}$ represents a point in the positive Grassmannian
$\mathrm{Gr}^{+}_{2,6}$ on condition that all weights $a,\dots,f,x,y,u$ are
positive. Conversely any element of $\mathrm{Gr}^{+}_{2,6}$ is equivalent
(after column operations) to (8) with some choice of positive weights
$a,\dots,f,x,y,u$. Letting these weights vary over ${\mathbb{R}}$ the above
represents a generic element of $\mathrm{Gr}_{2,6}$.

Likewise putting edge weights $1$ on the gadget $H_{2,6}^{*}$ leads to its
boundary measurement matrix

$\begin{pmatrix}1&1&0&-1&-1&-2\\\ 0&1&1&2&1&1\end{pmatrix},$ (9)

the transpose of that for $H_{2,6}$ (with all variables set to $1$). Finally,
we need to find edge weights $a,b,\dots,u$ so that the matrix $\phi_{bw}$ on
${\mathcal{G}}$ corresponds to the $2\times 2$ matrix obtained by multiplying
the first two rows of the $H_{2,6}$ gadget at $b$ with the first two columns
of the $H_{2,6}^{*}$ gadget at $w$: in this case

$\phi_{bw}=\begin{pmatrix}u&0\\\ bu&c\end{pmatrix}\begin{pmatrix}1&1\\\
0&1\end{pmatrix}=\begin{pmatrix}u&u\\\ bu&bu+c\end{pmatrix}.$

A similar expression holds for the other edges of ${\mathcal{G}}$.

Given matrices $\phi_{bw}$ on edges of ${\mathcal{G}}$, we therefore need to
find edge weights on the gadget $H_{2,6}$ at each black vertex so that the
appropriate rows of the Grassmannian element there, when multiplied by the
corresponding columns of (9), makes the $\phi_{bw}$.

We can guarantee this if we show that in fact for any three matrices $A,B,C\in
GL_{2}({\mathbb{R}})$, the Grassmannian element $(A~{}B~{}C)^{t}$ can be
realized with some choice of edge weights on $H_{2,6}$. For a given generic
Grassmannian element $(A~{}B~{}C)^{t}$, the edge weights on $H_{2,6}$ can be
given explicitly in terms of the matrices $A,B,C$ as follows. We have

$x=\Delta_{35},\quad\quad\quad y=\Delta_{15},\quad\quad\quad u=\Delta_{13}$
$a=\frac{\Delta_{56}}{\Delta_{15}},\quad\quad\quad
c=\frac{\Delta_{12}}{\Delta_{13}},\quad\quad\quad
e=\frac{\Delta_{34}}{\Delta_{35}}$
$b=\frac{\Delta_{23}}{\Delta_{13}},\quad\quad\quad
d=\frac{\Delta_{45}}{\Delta_{35}},\quad\quad\quad
f=\frac{\Delta_{16}}{\Delta_{15}}$

These formulas hold for generic $A,B,C$. If some minors of $(A~{}B~{}C)^{t}$
are zero, we can take a limit of the corresponding generic cases, resulting in
a simpler gadget with fewer edges and vertices. More concretely, we find
ourselves in a lower-dimensional positroid cell; Postnikov [Pos06] shows how
to assign a simpler gadget, depending on which minors are zero, which fully
parameterizes such matrices. The analogs of the above formulas, for the edge
weights in terms of the minors, for any positroid cell in
$\mathrm{Gr}^{\geq}_{k,n}$ were obtained in [Tal08].

###  5.2 Gadgets

We define $H_{k,n}$ to be a reduced plabic graph representing the top-
dimensional cell in $\mathrm{Gr}^{\geq}_{k,n}$ with white boundary vertices,
and $H_{k,n}^{*}$ the same graph but with the colors reversed. We will refer
to these $H_{k,n}$ and $H_{k,n}^{*}$ graphs as “gadgets”. Some examples can be
seen in Figures 18 and 19.

Figure 19: Gadgets $H^{*}_{2,8}$ and $H^{*}_{2,10}$. These are constructed
starting from a triangulation of a square and pentagon, shown in dotted red.
Gadgets for $H_{2,2n}^{*}$ can be constructed similarly from a $2n$-gon
surrounding a triangulation of an $n$-gon; equivalent gadgets are obtained by
choosing other triangulations.

We start with a graph ${\mathcal{G}}$ with vertex multiplicities $\mathbf{n}$
and a connection $\Phi$, and we will describe how to construct
$\hat{\mathcal{G}}$ by replacing each vertex with a gadget, and replacing each
edge by multiple parallel edges connecting the gadgets. The rule is different
at white vertices and black vertices. Each edge $bw$ of ${\mathcal{G}}$ is
replaced by $n_{w}$ parallel edges, where $n_{w}$ is the multiplicity at the
white endpoint of the edge. Each white vertex $w$ of ${\mathcal{G}}$ with $d$
neighbors is replaced by a gadget $H^{*}_{n_{w},dn_{w}}$; each black vertex
$b$ is replaced by a gadget $H_{n_{b},N}$ where $N=\sum_{w\sim b}n_{w}$. In
this way for each edge $bw$ the gadget at $b$ has an ordered subset of
boundary vertices, of size $n_{w}$, corresponding and connected via $n_{w}$
parallel edges, to the subset of size $n_{w}$ of the gadget at $w$.

We will assign all edges in gadgets $H_{n_{w},dn_{w}}^{*}$ at white vertices
of ${\mathcal{G}}$ a weight of $1$. Let $M_{w}$ be the reduced Kasteleyn
matrix of the gadget at $w$; this is an $n_{w}\times dn_{w}$ matrix. We write
$M_{w}=[M_{b_{1}w},M_{b_{2}w},\dots,M_{b_{d}w}]$ its block decomposition
according to the $d$ neighbors.

If an edge $bw$ of ${\mathcal{G}}$ has connection matrix $\phi_{bw}$, and if
the corresponding block of $M_{w}$ at $w$ is the matrix $M_{bw}$, then we
assign the edge weights in the gadget at $b$ so that the appropriate block of
its boundary measurement matrix $M_{b}$ is the matrix
$\phi_{bw}^{\prime}:=M_{bw}^{-1}\phi_{bw}$. That is, we write
$\phi_{bw}=\phi_{bw}^{\prime}M_{bw}$ where $M_{bw}$ is the fixed matrix coming
from the gadget at $w$.

As in the basic example from the previous section, there is a mapping from
dimer covers of the new graph $\hat{\mathcal{G}}$ onto the set of
$\mathbf{n}$-multiwebs of the original graph ${\mathcal{G}}$ (pictured in
Figure 17). The image is the unique $\mathbf{n}$-multiweb on $\mathcal{G}$
whose multiplicities $m_{e}$ are the number of _unoccupied_ edges of the set
of parallel edges of $\hat{\mathcal{G}}$ between $w$ and $b$.

###  5.3 Gauge Transformations

Let $\mathcal{G}$ be a bipartite graph with vertex labels $\mathbf{n}$ and a
connection $\Phi=\\{\phi_{bw}\\}$. For a black vertex $b$ with
$\mathbf{n}(b)=n_{b}$, and an invertible matrix $g\in\mathrm{GL}_{n_{b}}$, a
_gauge transformation_ at $b$ changes the connection matrices on all incident
edges $bw$ by the rule $\phi_{bw}\mapsto\phi_{bw}g$. Similarly, for a white
vertex $w$ with $\mathbf{n}(w)=n_{w}$ and some $g\in\mathrm{GL}_{n_{w}}$, the
corresponding gauge transformation alters weights of incident edges by
$\phi_{bw}\mapsto g\phi_{bw}$. The trace of every multiweb on $\mathcal{G}$ is
then multiplied by the scalar $\det(g)$.

Recall that for a $k$-valent white vertex $w$ with $\mathbf{n}(w)=n_{w}$, the
corresponding gadget is $H^{*}_{n_{w},kn_{w}}$, whose boundary measurement
matrix is a block matrix $K_{w}=(M_{1},M_{2},\dots,M_{k})$. Gauge
transformations at $w$ therefore correspond to the left action of
$\mathrm{GL}_{n_{w}}$ (i.e. $K_{w}\mapsto gK_{w}$). This agrees with the idea
that the boundary measurement matrix should be thought of as an element of the
Grassmannian $\mathrm{Gr}_{n_{w},kn_{w}}$.

Similarly, for a $k$-valent black vertex $b$ with $\mathbf{n}(b)=n_{b}$ and
$N=\sum_{w\sim b}n_{w}$, the corresponding gadget is $H_{n_{b},N}$, whose
boundary measurement matrix is the block matrix

$K_{b}=\begin{pmatrix}M_{1}^{-1}\phi_{b,w_{1}}\\\ M_{2}^{-1}\phi_{b,w_{2}}\\\
\vdots\\\ M_{k}^{-1}\phi_{b,w_{k}}\end{pmatrix}$

The gauge action at $b$ therefore corresponds to right multiplication (i.e.
$K_{b}\mapsto K_{b}\cdot g$).

###  5.4 Measure

For a planar graph ${\mathcal{G}}$ with vertex multiplicities $\mathbf{n}$ and
connection $\Phi$, we constructed above a graph $\hat{\mathcal{G}}$, the graph
with gadgets, and a map
$\Omega_{1}(\hat{\mathcal{G}})\to\Omega_{\mathbf{n}}({\mathcal{G}})$.

######  Theorem 5.

The map $\Omega_{1}(\hat{\mathcal{G}})\to\Omega_{\mathbf{n}}({\mathcal{G}})$
preserves the measure. More precisely, the sum of weights of dimer covers of
$\hat{\mathcal{G}}$ whose image is a given multiweb $m$, is $\mathrm{Tr}(m)$.

###### Proof.

Let $m\in\Omega_{\mathbf{n}}({\mathcal{G}})$. As before we split each edge $e$
of ${\mathcal{G}}$ into $m_{e}$ parallel edges, so that $m$ is simple, that
is, all edge multiplicities are $1$ or $0$.

It is convenient to put two new vertices on each edge of ${\mathcal{G}}$, both
with multiplicity $n_{w}$, so that original edge $bw$ with parallel transport
$\phi_{bw}$ becomes $bw^{\prime}b^{\prime}w$, with parallel transports
$\phi^{\prime}_{bw},I,$ and $M_{bw}$ on subedges
$bw^{\prime},w^{\prime}b^{\prime}$ and $b^{\prime}w$ respectively. Since
$\phi_{bw}=\phi^{\prime}_{bw}M_{bw}$ this operation does not change
$\mathrm{Tr}(m)$ for any multiweb $m$.

Recall that in Section 2.3 we defined $\mathrm{Tr}(m)$ as the contraction of a
certain tensor product of codeterminants. At a black vertex $b$, let $M_{b}$
be the element of $\mathrm{Gr}^{+}_{n_{b},N}$ there, written as an $N\times
n_{b}$ matrix (that is, as the transpose of the usual representation). Let
$v_{b}$ be the codeterminant, for the web $m$, at $b$. Associated to the web
$m$ is a submatrix $M_{b}(m)$ of $M_{b}$: the submatrix with rows
corresponding to edges of $m$ (and remember that each edge of $m$ corresponds
to multiple rows of $M_{b}$). We can apply $M_{b}(m)$ to the codeterminant
$v_{b}$, which means applying, for each edge $bw^{\prime}$ of $m$, the linear
map $\phi^{\prime}_{bw}$ to the entries of the tensor factor of $v_{b}$ for
that edge. This results in a “modified codeterminant” $M_{b}(m)v_{b}$.

This modified codeterminant can be written as follows. Let $n=n_{b}$. Starting
from (1), apply $A=\phi^{\prime}_{bw}$ to get

$\displaystyle M_{b}(m)v_{b}$ $\displaystyle=\sum_{\sigma\in
S_{n}}(-1)^{\sigma}Au_{\sigma(1)}\otimes\dots\otimes Au_{\sigma(n)}$
$\displaystyle=\sum_{\sigma\in
S_{n}}(-1)^{\sigma}(\sum_{j_{1}}A_{j_{1},\sigma(1)}u_{j_{1}})\otimes\dots\otimes(\sum_{j_{k}}A_{j_{n},\sigma(n)}u_{j_{n}}).$

The coefficient of $u_{i_{1}}\otimes\dots\otimes u_{i_{n}}$ in $M_{b}(m)v_{b}$
is thus

$\sum_{\sigma\in S_{k}}(-1)^{\sigma}A_{i_{1},\sigma(1)}\dots
A_{i_{n},\sigma(k)}=\det[(M_{b})_{\\{i_{1},\dots,i_{n}\\}}],$

the maximal minor of $M_{b}$ corresponding to rows $\\{i_{1},\dots,i_{n}\\}$.

Similarly at a white vertex $w$ we let $M_{w}$ be the element of
$\mathrm{Gr}^{+}_{n_{w},dn_{w}}$ there, written this time as an $n_{w}\times
dn_{w}$ matrix. It has a submatrix $M_{w}(m)$ with columns corresponding to
edges in $m$ (and each edge of $m$ corresponds to multiple columns). We apply
$M_{w}(m)$ to the codeterminant $v_{w}$, by applying $(M_{bw})^{t}$ to the
entries in the tensor factor for edge $b^{\prime}w$ (we use the transpose
because this is the action on the dual space). This results in a modified
codeterminant $M_{w}(m)v_{w}$. We contract the tensor product of all the
$M_{b}(m)v_{b}$ with the tensor product of all the $M_{w}(m)v_{w}$’s to get
the trace of $m$.

Now consider a dimer cover $\pi\in\Omega_{1}(\hat{\mathcal{G}})$. For each
edge $e$ of ${\mathcal{G}}$, it uses a subset $S(e,\pi)$ of edges of
$\hat{\mathcal{G}}$ lying over the middle subedge of $e$. Consider the set
$[\pi]\subset\Omega_{1}(\hat{\mathcal{G}})$ of all dimer covers $\pi^{\prime}$
using the same sets $S(e,\pi)$ for all edges $e$. This set is obtained by
completing the dimer cover within each gadget. For a black vertex $b$ of
${\mathcal{G}}$, suppose $\pi$ uses middle edges (that is, inputs) with
indices $S_{b}=\\{i_{1},\dots\,i_{n}\\}$ around the gadget at $b$. The sum of
weights of dimer covers completing the gadget is the corresponding minor
$I=\det[(M_{b})_{\\{i_{1},\dots,i_{n}\\}}]$. This is, as explained above, the
coefficient of $u_{i_{1}}\otimes\dots\otimes u_{i_{n}}$ in the modified
codeterminant $M_{b}(m)v_{b}$.

Likewise at a white vertex the sum of weights of dimer covers completing the
gadget and using inputs $\\{j_{1},\dots,j_{\ell}\\}$ is the coefficient of
$f_{j_{1}}\otimes\dots\otimes f_{j_{\ell}}$ in the modified codeterminant
$M_{w}(m)v_{w}$.

When contracting the modified codeterminants we get a nonzero contribution for
every term in which the indices on each edge of ${\mathcal{G}}$ match, in
which case the contribution to the trace is the product of the coefficients,
which is the product of the corresponding minors of the $M_{v}$.

This completes the proof. ∎

##  6 $2$-multiwebs on honeycomb graphs

Let $H$ be the bi-infinite honeycomb graph in ${\mathbb{R}}^{2}$, with
constant vertex multiplicities $\mathbf{n}\equiv 2$ and with periodic
$\text{GL}_{2}$-connection having matrices $A,B,C$ on the three directions of
edges. Let $\mathcal{H}$ be a finite subgraph of $H$. Under what conditions on
$A,B,C$ do all $2$-multiwebs in $\mathcal{H}$ have positive trace? We give a
large class of examples here (see Theorem 6).

###  6.1 The parameter space

Consider the quotient ${\mathcal{G}}$ of $H$ by the lattice of its
translational symmetries. This is the graph ${\mathcal{G}}$ on the torus with
two vertices as pictured in Figure 20, with multiplicity $n=2$ at both
vertices.

$I$$B$$A$ Figure 20: A simple bipartite graph on a torus

Up to gauge transformations, we can assume the central edge has the identity
connection, and the remaining edges have connection given by $2\times 2$
matrices $A$ and $B$, as in Figure 20. For generic $A$ we can further apply a
gauge transformation diagonalizing $A$. That is, replace $A,B$ with
$SAS^{-1},SBS^{-1}$ where $SAS^{-1}$ is diagonal. Finally, conjugating $B$ by
any matrix commuting with $A$ (that is, any diagonal matrix) removes one more
degree of freedom from the choice of $B$. This leaves a $5$ dimensional space
of matrix pairs $A,B$ which determine the system up to gauge.

Associated to ${\mathcal{G}}$ is a polynomial

$P(z,w)=\det\tilde{K}=\det(I+zA+wB),$

the _characteristic polynomial_ , see [KOS06]. We have

$P(z,w)=1+z^{2}\det A+w^{2}\det
B+z\,\mathrm{tr}A+w\,\mathrm{tr}B+zw\,\mathrm{tr}AB^{*}$ (10)

where $B^{*}=B^{-1}\det B$.

The characteristic polynomial describes the equivalent single-dimer system:
see below. It depends on a choice of basis for
$H_{1}({\mathbb{T}}^{2},{\mathbb{R}})$: $z$ and $w$ are the monodromies along
the two basis cycles. Moreover it is only well-defined up to sign changes
$z\mapsto\pm z$ and $w\mapsto\pm w$: these operations corresponds to different
choices of Kasteleyn signs for the torus.

###  6.2 Scalarization

The scalarization $\hat{\mathcal{G}}$ is the one pictured in Figure 16 and
Figure 17. This scalarization induces a scalarization of any subgraph
$\mathcal{H}$ of the honeycomb $H$; on boundary vertices of $\mathcal{H}$ we
“cap off” unused edges by removing them and their adjacent vertices from the
corresponding gadgets.

The boundary measurement matrix at the right/white gadget (which has all its
edge weights equal to 1) is

$M_{w}=\left(\begin{array}[]{cc|cc|cc}1&0&-1&-3&-2&-3\\\
0&1&1&2&1&1\end{array}\right)=(X_{1}|X_{2}|X_{3}).$

(This differs from the one of (9) mentioned in Section 5.1 by an
$\mathrm{SL}_{2}$ change of basis, but this does not change any of the
$2\times 2$ minors.) The boundary measurement matrix of the left/black gadget,
using the edge weights as in Figure 18, is

$M_{b}=\begin{pmatrix}1&0\\\ 0&1\\\ \hline\cr-bu&u\\\ -(cex+bdu+bey)&du+ey\\\
\hline\cr-(cx+by)&y\\\ -(acu+cfx+bfy)&fy\end{pmatrix}=\begin{pmatrix}Y_{1}\\\
\hline\cr Y_{2}\\\ \hline\cr Y_{3}\end{pmatrix}$

Again, this differs from the one of (8) presented in the earlier section by an
$\mathrm{SL}_{2}$ change of basis, and further using the assumption that
$cu=1$.

The assumption that $\hat{\mathcal{G}}$ is the scalarization of $\mathcal{G}$
is equivalent to saying that $X_{3}Y_{2}=A$ and $X_{2}Y_{3}=B$. This means the
edge weights can be expressed as functions of the minors of $M$ (as given in
Section 5.1), whose blocks are $Y_{1}=\mathrm{Id}$, $Y_{2}=X_{3}^{-1}A$ and
$Y_{3}=X_{2}^{-1}B$.

######  Example 7.

If all edge weights in both the white and black gadgets are set equal to $1$,
then $M_{b}=M_{w}^{\top}$, and we get

$A=\begin{pmatrix}11&-8\\\ -4&3\end{pmatrix},\quad\quad
B=A^{\top}=\begin{pmatrix}11&-4\\\ -8&3\end{pmatrix}$

The characteristic polynomial (10) in this case is

$P(z,w)=1+z^{2}+w^{2}+14z+14w-14zw.$

######  Example 8.

The trivial connection (when $A=B=\mathrm{Id}$) is realized in
$\hat{\mathcal{G}}$ by the choice of weights (some of which are negative)

$a=c=\frac{1}{3},\quad b=d=e=f=-\frac{1}{3},\quad x=-3,\quad y=u=3$

Its characteristic polynomial (10) in this case is

$P(z,w)=1+z^{2}+w^{2}+2z+2w+2zw=(1+z+w)^{2}.$

###  6.3 Positive weights

If the scalarization of ${\mathcal{G}}$ has positive edge weights, then all
multiwebs on $\mathcal{H}$ will have positive traces, by Theorem 5. This is a
sufficient condition for positive traces (but not necessary; see example 8
above). We can determine when this happens as follows.

The graph of Figure 16 is not _reduced_ ; by [GK13] it is equivalent, under
local operations and gauge transformations, to a reduced graph with the same
characteristic polynomial, up to a multiplicative factor. The reduction of
this graph is the one pictured in Figure 21, which has $6$ parameters
$X_{1},X_{2},X_{3},X_{4},z,w$, with the single relation
$X_{1}X_{2}X_{3}X_{4}=1$. This reduced graph thus realizes the fact that the
dimension of the space of parameters is indeed 5.

Figure 21: Reduced graph corresponding to scalarization $\hat{\mathcal{G}}$ of
the graph $\mathcal{G}$ from Figure 20.

We can compute the reduction of the graph ${\mathcal{G}}$ directly by
comparing the characteristic polynomials of ${\mathcal{G}}$ and the graph in
Figure 21. The partition function $\det\widetilde{K}$ must be equal, after
rescaling $z,w$, and possibly changing their signs, to the characteristic
polynomial $P_{\mathrm{red}}(z,w)$ of the graph of Figure 21, that is, to

$P_{\mathrm{red}}(z,w)=1+(1+X_{1}X_{3})z+X_{1}X_{3}z^{2}+(1+X_{1}X_{2})w+X_{1}X_{2}w^{2}-(X_{1}+X_{1}X_{2}X_{3})zw.$

Comparing coefficients of $P(z,w)$ from (10) with $P_{\mathrm{red}}(\pm az,\pm
bw)$ determines $A$ and $B$ up to gauge as functions of the five positive
parameters $X_{1},X_{2},X_{3},a,b$. We should allow for all four choices of
signs. For convenience let $\alpha=\pm a,\beta=\pm b$, so that
$P_{\mathrm{red}}(\pm az,\pm bw)=P(\alpha z,\beta w)$. We find in particular
the eigenvalues of $A$, $B$, and $AB^{-1}$ as functions of these parameters:
the eigenvalues of $A$ are $\alpha$ and $\alpha X_{1}X_{3}$, the eigenvalues
of $B$ are $\beta$ and $\beta X_{1}X_{2}$, and the eigenvalues of $AB^{-1}$
are $-\frac{\alpha}{\beta}X_{2}^{-1}$ and $-\frac{\alpha}{\beta}X_{3}$. This
allows for the following observation.

######  Theorem 6.

Let $\mathcal{H}$ be a finite balanced subgraph of the honeycomb graph $H$,
with matrices $A,B$ and $I$ on the edges as in Figure 20, and periodic cilia
as shown. Then the traces of all 2-multiwebs in $\mathcal{H}$ are positive if
among the three matrices $A,B$ and $AB^{-1}$, either two have positive
eigenvalues and one has negative eigenvalues, or all three have negative
eigenvalues.

###### Proof.

Up to changing sign of $\alpha$ and/or $\beta$ (which corresponds to changing
the sign of $A$ and/or $B$) we can suppose the eigenvalues of $A,B$ are
positive and those of $AB^{-1}$ are negative. Then by the discussion above, we
can choose positive values for the five parameters $X_{1}$, $X_{2}$, $X_{3}$,
$a$, and $b$, such that $\hat{\mathcal{H}}$ has the same dimer partition
function as $\mathcal{H}$. By the measure-preserving property (see Section
5.4), the trace of any 2-multiweb in $\mathcal{H}$ is equal to the weighted
sum of dimer covers of $\hat{\mathcal{H}}$ with the same topological type as
the multiweb. Since all weights are positive, this means all multiweb traces
are positive. ∎

There are other choices of matrices $A,B$ leading to positive traces, not
covered by the theorem: for example if $A,B$ are upper triangular with
positive determinants. We conjecture that these are the only two cases:

######  Conjecture 1.

Traces of all $2$-webs in $H$ are positive if and only if, up to gauge
equivalence, one of the following conditions holds:

  1. 1.

$A$ and $B$ are upper triangular with positive determinants, or

  2. 2.

Among the three matrices $A,B$ and $AB^{-1}$, either two have positive
eigenvalues and one has negative eigenvalues, or all three have negative
eigenvalues.

######  Example 9.

In the case from Example 7, where all edge weights are set equal to 1, the
eigenvalues of $A$ and $B$ are both $7\pm 4\sqrt{3}$, which are positive, and
the eigenvalues of $AB^{-1}$ are $-7\pm 4\sqrt{3}$, which are both negative.
This corresponds to the network in Figure 21 with the weights

$a=X_{1}=X_{2}=X_{4}=7+4\sqrt{3}\approx 13.928,\quad b=7-4\sqrt{3}\approx
0.0718$ $X_{3}=(7-4\sqrt{3})^{3}=1351-780\sqrt{3}\approx 0.00037$

######  Example 10.

Similar to the previous example, we could instead set all the parameters in
Figure 21 equal to 1, so that $X_{1}=\cdots=X_{4}=a=b=1$. In this case the
representative matrices are

$A=\begin{pmatrix}1&0\\\ 4&1\end{pmatrix},\quad\quad B=\begin{pmatrix}1&1\\\
0&1\end{pmatrix},\quad\quad AB^{-1}=\begin{pmatrix}1&-1\\\ 4&-3\end{pmatrix}$

It is easy to see that $A$ and $B$ have eigenvalues $1,1$, and $AB^{-1}$ has
eigenvalues $-1,-1$. The fact that $\mathrm{tr}(A)=2$ reflects the fact that
there are exactly 2 perfect matchings of the graph in Figure 21 with 1 edge
crossing the $z$-boundary (and similarly for $B$ and $AB^{-1}$). The fact that
$\det(A)=1$ reflects the fact that there is one perfect matching using two
edges which cross the $z$-boundary (and similarly for $B$).

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