$\displaystyle
d^{\bm{\mathsf{l}}}(-k,m)-b^{\bm{\mathsf{l}}}(-k,m)=\mathsf{l}_{k}(m-1)-\mathsf{l}_{k}(m)-\big{(}\mathsf{l}_{k+1}(m-1)-\mathsf{l}_{k+1}(m)\big{)}=\Delta_{k}(m-1)-\Delta_{k}(m).$

Inserting these into 4.8 (with the $(i,j;u,v)$ there equal to $(i-1,j;u,v+1)$
here), yields

(5.4) $\displaystyle\begin{aligned}
\mathbb{P}[\bm{\mathsf{L}}=\bm{\mathsf{l}}]=\mathcal{Z}^{-1}&\cdot\displaystyle\prod_{k=i-1}^{j}\displaystyle\prod_{\begin{subarray}{c}m\in\llbracket
u,v+1\rrbracket\\\ m\leq
N\end{subarray}}x^{\mathsf{l}_{k}(m-1)-\mathsf{l}_{k}(m)}\displaystyle\prod_{\begin{subarray}{c}m\in\llbracket
u,v+1\rrbracket\\\
m>N\end{subarray}}y^{\mathsf{l}_{k}(m-1)-\mathsf{l}_{k}(m)}\\\
&\times\displaystyle\prod_{k=i-1}^{j}\displaystyle\prod_{m=u}^{v+1}\big{(}1-q^{\Delta_{k}(m-1)}\cdot\mathbbm{1}_{\Delta_{k}(m-1)-\Delta_{k}(m)=1}\big{)},\end{aligned}$

for some normalization constant $\mathcal{Z}>0$.

Next observe since $u_{0}=\max\\{u-1,N\\}$ and $v_{0}=\min\\{v+1,N\\}$ that

$\displaystyle\displaystyle\sum_{m=u}^{\min\\{v+1,N\\}}\big{(}\mathsf{l}_{k}(m-1)-\mathsf{l}_{k}(m)\big{)}=\big{(}\mathsf{l}_{k}(u-1)-\mathsf{l}_{k}(v_{0})\big{)}\cdot\mathbbm{1}_{u\leq
N};$
$\displaystyle\displaystyle\sum_{m=\max\\{N+1,u\\}}^{v+1}\big{(}\mathsf{l}_{k}(m-1)-\mathsf{l}_{k}(m)\big{)}=\big{(}\mathsf{l}_{k}(u_{0})-\mathsf{l}_{k}(v+1)\big{)}\cdot\mathbbm{1}_{v\geq
N}.$

As we conditioned on $\mathsf{L}_{i-1}(m)=\mathsf{l}_{i-1}(m)$ for all $m$,
and on $\mathsf{L}_{k}(m)=\mathsf{l}_{k}(m)$ and
$\mathsf{L}_{k}(m)=\mathsf{l}_{k}(m)$ for all $k$ and $m\notin\llbracket
u,v\rrbracket$, we may incorporate the factors
$x^{(\mathsf{l}_{i-1}(u-1)-\mathsf{l}_{i-1}(v_{0}))\cdot\mathbbm{1}_{u\leq
N}}\cdot
y^{(\mathsf{l}_{i-1}(u_{0})-\mathsf{l}_{i-1}(v+1))\cdot\mathbbm{1}_{v\geq N}}$
and $x^{\mathsf{l}_{k}(u-1)\cdot\mathbbm{1}_{u\leq
N}+\mathsf{l}_{k}(v_{0})\cdot\mathbbm{1}_{u>N}}\cdot
y^{-\mathsf{l}_{i}(v+1)\cdot\mathbbm{1}_{v\geq
N}-\mathsf{l}_{i}(u_{0})\cdot\mathbbm{1}_{v<N}}$ for $k\in\llbracket
i,j\rrbracket$ appearing in the right side of (5.4) into the normalization
constant $\mathcal{Z}$ (where we used the facts that if $u>N$ then $v>N$ and
so $v_{0}=N\leq u-1$, and that if $v<N$ then $u<N$ and so $u_{0}=N\geq v+1$).
This gives

$\displaystyle\mathbb{P}[\bm{\mathsf{L}}=\bm{\mathsf{l}}]=\mathcal{Z}^{-1}$
$\displaystyle\cdot\displaystyle\prod_{k=i}^{j}x^{-\mathsf{l}_{k}(v_{0})}y^{\mathsf{l}_{k}(u_{0})}\displaystyle\prod_{k=i-1}^{j}\displaystyle\prod_{m=u}^{v+1}\big{(}1-q^{\Delta_{k}(m-1)}\cdot\mathbbm{1}_{\Delta_{k}(m-1)-\Delta_{k}(m)=1}\big{)},$

after altering $\mathcal{Z}$ if necessary, which yields the proposition. ∎

The Gibbs property described by 5.1 coincides the Hall–Littlewood Gibbs
property introduced in [33, Definition 3.4].111111To properly observe the
match, one must read first our line ensemble in reverse (from right to left),
and then apply a gauge transformation to its weights that does not affect its
Gibbs property (namely, multiply them by
$(q;q)_{\Delta_{i}(m-1)}^{-1}(q;q)_{\Delta_{i-1}(m-1)}^{-1}(q;q)_{\Delta_{i}(m)}(q;q)_{\Delta_{i-1}(m)}$).
Moreover, setting $n=1$ in 4.7 yields that the law of $\mathsf{L}_{1}$ has the
same law as the height function of the (uncolored) stochastic six-vertex model
introduced in [46]; this had been shown earlier in [15, Theorem 5.5]. More
generally, it can be shown (although we do not do so here) that the $n=1$ case
of $\bm{\mathsf{L}}$ coincides (up to an affine transformation) with the
Hall–Littlewood Gibbsian line ensemble introduced in [33, Section 3.2]. This
is to be expected, since taking the $n=1$ case of our arguments in the
preceding sections would essentially yield the content described in [15,
Section 5.7] and [33, Proposition 3.9].

######  Remark 5.2.

Observe that (5.2) only depends on $x$ and $y$ through their ratio $x^{-1}y$.
Indeed, if $u-1\geq N$ or $N\geq v+1$, then the factor
$x^{-\mathsf{l}_{i}(v_{0})}y^{\mathsf{l}_{i}(u_{0})}=x^{-\mathsf{l}_{i}(v+1)}y^{\mathsf{l}_{i}(u-1)}$
on the right side of (5.2) is fixed by the conditioning and can therefore be
incorporated into the normalization constant. Otherwise, $N\in\llbracket
u,v\rrbracket$, and so the factor
$x^{-\mathsf{l}_{i}(v_{0})}y^{\mathsf{l}_{i}(u_{0})}=(x^{-1}y)^{\mathsf{l}_{i}(N)}$
only depends on $x^{-1}y$.

###  5.2. The Case $q=0$

Throughout this section, we set $q=0$, $s=0$, and sample the colored line
ensemble
$\bm{\mathsf{L}}=\big{(}\bm{\mathsf{L}}^{(1)},\bm{\mathsf{L}}^{(2)},\ldots,\bm{\mathsf{L}}^{(n)}\big{)}$
from the measure
$\mathbb{P}_{\operatorname{scL}}^{\sigma}=\mathbb{P}_{\operatorname{scL};n;0;\bm{x};\bm{y}}^{\sigma}$.
The following proposition explains the Gibbs property (4.8) for
$\bm{\mathsf{L}}$. Below, we restrict to the scenario when $\llbracket
u,v\rrbracket$ does not contain $N$, since the Gibbs property will be simplest
to state in this situation (as, analogously to 5.2, it will not depend on $x$
or $y$).

######  Proposition 5.3.

Adopt the above notation and assumptions, and let $j\geq i\geq 1$ and
$u,v\in\llbracket 0,M+N\rrbracket$ be integers such that $u\leq v$ and
$N\notin\llbracket u,v\rrbracket$. Then, the following two statements hold.

  1. (1)

For any $(c,k,m)\in\llbracket
1,n\rrbracket\times\mathbb{Z}_{>0}\times\llbracket 1,M+N\rrbracket$ such that
$\mathsf{L}_{k}^{(c+1)}(m)>\mathsf{L}_{k+1}^{(c+1)}(m)$, we almost surely have

(5.5)
$\displaystyle\mathsf{L}_{k}^{(c)}(m-1)-\mathsf{L}_{k}^{(c)}(m)=\mathsf{L}_{k}^{(c+1)}(m-1)-\mathsf{L}_{k}^{(c+1)}(m),$

  2. (2)

Condition on the curves $\mathsf{L}_{k}^{(c)}(m)$ for all $c\in\llbracket
1,n\rrbracket$ and $(k,m)\notin\llbracket i,j\rrbracket\times\llbracket
u,v\rrbracket$. Then the law of $\bm{\mathsf{L}}$ is uniform over all simple
colored line ensembles
$\bm{\mathsf{l}}=\big{(}\bm{\mathsf{l}}^{(1)},\bm{\mathsf{l}}^{(2)},\ldots,\bm{\mathsf{l}}^{(n)}\big{)}$
that are $\llbracket i,j\rrbracket\times\llbracket u,v\rrbracket$-compatible
with $\bm{\mathsf{L}}$ such that, for any $(c,k,m)\in\llbracket
1,n\rrbracket\times\mathbb{Z}_{>0}\times\llbracket 1,M+N\rrbracket$ with
$\mathsf{l}_{k}^{(c+1)}(m)>\mathsf{l}_{k+1}^{(c+1)}(m)$, we have

(5.6)
$\displaystyle\mathsf{l}_{k}^{(c)}(m-1)-\mathsf{l}_{k}^{(c)}(m)=\mathsf{l}_{k}^{(c+1)}(m-1)-\mathsf{l}_{k}^{(c+1)}(m).$

To prove 5.3, we require the next lemma that explains why
$\mathsf{l}_{k}^{(c+1)}(m)>\mathsf{l}_{k+1}^{(c+1)}(m)$ should imply (5.6).

######  Lemma 5.4.

Let
$\bm{\mathsf{l}}=\big{(}\bm{\mathsf{l}}^{(1)},\bm{\mathsf{l}}^{(2)},\ldots,\bm{\mathsf{l}}^{(n)}\big{)}$
denote a simple colored line ensemble on $\llbracket 0,M+N\rrbracket$;
$\mathcal{E}^{\bm{\mathsf{l}}}$ denote the associated higher spin path
ensemble; and
$\big{(}A^{\bm{\mathsf{l}}}(-k,m),b^{\bm{\mathsf{l}}}(-k,m);\bm{C}^{\bm{\mathsf{l}}}(-k,m),d^{\bm{\mathsf{l}}}(-k,m)\big{)}$
denote the arrow configuration in $\mathcal{E}^{\bm{\mathsf{l}}}$ at any
$(-k,m)\in\mathbb{Z}_{\geq 0}\times\llbracket 1,M+N\rrbracket$. Then, for any
$(-k,m)\in\mathbb{Z}_{\geq 0}\times\llbracket 1,M+N\rrbracket$, we have that
$L_{1;0}\big{(}\bm{A}^{\bm{\mathsf{l}}}(-k,m),b^{\bm{\mathsf{l}}}(-k,m);\bm{C}^{\bm{\mathsf{l}}}(-k,m),d^{\bm{\mathsf{l}}}(-k,m)\big{)}=1$
if and only if, for any $c\in\llbracket 1,n\rrbracket$ with
$\mathsf{l}_{k}^{(c+1)}(m)>\mathsf{l}_{k+1}^{(c+1)}(m)$, (5.6) holds.
Otherwise, we have that the weight
$L_{1;0}\big{(}\bm{A}^{\bm{\mathsf{l}}}(-k,m),b^{\bm{\mathsf{l}}}(-k,m);\bm{C}^{\bm{\mathsf{l}}}(-k,m),d^{\bm{\mathsf{l}}}(-k,m)\big{)}=0$.

###### Proof.

By Definition 4.4, we have for any $c\in\llbracket 1,n\rrbracket$ that

(5.7) $\displaystyle\begin{aligned}
&C_{c}^{\bm{\mathsf{l}}}(-k,m)=\mathsf{l}_{k}^{(c)}(m)-\mathsf{l}_{k+1}^{(c)}(m)-\big{(}\mathsf{l}_{k}^{(c+1)}(m)-\mathsf{l}_{k+1}^{(c+1)}(m)\big{)};\\\
&d^{\bm{\mathsf{l}}}=\max\big{\\{}d\in\llbracket
1,n\rrbracket:\mathsf{l}_{k}^{(d)}(m-1)-\mathsf{l}_{k}^{(d)}(m)=1\big{\\}},\end{aligned}$

with $d^{\bm{\mathsf{l}}}=0$ if no such $d\in\llbracket 1,n\rrbracket$ exists.
Further observe (see Figure 17) that for $q=0$, if
$\bm{A}+\bm{e}_{b}=\bm{C}+\bm{e}_{d}$, then

$\displaystyle
L_{1;0}(\bm{A},b;\bm{C},d)=1-\mathbbm{1}_{d>0}\cdot\mathbbm{1}_{C_{[d+1,n]}>0}.$

By (5.7), we have for any $d\in\llbracket 1,n\rrbracket$ that

$\displaystyle
C_{[d+1,n]}^{\bm{\mathsf{l}}}(-k,m)=\mathsf{l}_{k}^{(d+1)}(m)-\mathsf{l}_{k+1}^{(d+1)}(m).$

Also by (5.7), we have $d^{\bm{\mathsf{l}}}(-k,m)=d\geq 1$ if and only if
$\mathsf{l}_{k}^{(d)}(m-1)-\mathsf{l}_{k}^{(d)}(m)-\big{(}\mathsf{l}_{k}^{(d+1)}(m-1)-\mathsf{l}_{k}^{(d+1)}(m)\big{)}>0$,
that is, if and only if (5.6) holds.

Hence,
$L_{1;0}\big{(}\bm{A}^{\bm{\mathsf{l}}}(-k,m),b^{\bm{\mathsf{l}}}(-k,m);\bm{C}^{\bm{\mathsf{l}}}(-k,m),d^{\bm{\mathsf{l}}}(-k,m)\big{)}=1$
if and only if, for any $c\in\llbracket 1,n\rrbracket$ such that
$\mathsf{l}_{k}^{(c+1)}(m)>\mathsf{l}_{k+1}^{(c+1)}(m)$, (5.6) holds.
Otherwise, this weight is equal to $0$. ∎

Now we can establish 5.3.

###### Proof of 5.3.

Observe (see Figure 17) for any $\bm{A},\bm{C}\in\mathbb{Z}_{\geq 0}^{n}$ and
$b,d\in\llbracket 0,n\rrbracket$ with $\bm{A}+\bm{e}_{b}=\bm{C}+\bm{e}_{d}$
that

(5.8) $\displaystyle\begin{aligned}
&L_{x;0}(\bm{A},b;\bm{C},d)=x^{\mathbbm{1}_{d>0}}\cdot
L_{1;0}(\bm{A},b;\bm{C},d);\qquad
L_{1;0}(\bm{A},b;\bm{C},d)=1-\mathbbm{1}_{d>0}\cdot\mathbbm{1}_{C_{[d+1,n]}>0}.\end{aligned}$

Thus, since $\bm{\mathsf{L}}$ was sampled according to the measure
$\mathbb{P}_{\operatorname{scL}}^{\sigma}$, (2.7), (2.8), (3.2), and 4.5
together imply that

$\displaystyle\mathbb{P}_{\operatorname{scL}}^{\sigma}[\bm{\mathsf{L}}]=\mathcal{Z}_{\bm{x};\bm{y}}^{-1}$
$\displaystyle\cdot\displaystyle\prod_{k=1}^{\infty}\displaystyle\prod_{m=1}^{N}\widehat{L}_{x_{m};0}\big{(}\bm{A}^{\bm{\mathsf{L}}}(-k,m),b^{\bm{\mathsf{L}}}(-k,m);\bm{C}^{\bm{\mathsf{L}}}(-k,m),d^{\bm{\mathsf{L}}}(-k,m)\big{)}$
$\displaystyle\qquad\times\displaystyle\prod_{k=1}^{\infty}\displaystyle\prod_{m=N+1}^{M+N}L_{y_{m-N};0}\big{(}\bm{A}^{\bm{\mathsf{L}}}(-k,m),b^{\bm{\mathsf{L}}}(-k,m);\bm{C}^{\bm{\mathsf{L}}}(-k,m),d^{\bm{\mathsf{L}}}(-k,m)\big{)},$

which by the first statement of (5.8) (and the fact that $x,y\neq 0$) is
nonzero if and only if
$L_{1;0}(\bm{A}^{\bm{\mathsf{L}}}(-k,m),b^{\bm{\mathsf{L}}}(-k,m);\bm{C}^{\bm{\mathsf{L}}}(-k,m),d^{\bm{\mathsf{L}}}(-k,m)\big{)}\neq
0$ for all $(-k,m)\in\mathbb{Z}_{\leq 0}\times\llbracket 1,M+N\rrbracket$. By
5.4, this is true if and only if, for any $(c,k,m)\in\llbracket
1,n\rrbracket\times\mathbb{Z}_{>0}\times\llbracket 1,M+N\rrbracket$ such that
$\mathsf{L}_{k}^{(c+1)}(m)>\mathsf{L}_{k+1}^{(c+1)}(m)$, (5.5) holds. This
confirms the first statement of the proposition.

To verify the second, we first apply 4.8 (with the $(i,j;u,v)$ there equal to
$(i-1,j;u,v+1)$ here) and (5.8) to deduce for some normalization constant
$\mathcal{Z}>0$ that

(5.9) $\displaystyle\begin{aligned}
\mathbb{P}_{\operatorname{scL}}^{\sigma}[\bm{\mathsf{L}}=\bm{\mathsf{l}}]=\mathcal{Z}^{-1}&\cdot\displaystyle\prod_{k=i-1}^{j}\displaystyle\prod_{\begin{subarray}{c}m\in\llbracket
u,v+1\rrbracket\\\ m\leq
N\end{subarray}}x^{\mathbbm{1}_{d^{\bm{\mathsf{l}}}(-k,m)>0}}\cdot
L_{1;0}\big{(}\bm{A}^{\bm{\mathsf{l}}}(-k,m),b^{\bm{\mathsf{l}}}(-k,m);\bm{C}^{\bm{\mathsf{l}}}(-k,m),d^{\bm{\mathsf{l}}}(-k,m)\big{)}\\\
&\times\displaystyle\prod_{k=i-1}^{j}\displaystyle\prod_{\begin{subarray}{c}m\in\llbracket
u,v+1\rrbracket\\\
m>N\end{subarray}}y^{\mathbbm{1}_{d^{\bm{\mathsf{l}}}(-k,m)>0}}\cdot
L_{1;0}\big{(}\bm{A}^{\bm{\mathsf{l}}}(-k,m),b^{\bm{\mathsf{l}}}(-k,m);\bm{C}^{\bm{\mathsf{l}}}(-k,m),d^{\bm{\mathsf{l}}}(-k,m)\big{)}.\end{aligned}$

Moreover, by Definition 4.4 (with the fact that $\bm{\mathsf{l}}^{(c)}$ is
simple), we have
$\mathbbm{1}_{d^{\bm{\mathsf{l}}}(-k,m)>0}=\mathsf{l}_{k+1}^{(1)}(m-1)-\mathsf{l}_{k+1}^{(1)}(m)$.
Hence, setting $u_{0}=\max\\{u-1,N\\}$ and $v_{0}=\min\\{v+1,N\\}$, we have

$\displaystyle\displaystyle\prod_{\begin{subarray}{c}m\in\llbracket
u,v+1\rrbracket\\\ m\leq
N\end{subarray}}x^{\mathbbm{1}_{d^{\bm{\mathsf{l}}}(-k,m)>0}}=x^{(\mathsf{l}_{k}^{(1)}(u-1)-\mathsf{l}_{k}^{(1)}(v_{0}))\cdot\mathbbm{1}_{u\leq
N}};\quad\displaystyle\prod_{\begin{subarray}{c}m\in\llbracket
u,v+1\rrbracket\\\
m>N\end{subarray}}y^{\mathbbm{1}_{d^{\bm{\mathsf{l}}}(-k,m)>0}}=y^{(\mathsf{l}_{k}^{(1)}(u_{0})-\mathsf{l}_{k}^{(1)}(v+1))\cdot\mathbbm{1}_{v\geq
N}}.$

Since $u-1,u_{0},v_{0},v+1\notin\llbracket u,v\rrbracket$ (as
$N\notin\llbracket u,v\rrbracket$), the above factors are fixed by the
conditioning. Thus, on the right side of (5.9), we may incorporate them into
the normalization constant $\mathcal{Z}$ to obtain (after altering
$\mathcal{Z}$ if necessary)

$\displaystyle\mathbb{P}_{\operatorname{scL}}^{\sigma}[\bm{\mathsf{L}}=\bm{\mathsf{l}}]=\mathcal{Z}^{-1}$
$\displaystyle\cdot\displaystyle\prod_{k=i-1}^{j}\displaystyle\prod_{\begin{subarray}{c}m\in\llbracket
u,v+1\rrbracket\\\ m\leq
N\end{subarray}}L_{1;0}\big{(}\bm{A}^{\bm{\mathsf{l}}}(-k,m),b^{\bm{\mathsf{l}}}(-k,m);\bm{C}^{\bm{\mathsf{l}}}(-k,m),d^{\bm{\mathsf{l}}}(-k,m)\big{)}$
$\displaystyle\times\displaystyle\prod_{k=i-1}^{j}\displaystyle\prod_{\begin{subarray}{c}m\in\llbracket
u,v+1\rrbracket\\\
m>N\end{subarray}}L_{1;0}\big{(}\bm{A}^{\bm{\mathsf{l}}}(-k,m),b^{\bm{\mathsf{l}}}(-k,m);\bm{C}^{\bm{\mathsf{l}}}(-k,m),d^{\bm{\mathsf{l}}}(-k,m)\big{)}.$

By 5.4, the above product is equal to $1$ if and only if, for any
$(c,k,m)\in\llbracket 1,n\rrbracket\times\mathbb{Z}_{>0}\times\llbracket
1,M+N\rrbracket$ with $\mathsf{l}_{k}^{(c+1)}(m)>\mathsf{l}_{k+1}^{(c+1)}(m)$,
(5.6) holds; otherwise it is equal to $0$. This confirms the second part of
the proposition. ∎

Now we can quickly establish 1.6.

###### Proof of 1.6.

The first part of this theorem follows from the $q=0$ case of 4.7; the second
and third follow from 5.3. ∎

##  6\. Fusion

Until now, we have used colored six-vertex or higher spin path ensembles; they
only allowed at most one arrow to occupy any horizontal edge. In this section
we remove that restriction using the fusion procedure originally introduced in
[62], and describe the counterparts to the statements from Section 2 and
Section 3 when horizontal edges may accommodate more than one arrow. Since
such ideas have been used repeatedly throughout the literature, and since many
statements in this section are similar to those in Section 2 and Section 3, we
will sometimes only outline (or omit) the proofs of the below results.
Throughout this section, we fix an integer $n\geq 1$.

###  6.1. Stochastic Fused Vertex Models

A _colored fused arrow configuration_ is a quadruple
$(\bm{A},\bm{B};\bm{C},\bm{D})$ of elements in $\mathbb{Z}_{\geq 0}^{n}$. We
view this as an assignment of directed up-right colored arrows to a vertex
$v\in\mathbb{Z}^{2}$, in which a (horizontal or vertical) edge can accommodate
arbitrarily many arrows. In particular, for each $k\in\llbracket
1,n\rrbracket$, the numbers $A_{k}$, $B_{k}$, $C_{k}$, and $D_{k}$ denote the
numbers of arrows of color $k$ vertically entering, horizontally entering,
vertically exiting, and horizontally exiting $v$, respectively.

As in Section 1.2 and Section 2.2, a _colored fused path ensemble_ on a domain
$\mathcal{D}\subseteq\mathbb{Z}^{2}$ is a consistent assignment of a colored
fused arrow configuration
$\big{(}\bm{A}(v),\bm{B}(v);\bm{C}(v),\bm{D}(v)\big{)}$ to each vertex
$v\in\mathcal{D}$. Observe that the arrows in a colored fused path ensemble on
$\mathcal{D}$ form colored up-right directed paths (that can share horizontal
and vertical edges) connecting vertices of $\mathcal{D}$. Associated with a
colored fused path ensemble $\mathcal{E}$ on some domain
$\mathcal{D}\subseteq\mathbb{Z}^{2}$ are height functions
$\mathfrak{h}_{c}^{\rightarrow},\mathfrak{h}_{c}^{\leftarrow},\mathfrak{h}_{\geq
c}^{\rightarrow},\mathfrak{h}_{\leq
c}^{\leftarrow}:\mathbb{Z}^{2}\rightarrow\mathbb{Z}$, which are defined in the
same way as in Section 2.2.

The probability measure on random colored fused path ensembles on
$\mathbb{Z}_{>0}^{2}$ that we next define depends on four sequences of complex
parameters $\bm{x}=(x_{1},x_{2},\ldots)$; $\bm{r}=(r_{1},r_{2},\ldots)$;
$\bm{y}=(y_{1},y_{2},\ldots)$; and $\bm{s}=(s_{1},s_{2},\ldots)$. We view
$x_{j}$ and $r_{j}$ as associated with the $j$-th row, so they are called row
rapidity and spin parameters, respectively; we view $y_{i}$ and $s_{i}$ as
associated with the $i$-th column, so they are called column rapidity and
fusion parameters, respectively. The specific forms of these probability
measures are expressed through certain weights
$U_{x_{j}/y_{i};r_{j},s_{i}}(\bm{A},\bm{B};\bm{C},\bm{D})$ associated with
each vertex $v=(i,j)\in\mathbb{Z}_{>0}^{2}$ (analogously to Definition 1.1).
We use the following ones, due to [26, Equation (7.10)] (though our notation
is closer to [25]121212The $(r,s)$ in Definition 6.1 are the
$(q^{-\mathsf{L}/2},q^{-\mathsf{M}/2})$ from [25].), that satisfy the
Yang–Baxter equation (see 6.6 below). In the following, we define the function
$\varphi:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$ by setting

$\displaystyle\varphi(\bm{X},\bm{Y})=\displaystyle\sum_{1\leq i<j\leq
n}X_{i}Y_{j},\qquad\text{for any $\bm{X},\bm{Y}\in\mathbb{R}^{n}$}.$

######  Definition 6.1 ([25, Equation (C.1.4)]).

Fix $r,s,z\in\mathbb{C}$ and $\bm{A},\bm{B},\bm{C},\bm{D}\in\mathbb{Z}_{\geq
0}^{n}$, and let $t=|\bm{T}|$ for each index $T\in\\{A,B,C,D\\}$. Define the
vertex weight

(6.1) $\displaystyle\begin{aligned} &U_{z;r,s}(\bm{A},\bm{B};\bm{C},\bm{D})\\\
&\quad=z^{d-b}r^{2(c-a)}s^{2d}q^{\varphi(\bm{D},\bm{C})}\displaystyle\frac{(r^{2};q)_{d}}{(r^{2};q)_{b}}\displaystyle\sum_{p=0}^{\min\\{b,c\\}}(r^{-2}z)^{p}\displaystyle\frac{(r^{-2}s^{2}z;q)_{c-p}(r^{2}z^{-1};q)_{p}(z;q)_{b-p}}{(s^{2}z;q)_{c+d-p}}\\\
&\qquad\qquad\quad\times\mathbbm{1}_{\bm{A}+\bm{B}=\bm{C}+\bm{D}}\cdot\displaystyle\sum_{\begin{subarray}{c}\bm{P}\leq\bm{B},\bm{C}\\\
|\bm{P}|=p\end{subarray}}q^{\varphi(\bm{B}-\bm{D}-\bm{P},\bm{P})}\displaystyle\prod_{i=1}^{n}\displaystyle\frac{(q;q)_{C_{i}+D_{i}-P_{i}}}{(q;q)_{D_{i}}(q;q)_{C_{i}-P_{i}}}\displaystyle\frac{(q;q)_{B_{i}}}{(q;q)_{P_{i}}(q;q)_{B_{i}-P_{i}}},\end{aligned}$

where the sum is over all $\bm{P}=(P_{1},\ldots,P_{n})\in\mathbb{Z}_{\geq
0}^{n}$ such that $|\bm{P}|=p$ and $0\leq P_{i}\leq\min\\{B_{i},C_{i}\\}$ for
each $i\in\llbracket 1,n\rrbracket$.

######  Remark 6.2.

As in 1.2, the $U$-weights from Definition 6.1 are stochastic in the sense
that $\sum_{\bm{C},\bm{D}\in\mathbb{Z}_{\geq
0}^{n}}U_{z;r,s}(\bm{A},\bm{B};\bm{C},\bm{D})=1$, for each
$r,s,z\in\mathbb{C}$ and $\bm{A},\bm{B}\in\mathbb{Z}_{\geq 0}^{n}$; see [25,
Proposition C.1.2].

We can now use these $U$-weights to describe (similarly to Section 1.2) how to
sample a random colored fused path ensemble on $\mathbb{Z}_{>0}^{2}$. We first
define probability measures $\mathbb{P}_{\operatorname{FV}}^{n}$ on the set of
colored fused ensembles whose vertices are all contained in the triangles
$\mathbb{T}_{N}=\big{\\{}(x,y)\in\mathbb{Z}_{>0}^{2}:x+y\leq N\big{\\}}$. The
initial measure $\mathbb{P}_{\operatorname{FV}}^{0}$ is supported by the empty
ensemble.

For each integer $N\geq 1$, we will define
$\mathbb{P}_{\operatorname{FV}}^{N+1}$ from
$\mathbb{P}_{\operatorname{FV}}^{N}$ by first using
$\mathbb{P}_{\operatorname{FV}}^{N}$ to sample a colored fused path ensemble
$\mathcal{E}_{N}$ on $\mathbb{T}_{N}$. This yields arrow configurations for
all vertices in the triangle $\mathbb{T}_{N-1}$. To extend this to a colored
six-vertex ensemble on $\mathbb{T}_{N+1}$, we must prescribe arrow
configurations to all vertices on the diagonal
$\mathbb{D}_{N}=\big{\\{}(x,y)\in\mathbb{Z}_{>0}^{2}:x+y=N\big{\\}}$. Since
$\mathcal{E}_{N}$ and the initial data prescribe the first two coordinates
$(\bm{A},\bm{B})$ of the arrow configuration to each vertex in
$\mathbb{D}_{N}$, it remains to explain how to assign the second two
coordinates $(\bm{C},\bm{D})$ of the arrow configuration at any vertex
$(i,j)\in\mathbb{D}_{N}$, given its first two coordinates $(\bm{A},\bm{B})$.
This is done according to the transition probability
$\mathbb{P}_{\operatorname{FV}}^{N}\big{[}(\bm{C},\bm{D})\big{|}(\bm{A},\bm{B})\big{]}=U_{y_{i}/x_{j};r_{j},s_{i}}(\bm{A},\bm{B};\bm{C},\bm{D})$.
We assume that the parameters $(\bm{x};\bm{y};\bm{r};\bm{s};q)$ are chosen so
that these probabilities are all nonnegative; the stochasticity of the
$U$-weights (6.2) then ensures that $\mathbb{P}_{\operatorname{FV}}^{N}$
indeed defines a probability measure.

Choosing $(\bm{C},\bm{D})$ according to the above transition probabilities
yields a random colored fused path ensemble $\mathcal{E}_{N+1}$, now defined
on $\mathbb{T}_{N+1}$; the probability distribution of $\mathcal{E}_{N+1}$ is
then denoted by $\mathbb{P}_{\operatorname{FV}}^{N+1}$. Taking the limit as
$n$ tends to $\infty$ yields a probability measure on colored fused path
ensembles on the quadrant. We refer to it as the _colored stochastic fused
vertex model_ ; observe that it may also be sampled on any rectangle
$\mathcal{D}\subset\mathbb{Z}^{2}$ in the same way as it was above on the
quadrant.

###  6.2. Yang–Baxter Equation for Fused Weights

In this section we state the Yang–Baxter equation for the $U$-weights from
Definition 6.1, with another family of weights given by the $W$ and
$\widehat{W}$ ones below (which serve as analogs of the $L$ and $\widehat{L}$
weights from Definition 2.1).

######  Definition 6.3.

Adopting the notation of Definition 6.1, define the weight

$\displaystyle W_{x;r,s}(\bm{A},\bm{B};\bm{C},\bm{D})=(-s)^{-d}\cdot
U_{x/s;r,s}(\bm{A},\bm{B};\bm{C},\bm{D}),$

so that

(6.2) $\displaystyle\begin{aligned} &W_{x;r,s}(\bm{A},\bm{B};\bm{C},\bm{D})\\\
&\quad=(-1)^{d}x^{d-b}r^{2(c-a)}s^{b}q^{\varphi(\bm{D},\bm{C})}\displaystyle\frac{(r^{2};q)_{d}}{(r^{2};q)_{b}}\displaystyle\sum_{p=0}^{\min\\{b,c\\}}(r^{2}s)^{-p}x^{p}\displaystyle\frac{(r^{-2}sx;q)_{c-p}(r^{2}sx^{-1};q)_{p}(s^{-1}x;q)_{b-p}}{(sx;q)_{c+d-p}}\\\
&\qquad\qquad\quad\times\mathbbm{1}_{\bm{A}+\bm{B}=\bm{C}+\bm{D}}\cdot\displaystyle\sum_{\begin{subarray}{c}\bm{P}\leq\bm{B},\bm{C}\\\
|\bm{P}|=p\end{subarray}}q^{\varphi(\bm{B}-\bm{D}-\bm{P},\bm{P})}\displaystyle\prod_{i=1}^{n}\displaystyle\frac{(q;q)_{C_{i}+D_{i}-P_{i}}}{(q;q)_{D_{i}}(q;q)_{C_{i}-P_{i}}}\displaystyle\frac{(q;q)_{B_{i}}}{(q;q)_{P_{i}}(q;q)_{B_{i}-P_{i}}}.\end{aligned}$

We also set $W_{x;r,0}(\bm{A},\bm{B};\bm{C},\bm{D})=\lim_{s\rightarrow
0}W_{x;r,s}(\bm{A},\bm{B},\bm{C},\bm{D})$, where the existence (and an
explicit form) of this limit follows from 6.7 below. Additionally, if there
exists an integer $\mathrm{R}\geq 1$ for which $r=q^{-\mathrm{R}/2}$, then set

(6.3)
$\displaystyle\widehat{W}_{x;r,s}(\bm{A},\bm{B};\bm{C},\bm{D})=W_{x;r,s}(\bm{A},\bm{B};\bm{C},\bm{D})\cdot(-s)^{-\mathrm{R}}\displaystyle\frac{(sx;q)_{\mathrm{R}}}{(s^{-1}x;q)_{\mathrm{R}}}.$

######  Remark 6.4.

Observe for any $r,s,z\in\mathbb{C}$ and $\bm{B}\in\mathbb{Z}_{\geq 0}^{n}$
that

$\displaystyle
U_{z;r,s}(\bm{e}_{0},\bm{B};\bm{e}_{0},\bm{B})=s^{2b}\cdot\displaystyle\frac{(z;q)_{b}}{(s^{2}z;q)_{b}};\qquad
W_{z;r,s}(\bm{e}_{0},\bm{B};\bm{e}_{0},\bm{B})=(-s)^{b}\cdot\displaystyle\frac{(s^{-1}z;q)_{b}}{(sz;q)_{b}},$

which quickly follow from the fact that the sums in (6.1) and (6.2) are
supported on the term $\bm{P}=\bm{e}_{0}$ (as $\bm{C}=\bm{e}_{0}$).

######  Remark 6.5.

Let us explain the sense in which the $\widehat{W}$-weight from Definition 6.3
is analogous to the $\widehat{L}$ one from (2.3). The latter was chosen to be
the (unique) normalization of $L_{x}$ such that
$\widehat{L}_{x}(\bm{e}_{0},k;\bm{e}_{0},k)=1$ for each $k\in\llbracket
1,n\rrbracket$; the arrow configuration $(\bm{e}_{0},k;\bm{e}_{0},k)$ could be
viewed as “horizontally saturated,” since horizontal edges could accommodate
at most one arrow under the $L$-weights. The analog of this constraint would
be to make $\widehat{W}_{z;r,s}$ a normalization of $W_{z;r,s}$ such that
$\widehat{W}_{z;r,s}=1$ at a fused arrow configuration that is “horizontally
saturated” in one color. One way to make sense of “horizontal saturation” for
fused arrow configurations is to impose a threshold
$\mathrm{R}\in\mathbb{Z}_{>0}$ for the number of arrows that can occupy a
horizontal edge; this is done by setting $r=q^{-\mathrm{R}/2}$ (as then the
factor of $(r^{2};q)_{d}$ in the $W$-weight (6.2) is equal to $0$ if
$d>\mathrm{R}$). In this case, the normalization condition would be for

(6.4)
$\displaystyle\widehat{W}_{z;r,s}(\bm{e}_{0},\mathrm{R}\bm{e}_{k};\bm{e}_{0},\mathrm{R}\bm{e}_{k})=1,\qquad\text{for
each $k\in\llbracket 1,n\rrbracket$},$

so that
$\widehat{W}_{z;r,s}(\bm{A},\bm{B};\bm{C},\bm{D})=W_{z;r,s}(\bm{A},\bm{B};\bm{C},\bm{D})\cdot
W_{z;r,s}(\bm{e}_{0},\mathrm{R}\bm{e}_{k};\bm{e}_{0},\mathrm{R}\bm{e}_{k})^{-1}$,
which by 6.4 yields (6.3).

Let us also briefly mention that another way of imposing “horizontal
saturation” would be to have infinitely many arrows of some color
$k\in\llbracket 1,n\rrbracket$ travel along rows of the model. One should then
track how many arrows of color $k$ leave a row (as well as how many arrows of
the other colors enter it), that is, one “complements” the arrow configuration
in the color $k$. This should enable one to remove the restriction that
$r=q^{-\mathrm{R}/2}$ for some integer $\mathrm{R}\geq 1$; similar ideas were
also used in [2, Section 3.1.3] and [5, Section 17.7.3]. However, we will not
pursue this direction here,131313See, however, Appendix A below, which
implements a version of this complementation to degenerate the colored
stochastic fused vertex model to the log-gamma polymer. and keep ourselves
constrained to the case when $r^{2}\in q^{\mathbb{Z}_{<0}}$ whenever using the
$\widehat{W}$-weights.

The following proposition states that the $U$ and $W$ weights (of Definition
6.1 and 6.5) satisfy the Yang-Baxter equation; it is due to [26, Equation
(3.20)] (though, as stated below, it appears in [25]141414In [25, Theorem
C.1.1], it was assumed that $r^{2},s^{2},t^{2}\in q^{\mathbb{Z}_{<0}}$, but
this assumption can be removed by using analytic continuation (with the fact
that the $U$-weights are rational in $r$, $s$, and $t$).); it is a fused
generalization of 2.2.

######  Lemma 6.6 ([25, Theorem C.1.1]).

Fix $x,y,z,r,s,t\in\mathbb{C}$ and
$\bm{I}_{1},\bm{J}_{1},\bm{K}_{1},\bm{I}_{3},\bm{J}_{3},\bm{K}_{3}\in\mathbb{Z}_{\geq
0}$. Then,

$\displaystyle\begin{aligned}
&\displaystyle\sum_{\bm{I}_{2},\bm{J}_{2},\bm{K}_{2}\in\mathbb{Z}_{\geq
0}^{n}}U_{x/y;r,s}(\bm{I}_{1},\bm{J}_{1};\bm{I}_{2},\bm{J}_{2})U_{x/z;r,t}(\bm{K}_{1},\bm{J}_{2};\bm{K}_{2},\bm{J}_{3})U_{y/z;s,t}(\bm{K}_{2},\bm{I}_{2};\bm{K}_{3},\bm{I}_{3})\\\
&\quad=\displaystyle\sum_{\bm{I}_{2},\bm{J}_{2},\bm{K}_{2}\in\mathbb{Z}_{\geq
0}^{n}}U_{y/z;s,t}(\bm{K}_{1},\bm{I}_{1};\bm{K}_{2},\bm{I}_{2})U_{x/z;r,t}(\bm{K}_{2},\bm{J}_{1};\bm{K}_{3},\bm{J}_{2})U_{x/y;r,s}(\bm{I}_{2},\bm{J}_{2};\bm{I}_{3},\bm{J}_{3}).\end{aligned}$

Therefore, if there exists an integer $\mathrm{R}>0$ such that
$r=q^{-\mathrm{R}/2}$, then

$\displaystyle\begin{aligned}
&\displaystyle\sum_{\bm{I}_{2},\bm{J}_{2},\bm{K}_{2}\in\mathbb{Z}_{\geq
0}^{n}}U_{x/y;r,s}(\bm{I}_{1},\bm{J}_{1};\bm{I}_{2},\bm{J}_{2})\widehat{W}_{x/z;r,t}(\bm{K}_{1},\bm{J}_{2};\bm{K}_{2},\bm{J}_{3})W_{y/z;s}(\bm{K}_{2},\bm{I}_{2};\bm{K}_{3},\bm{I}_{3})\\\
&\quad=\displaystyle\sum_{\bm{I}_{2},\bm{J}_{2},\bm{K}_{2}\in\mathbb{Z}_{\geq
0}^{n}}U_{y/z;s,t}(\bm{K}_{1},\bm{I}_{1};\bm{K}_{2},\bm{I}_{2})\widehat{W}_{x/z;r,t}(\bm{K}_{2},\bm{J}_{1};\bm{K}_{3},\bm{J}_{2})W_{x/y;r}(\bm{I}_{2},\bm{J}_{2};\bm{I}_{3},\bm{J}_{3}).\end{aligned}$

Before proceeding, we record the following results that evaluate the
$W_{x;r,0}$ weights.

######  Lemma 6.7.

Adopting the notation of Definition 6.3, we have

$\displaystyle\begin{aligned}
W_{z;r,0}(\bm{A},\bm{B};\bm{C},\bm{D})&=(-1)^{b-d}z^{d}r^{2(c-a)}q^{\varphi(\bm{D},\bm{C})+\binom{b}{2}}\displaystyle\frac{(r^{2};q)_{d}}{(r^{2};q)_{b}}\cdot\mathbbm{1}_{\bm{A}+\bm{B}=\bm{C}+\bm{D}}\\\
&\qquad\times\displaystyle\prod_{i=1}^{n}\Bigg{(}\displaystyle\sum_{p=0}^{\min\\{B_{i},C_{i}\\}}(-r^{-2})^{p}q^{\binom{p+1}{2}-p(B_{[i,n]}+D_{[1,i-1]})}\\\
&\qquad\qquad\qquad\times\displaystyle\frac{(q;q)_{C_{i}+D_{i}-p}}{(q;q)_{C_{i}-p}(q;q)_{D_{i}}}\displaystyle\frac{(q;q)_{B_{i}}}{(q;q)_{B_{i}-p}(q;q)_{p}}\Bigg{)}.\end{aligned}$

###### Proof.

Throughout this proof, we assume that $\bm{A}+\bm{B}=\bm{C}+\bm{D}$, as
otherwise $W_{z;r,0}(\bm{A},\bm{B};\bm{C},\bm{D})=0$ (as
$U_{z;r,s}(\bm{A},\bm{B};\bm{C},\bm{D})=0$ for any $s\in\mathbb{C}$, by
Definition 6.1). Inserting the equalities

$\displaystyle\displaystyle\lim_{s\rightarrow
0}s^{b-p}(s^{-1}z;q)_{b-p}=(-1)^{b-p}q^{\binom{b-p}{2}}z^{b-p};\qquad\displaystyle\lim_{s\rightarrow
0}\displaystyle\frac{(r^{-2}sz;q)_{c-p}(r^{2}sz^{-1};q)_{p}}{(sz;q)_{c+d-p}}=1,$

into (6.1), we obtain

$\displaystyle
W_{z;r,0}(\bm{A},\bm{B};\bm{C},\bm{D})=(-1)^{b-d}z^{d}r^{2(c-a)}q^{\varphi(\bm{D},\bm{C})}\displaystyle\frac{(r^{2};q)_{d}}{(r^{2};q)_{b}}$
$\displaystyle\displaystyle\sum_{p=0}^{\min\\{b,c\\}}(-r^{-2})^{p}q^{\binom{b-p}{2}}\displaystyle\sum_{\begin{subarray}{c}\bm{P}\leq\bm{B},\bm{C}\\\
|\bm{P}|=p\end{subarray}}q^{\varphi(\bm{B}-\bm{D}-\bm{P},\bm{P})}$
$\displaystyle\times\displaystyle\prod_{i=1}^{n}\displaystyle\frac{(q;q)_{C_{i}+D_{i}-P_{i}}}{(q;q)_{D_{i}}(q;q)_{C_{i}-P_{i}}}\displaystyle\frac{(q;q)_{B_{i}}}{(q;q)_{P_{i}}(q;q)_{B_{i}-P_{i}}}.$

Also since

$\displaystyle\binom{b-p}{2}$
$\displaystyle=\binom{b}{2}-bp+\binom{p+1}{2};\qquad\binom{p+1}{2}-\varphi(\bm{P},\bm{P})=\displaystyle\sum_{i=1}^{n}\binom{P_{i}+1}{2};$
$\displaystyle\varphi(\bm{B},\bm{P})-bp=-\displaystyle\sum_{1\leq i\leq j\leq
n}P_{i}B_{j}=-\displaystyle\sum_{i=1}^{n}B_{i}P_{i}-\varphi(\bm{P},\bm{B});$

and since
$\varphi(\bm{B}-\bm{D}-\bm{P},\bm{P})=\varphi(\bm{B},\bm{P})-\varphi(\bm{D},\bm{P})-\varphi(\bm{P},\bm{P})$
(by the bilinearity of $\varphi$), we have

(6.5) $\displaystyle\begin{aligned} W_{z;r,0}&(\bm{A},\bm{B};\bm{C},\bm{D})\\\
&=(-1)^{b-d}z^{d}r^{2(c-a)}q^{\varphi(\bm{D},\bm{C})+\binom{b}{2}}\displaystyle\frac{(r^{2};q)_{d}}{(r^{2};q)_{b}}\displaystyle\prod_{i=1}^{n}\displaystyle\frac{(q;q)_{B_{i}}}{(q;q)_{D_{i}}}\\\
&\quad\times\displaystyle\sum_{p=0}^{\min\\{b,c\\}}(-r^{-2})^{p}\displaystyle\sum_{\begin{subarray}{c}\bm{P}\leq\bm{B},\bm{C}\\\
|\bm{P}|=p\end{subarray}}q^{-\varphi(\bm{P},\bm{B})-\varphi(\bm{D},\bm{P})}\displaystyle\prod_{i=1}^{n}q^{\binom{P_{i}+1}{2}-B_{i}P_{i}}\displaystyle\frac{(q;q)_{C_{i}+D_{i}-P_{i}}}{(q;q)_{C_{i}-P_{i}}(q;q)_{P_{i}}(q;q)_{B_{i}-P_{i}}}.\end{aligned}$

Now observe for any complex number $w\in\mathbb{C}$; $n$-tuples
$\bm{X},\bm{Y},\bm{Z}\in\mathbb{Z}^{n}$; and functions
$f_{1},f_{2},\ldots,f_{n}:\mathbb{Z}_{\geq 0}\rightarrow\mathbb{C}$ we have

$\displaystyle\displaystyle\sum_{\begin{subarray}{c}\bm{P}\in\mathbb{Z}_{\geq
0}^{n}\\\
\bm{P}\leq\bm{Z}\end{subarray}}w^{|\bm{P}|}q^{\varphi(\bm{P},\bm{X})+\varphi(\bm{Y},\bm{P})}\displaystyle\prod_{i=1}^{n}f_{i}(P_{i})=\displaystyle\prod_{i=1}^{n}\Bigg{(}\displaystyle\sum_{p=0}^{Z_{i}}q^{p(X_{[i+1,n]}+Y_{[1,i-1]})}w^{p}f_{i}(p)\Bigg{)},$

by expanding the product on the right side. Applying this in (6.5) with

$\displaystyle
w=-r^{-2};\qquad\bm{X}=-\bm{B};\qquad\bm{Y}=-\bm{D};\qquad\bm{Z}=\min\\{\bm{B},\bm{C}\\};$
$\displaystyle\qquad\quad
f_{i}(k)=q^{\binom{k+1}{2}-B_{i}k}\displaystyle\frac{(q;q)_{C_{i}+D_{i}-k}}{(q;q)_{C_{i}-k}(q;q)_{k}(q;q)_{B_{i}-k}},$

yields the lemma (where $\min\\{\bm{B},\bm{C}\\}$ denotes the entry-wise
minimum of $\bm{B}$ and $\bm{C}$). ∎

######  Corollary 6.8.

Adopting the notation of Definition 6.3, we have
$W_{z;r,0}(\bm{A},\bm{B};\bm{A}+\bm{B},\bm{e}_{0})=1$.

###### Proof.

By 6.7 and the facts that
$(q;q)_{B_{i}}(q;q)_{B_{i}-p}^{-1}=(-1)^{p}q^{B_{i}p-\binom{p}{2}}(q^{-B_{i}};q)_{B_{i}}$;
that $B_{i}-B_{[i,n]}=-B_{[i+1,n]}$; and that $\binom{p+1}{2}-\binom{p}{2}=p$,
we have

(6.6) $\displaystyle
W_{z;r,0}(\bm{A},\bm{B};\bm{A}+\bm{B},\bm{e}_{0})=(-1)^{b}r^{2b}q^{\binom{b}{2}}(r^{2};q)_{b}^{-1}\cdot\displaystyle\prod_{i=1}^{n}\Bigg{(}\displaystyle\sum_{p=0}^{B_{i}}(r^{-2}q^{1-B_{[i+1,n]}})^{p}\displaystyle\frac{(q^{-B_{i}};q)_{p}}{(q;q)_{p}}\Bigg{)}.$

From the $q$-binomial theorem, we have for each $i\in\llbracket 1,n\rrbracket$
that

$\displaystyle\displaystyle\sum_{p=0}^{B_{i}}(r^{-2}q^{1-B_{[i+1,n]}})^{p}\displaystyle\frac{(q^{-B_{i}};q)_{p}}{(q;q)_{p}}=(r^{-2}q^{1-B_{[i,n]}};q)_{B_{i}},$

and hence

$\displaystyle\displaystyle\prod_{i=1}^{n}\Bigg{(}\displaystyle\sum_{p=0}^{B_{i}}(r^{-2}q^{1-B_{[i+1,n]}})^{p}\displaystyle\frac{(q^{-B_{i}};q)_{p}}{(q;q)_{p}}\Bigg{)}$
$\displaystyle=\displaystyle\prod_{i=1}^{n}(r^{-2}q^{1-B_{[i,n]}};q)_{B_{i}}$
$\displaystyle=(r^{-2};q^{-1})_{b}=(-1)^{b}r^{-2b}q^{-\binom{b}{2}}(r^{2};q)_{b}.$

Inserting this into (6.6) yields the corollary. ∎

######  Remark 6.9.

An alternative (and perhaps more conceptual) way of proving 6.8 would be
through fusion, using the fact that it holds at $\mathrm{R}=1$ by (3.11); we
will not provide further details on that route here.

###  6.3. Fused Nonsymmetric Functions

In this section we formulate the fused analogs of the functions $f$ and $G$
(from Definition 2.6), as well as some of their properties. We begin with the
following definition, which is parallel to Definition 2.3.

######  Definition 6.10.

Fix an integer $N\geq 1$; a complex number $s\in\mathbb{C}$; sequences of
complex numbers $\bm{r}=(r_{1},r_{2},\ldots,r_{N})$ and
$\bm{x}=(x_{1},x_{2},\ldots,x_{N})$; and a colored fused path ensemble
$\mathcal{E}$ on $\mathcal{D}_{N}=\mathbb{Z}_{\leq 0}\times\llbracket
1,N\rrbracket$, whose arrow configuration at any $v\in\mathcal{D}_{N}$ is
denoted by $\big{(}\bm{A}(v),\bm{B}(v);\bm{C}(v),\bm{D}(v)\big{)}$. Set

$\displaystyle
W_{\bm{x};\bm{r};s}(\mathcal{E})=\displaystyle\prod_{k=1}^{\infty}\displaystyle\prod_{j=1}^{N}W_{x_{j};r_{j},s}\big{(}\bm{A}(-k,j),\bm{B}(-k,j);\bm{C}(-k,j),\bm{D}(-k,j)\big{)}$
$\displaystyle\qquad\qquad\qquad\times\displaystyle\prod_{j=1}^{N}W_{x_{j};r_{j},0}\big{(}\bm{A}(0,j),\bm{B}(0,j);\bm{C}(0,j),\bm{D}(0,j)\big{)};$
$\displaystyle\widehat{W}_{\bm{x};\bm{r};s}(\mathcal{E})=\displaystyle\prod_{k=1}^{\infty}\displaystyle\prod_{j=1}^{N}\widehat{W}_{x_{j};r_{j},s}\big{(}\bm{A}(-k,j),\bm{B}(-k,j);\bm{C}(-k,j),\bm{D}(-k,j)\big{)}$
$\displaystyle\qquad\qquad\qquad\times\displaystyle\prod_{j=1}^{N}W_{x_{j};r_{j},0}\big{(}\bm{A}(0,j),\bm{B}(0,j);\bm{C}(0,j),\bm{D}(0,j)\big{)},$

where in the second equality it is assumed that $r_{j}^{2}\in
q^{\mathbb{Z}_{<0}}$, for each $j\in\llbracket 1,N\rrbracket$.

Now we can define the following (non)symmetric functions, in a way parallel to
Definition 2.6, but with three differences. First, the vertex models are fused
path ensembles, instead of higher spin ones. Second, they have weights the $W$
and $\widehat{W}$, instead of the $L$ and $\widehat{L}$. Third, in defining
$\mathbbm{f}$ below, we have multiple arrows entering all rows, instead of
only one, with the purpose of fully saturating them (see Figure 18).

######  Definition 6.11.

Fix an integer $N\geq 1$; a sequence of positive integers
$\bm{\mathrm{R}}=(\mathrm{R}_{1},\mathrm{R}_{2},\ldots,\mathrm{R}_{N})$; two
$n$-compositions $\mu,\nu\in\operatorname{Comp}_{n}$; and a function
$\sigma:\llbracket 1,N\rrbracket\rightarrow\llbracket 1,n\rrbracket$.

If $\ell(\mu)=\ell(\nu)+\mathrm{R}_{[1,N]}$, then let
$\mathfrak{P}_{\mathbbm{f}}(\mu/\nu;\sigma;\bm{\mathrm{R}})$ denote the set of
colored fused path ensembles on $\mathcal{D}_{N}=\mathbb{Z}_{\leq
0}\times\llbracket 1,N\rrbracket$ with the following boundary data.

  1. (1)

For each $j\in\llbracket 1,N\rrbracket$, $\mathrm{R}_{j}$ arrows of color
$\sigma(j)$ horizontally enters $\mathcal{D}_{N}$ through $(-\infty,j)$.

  2. (2)

For each $k\geq 0$ and $c\in\llbracket 1,n\rrbracket$,
$\mathfrak{m}_{k}\big{(}\nu^{(c)}\big{)}$ arrows of color $c$ vertically enter
$\mathcal{D}_{N}$ through $(-k,1)$.

  3. (3)

For each $k\geq 0$ and $c\in\llbracket 1,n\rrbracket$,
$\mathfrak{m}_{k}\big{(}\mu^{(c)}\big{)}$ arrows of color $c$ vertically exit
$\mathcal{D}_{N}$ through $(-k,N)$.

See the left side of Figure 18 for a depiction when $\mu=(7,2,0\mid
5,5,4,1,1,0\mid 5,2,2)$; $\nu=(\emptyset\mid 6\mid 6,5)$;
$\big{(}\sigma(1),\sigma(2),\sigma(3),\sigma(4),\sigma(5)\big{)}=(1,3,2,1,2)$;
and $\bm{\mathrm{R}}=(2,1,3,1,2)$. There, red, green, and blue are colors $1$,
$2$, and $3$, respectively.

Similarly, if $\ell(\mu)=\ell(\nu)$, then let
$\mathfrak{P}_{\mathbb{G}}(\mu/\nu)$ denote the set of colored fused path
ensembles on $\mathcal{D}_{N}=\mathbb{Z}_{\leq 0}\times\llbracket
1,N\rrbracket$, with the following boundary data.

  1. (1)

For each $j\in\llbracket 1,N\rrbracket$, no arrow horizontally enters or exits
$\mathcal{D}_{N}$ through the $j$-th row.

  2. (2)

For each $k\geq 0$ and $c\in\llbracket 1,n\rrbracket$,
$\mathfrak{m}_{k}\big{(}\mu^{(c)}\big{)}$ arrows of color $c$ vertically enter
$\mathcal{D}_{N}$ through $(-k,1)$.

  3. (3)

For each $k\geq 0$ and $c\in\llbracket 1,n\rrbracket$,
$\mathfrak{m}_{k}\big{(}\nu^{(c)}\big{)}$ arrows of color $c$ vertically exit
$\mathcal{D}_{N}$ through $(-k,N)$.

See the right side of Figure 18 for a depiction when $\mu=(7,5\mid 7,6\mid
6,4)$ and $\nu=(5,2\mid 5,1\mid 3,2)$.

For any complex number $s\in\mathbb{C}$, and sequences of complex numbers
$\bm{r}=(r_{1},r_{2},\ldots,r_{N})$ and $\bm{x}=(x_{1},x_{2},\ldots,x_{N})$,
let

(6.7)
$\displaystyle\mathbbm{f}_{\mu/\nu;s}^{\sigma}(\bm{x};\bm{r})=\displaystyle\sum_{\mathfrak{P}_{f}(\mu/\nu;\sigma;\bm{\mathrm{R}})}\widehat{W}_{\bm{x};\bm{r};s}(\mathcal{E});\qquad\mathbbm{G}_{\mu/\nu;s}(\bm{x};\bm{r})=\displaystyle\sum_{\mathfrak{P}_{G}(\mu/\nu)}W_{\bm{x};\bm{r};s}(\mathcal{E}),$

where in the first equality it is assumed that
$r_{j}^{-2}=q^{-\mathrm{R}_{j}}$ for each $j\in\llbracket 1,N\rrbracket$ (and
$\bm{r}$ can be arbitrary in the second). If $\nu=\emptyset$ is empty, we
write
$\mathbbm{f}_{\mu;s}^{\sigma}(\bm{x};\bm{r})=\mathbbm{f}_{\mu/\emptyset;s}^{\sigma}(\bm{x};\bm{r})$
and
$\mathbb{G}_{\mu;s}(\bm{x};\bm{r})=\mathbb{G}_{\mu/0^{N};s}(\bm{x};\bm{r})$.
If $N=1$, we may write $\mathbbm{f}_{\mu/\nu;s}^{\sigma(1)}$ in place of
$\mathbbm{f}_{\mu/\nu;s}^{\sigma}$.

$\sigma(1)$$\sigma(2)$$\sigma(N)$$(x_{1},r_{1})$$(x_{2},r_{2})$$\vdots$$(x_{N},r_{N})$$\nu$$\mu$$\mathbbm{f}_{\mu/\nu}$$(x_{1},r_{1})$$(x_{2},r_{2})$$\vdots$$(x_{N},r_{N})$$\mu$$\nu$$G_{\mu/\nu}$
Figure 18.  Depicted to the left and right are vertex models for
$\mathbbm{f}_{\mu/\nu;s}^{\sigma}$ and $\mathbb{G}_{\mu/\nu;s}$, respectively.

Observe that the quantity $\widehat{W}_{\bm{x};\bm{r};s}(\mathcal{E})$
appearing as the summand in (6.7) defining $\mathbbm{f}_{\mu/\nu}^{\sigma}$ is
bounded, since all but finitely many vertices in any ensemble
$\mathcal{E}\in\mathfrak{P}_{\mathbbm{f}}(\mu/\nu;\sigma;\bm{\mathrm{R}})$
have arrow configurations of the form
$(\bm{e}_{0},\mathrm{R}_{j}\bm{e}_{\sigma(j)};\bm{e}_{0},\mathrm{R}_{j}\bm{e}_{\sigma(j)})$
for some $j\in\llbracket 1,N\rrbracket$, and
$\widehat{W}_{x_{j};r_{j},s}(\bm{e}_{0},\mathrm{R}_{j}\bm{e}_{\sigma(j)};\bm{e}_{0},\mathrm{R}_{j}\bm{e}_{\sigma(j)})=1$
for $r_{j}=q^{-\mathrm{R}_{j}/2}$, by (6.4). Similarly,
$W_{\bm{x};\bm{r};s}(\mathcal{E})$ appearing as the summand in (6.7) defining
$\mathbb{G}_{\mu/\nu;s}$ is bounded, since all but finitely many vertices in
any $\mathcal{E}\in\mathfrak{P}_{\mathbb{G}}(\mu/\nu)$ have arrow
configurations of the form $(\bm{e}_{0},\bm{e}_{0};\bm{e}_{0},\bm{e}_{0})$,
and we have $W_{x;r,s}(\bm{e}_{0},0;\bm{e}_{0},0)=1$ by 6.4.

######  Remark 6.12.

It may be possible to analytically continue the $\mathbbm{f}$ functions in the
parameters $\bm{r}$ (so as to avoid imposing the assumption that each
$r_{j}\in q^{\mathbb{Z}_{<0}}$), by following the complementation procedure
outlined at the end of 6.5. However, we will not pursue this here.

###  6.4. Properties of $\mathbbm{f}$ and $\mathbb{G}$

In this section we provide properties of the $\mathbbm{f}$ and $\mathbb{G}$
functions from Definition 6.11. The first is the _fusion_ property that
relates these to the $f$ and $G$ functions from Definition 2.6, when the
parameters of the latter are specialized to unions of geometric progressions.
We omit its proof, as very similar statements have appeared repeatedly
throughout the literature; see [23, Section 6E], [20, Theorem 6.2], and [5,
Proposition 7.2.3] for references in the colored case and [22, Proposition
5.5] in the uncolored one.

######  Lemma 6.13.

Adopt the notation of Definition 6.11, and assume
$r_{j}=q^{-\mathrm{R}_{j}/2}$ for each $j\in\llbracket 1,N\rrbracket$. Define
$\omega:\llbracket 1,\mathrm{R}_{[1,N]}\rrbracket\rightarrow\llbracket
1,n\rrbracket$ by for each $j\in\llbracket 1,\mathrm{R}_{[1,N]}\rrbracket$
setting $\omega(j)=\sigma(j^{\prime})$, where $j^{\prime}\in\llbracket
1,N\rrbracket$ is the unique index satisfying
$\mathrm{R}_{[1,j^{\prime}-1]}+1\leq j\leq\mathrm{R}_{[1,j^{\prime}]}$; also
define the $\mathrm{R}_{[1,N]}$-tuple of complex numbers

$\displaystyle\bm{z}=(x_{1},qx_{1},\ldots,q^{\mathrm{R}_{1}-1}x_{1},x_{2},qx_{2},\ldots,q^{\mathrm{R}_{2}-1}x_{2},\ldots,x_{N},qx_{N},\ldots,q^{\mathrm{R}_{N}-1}x_{n}).$

Then, we have
$\mathbbm{f}_{\mu/\nu;s}^{\sigma}(\bm{x};\bm{r})=f_{\mu/\nu;s}^{\omega}(\bm{z})$
and $\mathbb{G}_{\mu/\nu;s}(\bm{x};\bm{r})=G_{\mu/\nu;s}(\bm{z})$.

We next provide symmetry properties, branching statements, and Cauchy
identities for the $\mathbbm{f}$ and $\mathbb{G}$ functions, which are
parallel to their counterparts (2.7, 2.8, and 2.9, respectively) for the $f$
and $g$ functions. They are quick consequences of the latter, together with
6.13 and analytic continuation.

######  Lemma 6.14.

Adopt the notation of Definition 6.11, and let $\varsigma:\llbracket
1,M\rrbracket\rightarrow\llbracket 1,M\rrbracket$ denote a permutation. We
have
$\mathbb{G}_{\mu/\nu;s}(\bm{y};\bm{r})=\mathbb{G}_{\mu/\nu;s}\big{(}\varsigma(\bm{y});\varsigma(\bm{r})\big{)}$,
where
$\varsigma(\bm{y})=(y_{\varsigma(1)},y_{\varsigma(2)},\ldots,y_{\varsigma(M)}\big{)}$
and
$\varsigma(\bm{r})=(r_{\varsigma(1)},r_{\varsigma(2)},\ldots,r_{\varsigma(N)})$.

###### Proof.

If there exist positive integers
$(\mathrm{R}_{1},\mathrm{R}_{2},\ldots,\mathrm{R}_{N})$ such that
$r_{j}=q^{-\mathrm{R}_{j}/2}$ for each $j\in\llbracket 1,N\rrbracket$, then
this follows from 6.13 and 2.7. The fact that the lemma holds for an arbitrary
set $\bm{r}$ of complex numbers then follows from uniqueness of analytic
continuation, as $\mathbb{G}$ is a rational function in $\bm{r}$ (since the
$W$-weights from Definition 6.3 are). ∎

######  Lemma 6.15.

Adopting the notation of Definition 6.11; letting $\ell=\ell(\nu)$; and fixing
an integer $k\in\llbracket 1,N\rrbracket$, we have

$\displaystyle\mathbbm{f}_{\mu/\nu;s}^{\sigma}(\bm{x};\bm{r})=\displaystyle\sum_{\kappa\in\operatorname{Comp}_{n}(\ell+k)}\mathbbm{f}_{\kappa/\nu;s}^{\sigma|_{\llbracket
1,k\rrbracket}}\big{(}\bm{x}_{[1,k]};\bm{r}_{[1,k]}\big{)}\mathbbm{f}_{\mu/\kappa;s}^{\sigma|_{\llbracket
k+1,N\rrbracket}}\big{(}\bm{x}_{[k+1,N]};\bm{r}_{[k+1,N]}\big{)};$
$\displaystyle\mathbb{G}_{\mu/\nu;s}(\bm{x};\bm{r})=\displaystyle\sum_{\kappa\in\operatorname{Comp}_{n}(\ell)}\mathbb{G}_{\mu/\kappa;s}\big{(}\bm{x}_{[1,k]};\bm{r}_{[1,k]}\big{)}\mathbb{G}_{\kappa/\nu;s}\big{(}\bm{x}_{[k+1,N]};\bm{r}_{[k+1,N]}\big{)}.$

where in the first equality we assume that $r_{j}=q^{-\mathrm{R}_{j}/2}$ for
each $j\in\llbracket 1,N\rrbracket$ (and $\bm{r}$ can be arbitrary in the
second). Here, we have defined the variable sets
$\bm{x}_{[1,k]}=(x_{1},x_{2},\ldots,x_{k})$,
$\bm{x}_{[k+1,N]}=(x_{k+1},x_{k+2},\ldots,x_{N})$,
$\bm{r}_{[1,k]}=(r_{1},r_{2},\ldots,r_{k})$, and
$\bm{r}_{[k+1,N]}=(r_{k+1},r_{k+2},\ldots,r_{N})$. For any interval
$I=\big{\llbracket}i_{0}+1,i_{0}+|I|\big{\rrbracket}\subset\llbracket
1,N\rrbracket$, we have also defined the function
$\sigma|_{I}:\big{\llbracket}1,|I|\big{\rrbracket}\rightarrow\llbracket
1,n\rrbracket$ by setting $\sigma|_{I}(i)=\sigma(i+i_{0})$ for each
$i\in\big{\llbracket}1,|I|\big{\rrbracket}$.

###### Proof.

The first equality in the lemma follows from 2.8 and 6.13. If
$r_{j}=q^{-\mathrm{R}_{j}/2}$ for each $j\in\llbracket 1,N\rrbracket$, the
second follows from the same two statements; for general $\bm{r}$, it then
follows from uniqueness of analytic continuation, as $\mathbb{G}$ is rational
in $\bm{r}$. ∎

######  Lemma 6.16.

Fix integers $n,M,N\geq 1$; a sequence of positive integers
$\bm{\mathrm{R}}=(\mathrm{R}_{1},\mathrm{R}_{2},\ldots,\mathrm{R}_{N})$;
sequences of complex variables $\bm{r}=(r_{1},r_{2},\ldots,r_{N})$,
$\bm{t}=(t_{1},t_{2},\ldots,t_{M})$, $\bm{x}=(x_{1},x_{2},\ldots,x_{N})$, and
$\bm{y}=(y_{1},y_{2},\ldots,y_{M})$; and a function $\sigma:\llbracket
1,N\rrbracket\rightarrow\llbracket 1,n\rrbracket$. Assume for each
$j\in\llbracket 1,N\rrbracket$ that $r_{j}=q^{-\mathrm{R}_{j}/2}$ and that

(6.8) $\displaystyle\displaystyle\max_{\begin{subarray}{c}1\leq i\leq M\\\
1\leq j\leq
N\end{subarray}}\displaystyle\sup_{\begin{subarray}{c}(b,b^{\prime})\in\mathbb{Z}_{\geq
0}\times\llbracket 0,\mathrm{R}_{j}\rrbracket\\\
(b,b^{\prime})\neq(0,\mathrm{R}_{j})\end{subarray}}\bigg{|}(-s)^{b}\displaystyle\frac{(s^{-1}y_{j};q)_{b}}{(sy_{j};q)_{b}}\cdot(-s)^{b^{\prime}-\mathrm{R}_{j}}\displaystyle\frac{(s^{-1}x_{i};q)_{b^{\prime}}}{(sx_{i};q)_{b^{\prime}}}\displaystyle\frac{(sx_{i};q)_{\mathrm{R}_{j}}}{(s^{-1}x_{i};q)_{\mathrm{R}_{j}}}\bigg{|}<1.$

Then,

$\displaystyle\displaystyle\sum_{\mu\in\operatorname{Comp}_{n}(\mathrm{R}_{[1,N]})}\mathbbm{f}_{\mu;s}^{\sigma}(\bm{x};\bm{r})\mathbb{G}_{\mu;s}(\bm{y};\bm{t})=\displaystyle\prod_{i=1}^{M}\displaystyle\prod_{j=1}^{N}\displaystyle\frac{(t_{i}^{2}x_{j}y_{i}^{-1};q)_{\mathrm{R}_{j}}}{t_{i}^{2\mathrm{R}_{j}}(x_{j}y_{i}^{-1};q)_{\mathrm{R}_{j}}}.$

###### Proof (Outline).

The proof is analogous to that of 2.9, so we only briefly outline it. We will
use the equality of partition functions depicted in Figure 19 (which is
similar to the equality of the partition functions depicted in Figure 7 and
Figure 8). On both sides of that figure, $\mathrm{R}_{j}$ arrows of color
$\sigma(j)$ enter horizontally through the $(M+j)$-th row (from the bottom)
for each $j\in\llbracket 1,N\rrbracket$, and all arrows exit vertically
through the $y$-axis. The weights on both sides are assigned as follows. In
the crosses, we use $U_{x_{j}/y_{i};r_{j},t_{i}}$ at the intersection of the
$j$-th row (from the bottom) and $i$-th column (from the left). Along the
$y$-axis, we use $W_{x_{j};r_{j},0}$ or $W_{y_{i};t_{i},0}$, depending on
whether the row is marked by $(x_{j},t_{j})$ or $(y_{i},s_{i})$ in Figure 19.
In $\mathbb{Z}_{<0}\times\llbracket 1,M+N\rrbracket$, we use
$\widehat{W}_{x_{j};r_{j},s}$ or $W_{y_{i};t_{i},s}$, again depending on the
marking of the row. The equality of partition functions depicted in Figure 19
is then a consequence of the Yang–Baxter equation 6.6; denote this partition
function by $\mathcal{Z}$.

The vertex model on the left side of Figure 19 is frozen; it is quickly
verified that it has weight

(6.9) $\displaystyle\mathcal{Z}=1,$

using 6.4 (at $b=0$), (6.4), and 6.8. To analyze the right side of Figure 19,
first observe by (6.8), 6.4, and (6.3) that, for any $(i,j)\in\llbracket
1,M\rrbracket\times\llbracket 1,N\rrbracket$, we have

$\displaystyle\displaystyle\sup_{\begin{subarray}{c}(b,b^{\prime})\in\mathbb{Z}_{\geq
0}\times\llbracket 0,\mathrm{R}_{j}\rrbracket\\\
(b,b^{\prime})\neq(0,\mathrm{R}_{j})\end{subarray}}\displaystyle\max_{\begin{subarray}{c}\bm{B},\bm{B}^{\prime}\in\mathbb{Z}_{\geq
0}^{n}\\\
|\bm{B}|=b,|\bm{B}^{\prime}|=b^{\prime}\end{subarray}}\bigg{|}\displaystyle\frac{\widehat{W}_{x_{j};r_{j},s}(\bm{e}_{0},\bm{B}^{\prime};\bm{e}_{0},\bm{B}^{\prime})}{\widehat{W}_{x_{j};r_{j},s}(\bm{e}_{0},\mathrm{R}_{j}\bm{e}_{\sigma(j)};\bm{e}_{0},\mathrm{R}_{j}\bm{e}_{\sigma(j)})}\cdot\displaystyle\frac{W_{y_{i};t_{i},s}(\bm{e}_{0},\bm{B};\bm{e}_{0},\bm{B})}{W_{y_{i};t_{i},s}(\bm{e}_{0},\bm{e}_{0};\bm{e}_{0},\bm{e}_{0})}\bigg{|}<1.$

Using this bound, one can verify (see, for example, the proof of [5,
Proposition 6.2.2] for a very similar argument) that the vertex model on the
right side of Figure 19 has nonzero weight only if all but finitely many
vertices in rows marked by $(y_{i},t_{i})$ for some $i\in\llbracket
1,M\rrbracket$ have arrow configuration
$(\bm{e}_{0},\bm{e}_{0};\bm{e}_{0},\bm{e}_{0})$. This implies that the cross
on the right side of Figure 19 is frozen to have arrow configuration
$(\bm{e}_{0},\mathrm{R}_{j}\bm{e}_{\sigma(j)};\bm{e}_{0},\mathrm{R}_{j}\bm{e}_{\sigma(j)}\big{)}$
at each vertex in its $j$-th row.

Hence, the weight of the cross is

$\displaystyle\displaystyle\prod_{i=1}^{M}\displaystyle\prod_{j=1}^{N}U_{x_{j}/y_{i};r_{j},t_{i}}(\bm{e}_{0},\mathrm{R}_{j}\bm{e}_{\sigma(j)};\bm{e}_{0},\mathrm{R}_{j}\bm{e}_{\sigma(j)})=\displaystyle\sum_{i=1}^{M}\displaystyle\prod_{j=1}^{N}\displaystyle\frac{t_{i}^{2\mathrm{R}_{j}}(x_{j}y_{i}^{-1};q)_{\mathrm{R}_{j}}}{(t_{i}^{2}x_{j}y_{i}^{-1};q)_{\mathrm{R}_{j}}}.$

Moreover, by (6.7), the weight of the part of the right side of Figure 19 on
$\mathbb{Z}_{\leq 0}\times\llbracket 1,M+N\rrbracket$ is given by
$\sum_{\mu\in\operatorname{Comp}_{n}(\mathrm{R}_{[1,N]})}\mathbbm{f}_{\mu}^{\sigma}(\bm{x};\bm{r})\mathbb{G}_{\mu}(\bm{y};\bm{t})$.
Hence,

$\displaystyle\mathcal{Z}=\displaystyle\prod_{i=1}^{M}\displaystyle\prod_{j=1}^{N}\displaystyle\frac{t_{i}^{2\mathrm{R}_{j}}(x_{j}y_{i}^{-1};q)_{\mathrm{R}_{j}}}{(t_{i}^{2}x_{j}y_{i}^{-1};q)_{\mathrm{R}_{j}}}\cdot\displaystyle\sum_{\mu\in\operatorname{Comp}_{n}(\mathrm{R}_{[1,N]})}\mathbbm{f}_{\mu}^{\sigma}(\bm{x};\bm{r})\mathbb{G}_{\mu}(\bm{y};\bm{t}).$

This, together with (6.9), yields the lemma. ∎

$(y_{M},t_{M})$$\cdots$$(y_{2},t_{2})$$(y_{1},t_{1})$$(x_{1},r_{1})$$\cdots$$(x_{N},r_{N})$$\mathrm{R}_{1}$$\vdots$$\mathrm{R}_{n}$$\sigma(1)$$\cdots$$\sigma(N)$$M$$=$$(y_{M},t_{M})$$\cdots$$(y_{2},t_{2})$$(y_{1},t_{1})$$\mathrm{R}_{1}$$\cdots$$\mathrm{R}_{N}$$\sigma(1)$$\cdots$$\sigma(N)$$(x_{1},t_{1})$$\vdots$$(x_{N},t_{N})$$M$
Figure 19.  Depicted above is the equality of partition functions used in the
proof of 6.16.

###  6.5. Ascending $\mathbbm{f}\mathbb{G}$ Measures

In this section we introduce the fused variants of the probability measures
from Definition 3.3. Throughout, we fix integers $M,N\geq 1$; a sequence of
positive integers
$\bm{\mathrm{R}}=(\mathrm{R}_{1},\mathrm{R}_{2},\ldots,\mathrm{R}_{N})$; a
composition $\bm{\ell}=(\ell_{1},\ell_{2},\ldots,\ell_{n})$ of
$\mathrm{R}_{[1,N]}$; and a function $\sigma:\llbracket
1,N\rrbracket\rightarrow\llbracket 1,n\rrbracket$ such that
$\ell_{i}=\sum_{j\in\sigma^{-1}(i)}\mathrm{R}_{j}$ for each $i\in\llbracket
1,n\rrbracket$. We must first introduce the relevant family of
$n$-compositions.

######  Definition 6.17.

A sequence $\bm{\mu}=\big{(}\mu(0),\mu(1),\ldots,\mu(M+N)\big{)}$ of
$n$-compositions is called _$(M;\sigma;\bm{\mathrm{R}})$ -ascending_ if the
following hold.

  1. (1)

We have $\mu(0)=(\emptyset\mid\cdots\mid\emptyset)$ and
$\mu(M+N)=(0^{\ell_{1}}\mid\cdots\mid 0^{\ell_{n}})$.

  2. (2)

    1. (a)

For all $j\in\llbracket 0,N\rrbracket$ and $c\in\llbracket 1,n\rrbracket$, we
have
$\ell\big{(}\mu^{(c)}(j)\big{)}=\sum_{k=1}^{j}\mathrm{R}_{k}\cdot\mathbbm{1}_{\sigma(k)=c}$.
Thus, $\ell\big{(}\mu(j)\big{)}=\mathrm{R}_{[1,j]}$.

    2. (b)

For all $i\in\llbracket N,M+N\rrbracket$ and $c\in\llbracket 1,n\rrbracket$,
we have $\ell\big{(}\mu^{(c)}(i)\big{)}=\ell_{c}$. Thus,
$\ell\big{(}\mu(i)\big{)}=\mathrm{R}_{[1,N]}$.

  3. (3)

For each $(c,k,i)\in\llbracket
1,n\rrbracket\times\mathbb{Z}_{>0}\times\llbracket 1,M+N\rrbracket$, we have

(6.10) $\displaystyle\mathfrak{Q}_{c}^{\bm{\mu}}(k,i)=\mathfrak{m}_{\leq
k-1}\big{(}\mu^{(c)}(i)\big{)}-\mathfrak{m}_{\leq
k-1}\big{(}\mu^{(c)}(i-1)\big{)}\geq 0.$

For each $(k,i)\in\mathbb{Z}_{>0}\llbracket 1,M+N\rrbracket$, we set
$\bm{\mathfrak{Q}}^{\bm{\mu}}(k,i)=\big{(}\mathfrak{Q}_{1}^{\bm{\mu}}(k,i),\mathfrak{Q}_{2}^{\bm{\mu}}(k,i),\ldots,\mathfrak{Q}_{n}^{\bm{\mu}}(k,i)\big{)}\in\mathbb{Z}_{\geq
0}^{n}$.

Let us also define the sequence
$\mathscr{Q}(\bm{\mu})=\big{(}\bm{\mathfrak{Q}}^{\bm{\mu}}(1,1),\bm{\mathfrak{Q}}^{\bm{\mu}}(1,2),\ldots,\bm{\mathfrak{Q}}^{\bm{\mu}}(1,M+N)\big{)}$.

######  Remark 6.18.

Given an $(M;\sigma;\bm{\mathrm{R}})$-ascending sequence of compositions
$\bm{\mu}$ as in Definition 6.17, we will often view the $n$-composition
$\mu(i)$ as indexing the positions (as in 2.5) of the colored arrows exiting
the row $\\{y=i\\}$, in a vertex model on $\mathbb{Z}_{\leq 0}\times\llbracket
1,M+N\rrbracket$. This yields a colored fused path ensemble on
$\mathbb{Z}_{\leq 0}\times\llbracket 1,M+N\rrbracket$, that we will denote by
$\mathcal{E}_{\bm{\mu}}$. In this way, the $c$-th entry in
$\bm{\mathfrak{Q}}^{\bm{\mu}}(k,i)$ denotes the number of color $c$ arrows in
$\mathcal{E}_{\bm{\mu}}$ along the edge connecting $(-k,i)$ to $(1-k,i)$.
Therefore, $\bm{\mathfrak{Q}}$ records the colors of the arrows (from top to
bottom) along the horizontal edges in $\mathcal{E}_{\bm{\mu}}$ joining the
$(-1)$-st column to the $0$-th one.

The boundary data for this ensemble is described as follows. For each
$j\in\llbracket 1,N\rrbracket$, it has $\mathrm{R}_{j}$ arrows of color
$\sigma(j)$ horizontally entering the row $\\{y=j\\}$, and it has no other
arrows horizontally entering or exiting any other row of the model. For each
$c\in\llbracket 1,n\rrbracket$, it has $\ell_{c}$ arrows of color $c$
vertically exiting the $y$-axis $\\{x=0\\}$, and it has no other arrows
horizontally entering or exiting any other column of the model. We denote by
$\mathfrak{P}_{\mathbbm{f}\mathbb{G}}(M;\sigma;\bm{\mathrm{R}})$ the set of
colored fused path ensembles on $\mathbb{Z}_{\leq 0}\times\llbracket
1,M+N\rrbracket$ with these boundary conditions, as any
$\mathcal{E}_{\bm{\mu}}\in\mathfrak{P}_{\mathbbm{f}\mathbb{G}}(M;\sigma;\bm{\mathrm{R}})$
can be thought of an ensemble from $\mathfrak{P}_{\mathbb{G}}(\mu/\emptyset)$
that is juxtaposed above one from
$\mathfrak{P}_{\mathbbm{f}}(\mu/\emptyset;\sigma;\bm{\mathrm{R}})$ (recall
Definition 6.11), for some $n$-composition
$\mu\in\operatorname{Comp}_{n}(\mathrm{R}_{[1,N]})$. It is quickly verified
that the above procedure is a bijection between
$\mathfrak{P}_{\mathbbm{f}\mathbb{G}}(M;\sigma;\bm{\mathrm{R}})$ and
$(M;\sigma;\bm{\mathrm{R}})$-ascending sequences $\bm{\mu}$ of
$n$-compositions.

See Figure 20 for a depiction, where there $(n,M,N)=(2,3,4)$,
$(\mathrm{R}_{1},\mathrm{R}_{2},\mathrm{R}_{3},\mathrm{R}_{4})=(3,1,2,2)$,
$(\ell_{1},\ell_{2})=(5,3)$,
$\big{(}\sigma(1),\sigma(2),\sigma(3),\sigma(4)\big{)}=(1,2,2,1)$, and

$\displaystyle\mu(1)=(2,2,0\mid\emptyset),\qquad\mu(2)=(2,0,0\mid
3),\qquad\mu(3)=(2,0,0\mid 4,1,0),$ $\displaystyle\mu(4)=($ $\displaystyle
5,3,0,0,0\mid 3,0,0),\quad\mu(5)=(2,1,0,0,0\mid
3,0,0),\quad\mu(6)=(1,0,0,0,0\mid 1,0,0).$

$\cdots$$-5$$-4$$-3$$-2$$-1$$\sigma(1)$$\sigma(2)$$\cdots$$\sigma(N)$ Figure
20.  Depicted above is the colored fused path ensemble associated with the
sequence $\bm{\mu}$ in the example at the end of 6.18. Here, red and blue are
colors $1$ and $2$, respectively.

Next we define the following probability measure on sequences of ascending
compositions.

######  Definition 6.19.

Fix a complex number $s\in\mathbb{C}$ and four sequences
$\bm{r}=(r_{1},r_{2},\ldots,r_{N})$; $\bm{t}=(t_{1},t_{2},\ldots,t_{M})$;
$\bm{x}=(x_{1},x_{2},\ldots,x_{N})$; and $\bm{y}=(y_{1},y_{2},\ldots,y_{M})$
of complex numbers, with $r_{j}=q^{-\mathrm{R}_{j}/2}$ for each
$j\in\llbracket 1,n\rrbracket$. Define the probability measure
$\mathbb{P}_{\mathbbm{f}\mathbb{G}}^{\sigma}=\mathbb{P}_{\mathbbm{f}\mathbb{G};n;s;\bm{x};\bm{y};\bm{r};\bm{t}}^{\sigma}$
on $(M;\sigma;\bm{\mathrm{R}})$-ascending sequences of $n$-compositions, by
setting

(6.11)
$\displaystyle\mathbb{P}_{\mathbbm{f}\mathbb{G}}^{\sigma}[\bm{\mu}]=\mathcal{Z}_{\bm{x};\bm{y};\bm{r};\bm{t}}^{-1}\cdot\displaystyle\prod_{j=1}^{N}\mathbbm{f}_{\mu(j)/\mu(j-1);s}^{\sigma(j)}(x_{j};r_{j})\displaystyle\prod_{i=N+1}^{M+N}\mathbb{G}_{\mu(i-1)/\mu(i);s}(y_{i-N};t_{i-N}).$

for each $(M;\sigma;\bm{\mathrm{R}})$-ascending sequence
$\bm{\mu}=\big{(}\mu(0),\mu(1),\ldots,\mu(M+N)\big{)}$, where

(6.12)
$\displaystyle\mathcal{Z}_{\bm{x};\bm{y};\bm{r};\bm{t}}=\displaystyle\prod_{i=1}^{M}\displaystyle\prod_{j=1}^{N}\displaystyle\frac{(t_{i}^{2}x_{j}y_{i}^{-1};q)_{\mathrm{R}_{j}}}{t_{i}^{2\mathrm{R}_{j}}(x_{j}y_{i}^{-1};q)_{\mathrm{R}_{j}}}.$

Here, we implicitly assume that $s$, $\bm{x}$, $\bm{y}$, $\bm{\mathrm{R}}$,
and $\bm{t}$ are such that the left side of (6.11) is nonnegative and (6.8)
holds. The fact that these probabilities sum to one follows from 6.20 below.

The proof of the below lemma, given 6.15 and 6.16, is entirely analogous to
that of 3.4, given 2.8 and 2.9; it is therefore omitted.

######  Lemma 6.20.

Under the notation and assumptions of Definition 6.19, we have

$\displaystyle\displaystyle\sum_{\bm{\mu}}\displaystyle\prod_{j=1}^{N}\mathbbm{f}_{\mu(j)/\mu(j-1);s}^{\sigma(j)}(x_{j};r_{j})\cdot\displaystyle\prod_{i=N+1}^{M+N}\mathbb{G}_{\mu(i-1)/\mu(i);s}(y_{i-N};t_{i-N})=\mathcal{Z}_{\bm{x};\bm{y};\bm{r};\bm{s}},$

where the sum on the left side is over all
$(M;\sigma;\bm{\mathrm{R}})$-ascending sequences of $n$-compositions
$\bm{\mu}$.

###  6.6. Matching Between Colored Stochastic Six-Vertex Models and
$\mathbb{P}_{\mathbbm{f}\mathbb{G}}^{\sigma}$

In this section we provide a matching between the law of the $(M+N)$-tuple
$\mathscr{Q}(\bm{\mu})$ of elements in $\mathbb{Z}_{\geq 0}^{n}$ (recall
Definition 6.17) associated with a sequence of compositions sampled from
$\mathbb{P}_{\mathbbm{f}\mathbb{G}}^{\sigma}$ (from Definition 6.19), with a
certain random variable associated with the colored fused vertex model (from
Section 6.1). We begin by defining the latter, which is analogous to
Definition 3.5. Throughout this section, we fix integers $M,N\geq 1$; a
complex number $s\in\mathbb{C}$; sequence of positive integers
$\bm{\mathrm{R}}=(\mathrm{R}_{1},\mathrm{R}_{2},\ldots,\mathrm{R}_{N})$;
sequences of complex numbers $\bm{r}=(r_{1},r_{2},\ldots,r_{N})$,
$\bm{t}=(t_{1},t_{2},\ldots,t_{M})$, $\bm{x}=(x_{1},x_{2},\ldots,x_{N})$, and
$\bm{y}=(y_{1},y_{2},\ldots,y_{M})$, satisfying (6.8) and
$r_{j}=q^{-\mathrm{R}_{j}/2}$ for each $j\in\llbracket 1,N\rrbracket$; and a
function $\sigma:\llbracket 1,N\rrbracket\rightarrow\llbracket 1,n\rrbracket$.

######  Definition 6.21.

Let $\mathcal{E}$ denote a colored fused path ensemble on the rectangle domain
$\mathcal{D}_{M;N}=\llbracket 1,M\rrbracket\times\llbracket 1,N\rrbracket$.
For each integer $i\in\llbracket 1,M\rrbracket$, let
$\bm{C}(i)=\bm{C}^{\mathcal{E}}(i)\in\mathbb{Z}_{\geq 0}^{n}$ be such that the
$k$-th entry $C_{k}(i)$ of $\bm{C}(i)$ denotes the number of color $k$ arrows
in $\mathcal{E}$ vertically exiting $\mathcal{D}_{M;N}$ through $(i,N)$, for
each $k\in\llbracket 1,n\rrbracket$. For each integer $j\in\llbracket
1,N\rrbracket$, let $\bm{D}(j)=\bm{D}^{\mathcal{E}}(j)\in\mathbb{Z}_{\geq
0}^{n}$ be such that the $k$-th entry $D_{k}(j)$ of $\bm{D}(j)$ denotes the
number of color $k$ arrows in $\mathcal{E}$ horizontally exiting
$\mathcal{D}_{M;N}$ through $(M,j)$. Then set
$\mathscr{C}(\mathcal{E})=\big{(}\bm{C}(1),\bm{C}(2),\ldots,\bm{C}(M)\big{)}\in(\mathbb{Z}_{\geq
0}^{n})^{M}$ and
$\mathscr{D}(\mathcal{E})=\big{(}\bm{D}(1),\bm{D}(2),\ldots,\bm{D}(N)\big{)}\in(\mathbb{Z}_{\geq
0}^{n})^{N}$.

We next require notation for the colored stochastic fused vertex model
(defined at the end of Section 6.1) with specific boundary data.

######  Definition 6.22.

We say that a colored fused path ensemble on the rectangle domain
$\mathcal{D}_{M;N}=\llbracket 1,M\rrbracket\times\llbracket 1,N\rrbracket$ has
_$(\sigma;\bm{\mathrm{R}})$ -entrance data_ if the following holds. For each
$j\in\llbracket 1,N\rrbracket$, $\mathrm{R}_{j}$ paths of color $\sigma(j)$
horizontally enters $\mathcal{D}_{M;N}$ from the site $(j,0)$ on the $y$-axis,
and no path horizontally enters $\mathcal{D}_{M;N}$ from any site on the
$x$-axis. Let
$\mathbb{P}_{\operatorname{FV}}^{\sigma}=\mathbb{P}_{\operatorname{FV};\bm{x};\bm{y};\bm{r};\bm{t}}^{\sigma}$
denote the measure on colored fused path ensembles on $\mathcal{D}_{M;N}$
obtained by running the colored stochastic fused vertex model on
$\mathcal{D}_{M;N}$ under $(\sigma;\bm{\mathrm{R}})$-boundary data, with
weight $U_{y_{i}/x_{j};r_{j},t_{i}}$ (recall Definition 6.1) at any vertex
$(i,j)\in\mathcal{D}_{M;N}$.

The following proposition now provides a matching between the $(M+N)$-tuple
$\mathscr{Q}(\bm{\mu})$ (recall Definition 6.17) sampled under
$\mathbb{P}_{\mathbbm{f}\mathbb{G}}^{\sigma}$ of Definition 6.19 and the
$(M+N)$-tuple
$\mathscr{D}(\mathcal{E})\cup\overleftarrow{\mathscr{C}}(\mathcal{E})$ sampled
under $\mathbb{P}_{\operatorname{FV}}^{\sigma}$ of Definition 6.22. We omit
its proof, which is very similar to that of 3.7 (with the few necessary
modifications already explained in the proof outline of 6.16).

######  Proposition 6.23.

Let
$\mathscr{Q}=\big{(}\bm{\mathfrak{Q}}(1),\bm{\mathfrak{Q}}(2),\ldots,\bm{\mathfrak{Q}}(M+N)\big{)}$
denote an $(M+N)$-tuple of elements in $\mathbb{Z}_{\geq 0}^{n}$, and define
the $M$-tuple
$\mathscr{C}=\big{(}\bm{\mathfrak{Q}}(M+N),\bm{\mathfrak{Q}}(M+N-1),\ldots,\bm{\mathfrak{Q}}(N+1)\big{)}$
and $N$-tuple
$\mathscr{D}=\big{(}\bm{\mathfrak{Q}}(1),\bm{\mathfrak{Q}}(2),\ldots,\bm{\mathfrak{Q}}(N)\big{)}$.
Then,

(6.13)
$\displaystyle\mathbb{P}_{\operatorname{FV}}^{\sigma}\Big{[}\\{\mathscr{C}(\mathcal{E})=\mathscr{C}\big{\\}}\cap\big{\\{}\mathscr{D}(\mathcal{E})=\mathscr{D}\big{\\}}\Big{]}=\mathbb{P}_{\mathbbm{f}\mathbb{G}}^{\sigma}\big{[}\mathscr{Q}(\bm{\mu})=\mathscr{Q}\big{]}.$

Here, on the left side of (6.13), the colored fused path ensemble
$\mathcal{E}$ is sampled under the colored stochastic fused vertex measure
$\mathbb{P}_{\operatorname{FV};\bm{x};\bm{y};\bm{r};\bm{t}}^{\sigma}$. On the
right side of (6.13), the $(M;\sigma;\bm{\mathrm{R}})$-ascending sequence
$\bm{\mu}$ of colored compositions is sampled under the measure
$\mathbb{P}_{\mathbbm{f}\mathbb{G};n;s;\bm{x};\bm{y};\bm{r};\bm{t}}^{\sigma}$.

The following corollary of 6.23 equates the joint law of the height functions
(recall Section 2.2) evaluated along the exit sites of an $M\times N$
rectangle, sampled under the colored stochastic fused vertex model, with the
joint law of the number of zero entries in a family $\bm{\mu}$ of
$n$-compositions, sampled under the
$\mathbb{P}_{\mathbbm{f}\mathbb{G}}^{\sigma}$ measure. We omit its proof,
which given 6.23 is entirely analogous to that of 3.8 given 3.7.

######  Corollary 6.24.

The joint law of all the height functions

(6.14) $\displaystyle\bigcup_{c=1}^{n}\big{(}\mathfrak{h}_{\geq
c}^{\rightarrow}(M,1),\mathfrak{h}_{\geq
c}^{\rightarrow}(M,2),\ldots,\mathfrak{h}_{\geq
c}^{\rightarrow}(M,N),\mathfrak{h}_{\geq
c}^{\rightarrow}(M-1,N),\ldots,\mathfrak{h}_{\geq
c}^{\rightarrow}(0,N)\big{)},$

is equal to the joint law of all zero-entry counts

(6.15) $\displaystyle\bigcup_{c=1}^{n}\Big{(}\mathfrak{m}_{0}^{\geq
c}\big{(}\mu(1)\big{)},\mathfrak{m}_{0}^{\geq
c}\big{(}\mu(2)\big{)},\ldots,\mathfrak{m}_{0}^{\geq
c}\big{(}\mu(N)\big{)},\mathfrak{m}_{0}^{\geq
c}\big{(}\mu(N+1)\big{)},\ldots,\mathfrak{m}_{0}^{\geq
c}\big{(}\mu(M+N)\big{)}\Big{)}.$

Here, the height functions in (6.14) are associated with a colored fused path
ensemble sampled under
$\mathbb{P}_{\operatorname{FV};\bm{x};\bm{y};\bm{r};\bm{t}}^{\sigma}$, and the
zero-entry counts in (6.15) are associated with a
$(M;\sigma;\bm{\mathrm{R}})$-ascending sequence of $n$-compositions
$\bm{\mu}=\big{(}\mu(0),\mu(1),\ldots,\mu(M+N)\big{)}$ sampled under
$\mathbb{P}_{\mathbbm{f}\mathbb{G};n;s;\bm{x};\bm{y};\bm{r};\bm{t}}^{\sigma}$.

##  7\. Colored Line Ensembles for Fused Vertex Models

This section may be viewed as the fused counterpart of Section 4, in which we
describe the colored line ensembles associated with fused stochastic vertex
models. Unlike in Section 4.1, the line ensembles we obtain will no longer be
simple, which is a manifestation of the fact that horizontal edges in the
associated stochastic fused vertex model can accommodate more than one arrow.
Outside of this difference, the content in this section will closely follow
that in Section 4. Throughout this section, we fix integers $n,M,N\geq 1$; a
complex number $s\in\mathbb{C}$; a sequence of positive integers
$\bm{\mathrm{R}}=(\mathrm{R}_{1},\mathrm{R}_{2},\ldots,\mathrm{R}_{N})$;
sequences of complex numbers $\bm{r}=(r_{1},r_{2},\ldots,r_{N})$,
$\bm{t}=(t_{1},t_{2},\ldots,t_{M})$, $\bm{x}=(x_{1},x_{2},\ldots,x_{N})$ and
$\bm{y}=(y_{1},y_{2},\ldots,y_{M})$, satisfying (6.8) and
$r_{j}=q^{-\mathrm{R}_{j}/2}$ for each $j\in\llbracket 1,N\rrbracket$; a
composition $\bm{\ell}=(\ell_{1},\ell_{2},\ldots,\ell_{n})$ of
$\mathrm{R}_{[1,N]}$; and a function $\sigma:\llbracket
1,N\rrbracket\rightarrow\llbracket 1,n\rrbracket$, such that
$\ell_{i}=\sum_{j\in\sigma^{-1}(i)}\mathrm{R}_{j}$ for each $i\in\llbracket
1,n\rrbracket$.

###  7.1. Fused Colored Line Ensembles and Ascending Sequences

In this section, given an $(M;\sigma;\bm{\mathrm{R}})$-ascending sequence
$\bm{\mu}$ of $n$-compositions, we associate a colored line ensemble (which,
unlike in Definition 4.1 and 4.2, need not be simple). This is done through
the following definition (where in the below we recall the notion of a colored
line ensemble from Definition 1.4).

######  Definition 7.1.

Let $\bm{\mu}=\big{(}\mu(0),\mu(1),\ldots,\mu(M+N)\big{)}$ denote an
$(M;\sigma;\bm{\mathrm{R}})$-ascending sequence of $n$-compositions. The
associated colored line ensemble
$\mathds{L}=\mathds{L}_{\bm{\mu}}=\big{(}\mathds{L}^{(1)},\mathds{L}^{(2)},\ldots,\mathds{L}^{(n)}\big{)}$
on $\llbracket 0,M+N\rrbracket$ is defined as follows. For each
$c\in\llbracket 1,n\rrbracket$ let
$\mathds{L}^{(c)}=\mathds{L}_{\bm{\mu}}^{(c)}=\big{(}\mathds{L}_{1}^{(c)},\mathds{L}_{2}^{(c)},\ldots\big{)}$,
where for each $k\geq 1$ the function
$\mathds{L}_{k}^{(c)}=\mathds{L}_{k;\bm{\mu}}^{(c)}:\llbracket
0,M+N\rrbracket\rightarrow\mathbb{Z}$ is prescribed by setting

(7.1) $\displaystyle\mathds{L}_{k}^{(c)}(i)=\ell_{[c,n]}-\mathfrak{m}_{\leq
k-1}^{\geq c}\big{(}\mu(i)\big{)},\qquad\text{for each $i\in\llbracket
0,M+N\rrbracket$}.$

The fact that this defines a colored line ensemble follows from 7.2 below. We
moreover set the differences
$\bm{\Lambda}^{(c)}=\big{(}\Lambda_{1}^{(c)},\Lambda_{2}^{(c)},\ldots\big{)}$
of $\mathds{L}$ by
$\Lambda_{k}^{(c)}(i)=\mathds{L}_{k}^{(c)}(i)-\mathds{L}_{k}^{(c+1)}(i)$, for
each $(c,k,i)\in\llbracket 1,n\rrbracket\times\mathbb{Z}_{>0}\times\llbracket
0,M+N\rrbracket$, where $\mathds{L}_{k}^{(n+1)}:\llbracket
0,M+N\rrbracket\rightarrow\mathbb{Z}$ is defined by setting
$\mathds{L}_{k}^{(n+1)}(i)=0$ for each
$(k,i)\in\mathbb{Z}_{>0}\times\llbracket 0,M+N\rrbracket$.

$0$$1$$2$$3$$0$$1$$2$$3$$4$$5$$6$$7$$\mathds{L}_{1}^{(2)}$$\mathds{L}_{5}^{(2)}$$0$$1$$2$$3$$4$$5$$6$$7$$0$$1$$2$$3$$4$$5$$6$$7$$8$$\mathds{L}_{1}^{(1)}$$\mathds{L}_{5}^{(1)}$
Figure 21.  Depicted above is the colored line ensemble associated with the
fused path ensemble in Figure 20.

######  Lemma 7.2.

Adopting the notation and assumptions of Definition 7.1,
$\mathds{L}_{\bm{\mu}}$ is a colored line ensemble, which satisfies the
following three properties for any $c\in\llbracket 1,n\rrbracket$,
$k\in\mathbb{Z}_{>0}$, and $i\in\llbracket 0,M+N\rrbracket$.

  1. (1)

We have $\mathds{L}_{1}^{(c)}(i)\geq\mathds{L}_{2}^{(c)}(i)\geq\cdots$ and
$\mathds{L}_{k}^{(1)}(i)\geq\mathds{L}_{k}^{(2)}(i)\geq\cdots$.

  2. (2)

We have
$\Lambda_{k}^{(c)}(i)-\Lambda_{k+1}^{(c)}(i)=\mathfrak{m}_{k}\big{(}\mu^{(c)}(i)\big{)}$.

  3. (3)

If $i\geq 1$, we have
$\Lambda_{k}^{(c)}(i-1)-\Lambda_{k}^{(c)}(i)=\mathfrak{Q}_{c}^{\bm{\mu}}(k,i)$.

###### Proof.

The proofs of the first two parts of the lemma are very similar to those of
4.2 and are therefore omitted. The third follows from the fact that

$\displaystyle\Lambda_{k}^{(c)}(i-1)-\Lambda_{k}^{(c)}(i)$
$\displaystyle=\big{(}\mathds{L}_{k}^{(c)}(i-1)-\mathds{L}_{k}^{(c)}(i)\big{)}-\big{(}\mathds{L}_{k}^{(c+1)}(i-1)-\mathds{L}_{k}^{(c+1)}(i)\big{)}$
$\displaystyle=\mathfrak{m}_{\leq k-1}^{\geq
c}\big{(}\mu(i-1)\big{)}-\mathfrak{m}_{\leq k-1}^{\geq
c+1}\big{(}\mu(i)\big{)}$
$\displaystyle=\mathfrak{Q}_{[c,n]}^{\bm{\mu}}(k,i)-\mathfrak{Q}_{[c+1,n]}^{\bm{\mu}}(k,i)=\mathfrak{Q}_{c}^{\bm{\mu}}(k,i)\geq
0,$

where in the first statement we used the definition of $\bm{\Lambda}^{(c)}$
from Definition 7.1; in the second we used (7.1); and in the third we used
(6.10). The second and third statements of the lemma together imply that each
$\bm{\Lambda}^{(c)}$ is a line ensemble, and thus $\mathds{L}_{\bm{\mu}}$ is a
colored line ensemble by Definition 1.4. ∎

######  Remark 7.3.

As in 6.18, we may interpret $\bm{\mu}$ as associated with a colored fused
path ensemble
$\mathcal{E}_{\bm{\mu}}\in\mathfrak{P}_{\mathbbm{f}\mathbb{G}}(M;\sigma;\bm{\mathrm{R}})$
on $\mathbb{Z}_{\leq 0}\times\llbracket 1,M+N\rrbracket$. Then
$\mathds{L}_{k}^{(c)}(i)=\mathfrak{h}_{\geq c}^{\leftarrow}(-k,i)$, where the
height function $\mathfrak{h}_{\geq c}^{\leftarrow}$ is with respect to
$\mathcal{E}_{\bm{\mu}}$; stated alternatively, $\mathds{L}_{k}^{(c)}(i)$
denotes the number of arrows with color at least $c$ that horizontally exit
the column $\\{x=-k\\}$ strictly above the vertex $(-k,i)$. See Figure 21 for
a depiction of the colored line ensemble in the example at the end of 6.18
(and shown in Figure 20).

By 7.2, Definition 7.1 associates a colored line ensemble to a given
$(M;\sigma;\bm{\mathrm{R}})$-ascending sequence of $n$-compositions. Since the
former are in bijection with colored higher spin path ensembles in
$\mathfrak{P}_{\mathbbm{f}\mathbb{G}}(M;\sigma;\bm{\mathrm{R}})$ by 6.18, this
associates a colored line ensemble with any element of
$\mathfrak{P}_{\mathbbm{f}\mathbb{G}}(M;\sigma;\bm{\mathrm{R}})$. The
following definition is towards the reverse direction; it associates a colored
fused path ensemble with a colored line ensemble $\mathds{L}$.

######  Definition 7.4.

Fix a colored line ensemble
$\mathds{L}=\big{(}\mathds{L}^{(1)},\mathds{L}^{(2)},\ldots,\mathds{L}^{(n)}\big{)}$,
and for each $c\in\llbracket 1,n\rrbracket$ denote
$\mathds{L}^{(c)}=\big{(}\mathds{L}_{1}^{(c)},\mathds{L}_{2}^{(c)},\ldots\big{)}$.
For any $v=(-k,i)\in\mathbb{Z}_{\leq 0}\times\llbracket 1,M+N\rrbracket$,
define the arrow configuration
$\big{(}\bm{A}^{\mathds{L}}(v),\bm{B}^{\mathds{L}}(v);\bm{C}^{\mathds{L}}(v),\bm{D}^{\mathds{L}}(v)\big{)}$
as follows. For each $c\in\llbracket 1,n\rrbracket$ and set

$\displaystyle
A_{c}^{\mathds{L}}(v)=\Lambda_{k}^{(c)}(i-1)-\Lambda_{k+1}^{(c)}(i-1)=\mathds{L}_{k}^{(c)}(i-1)-\mathds{L}_{k+1}^{(c)}(i-1)-\big{(}\mathds{L}_{k}^{(c+1)}(i-1)-\mathds{L}_{k+1}^{(c+1)}(i-1)\big{)};$
$\displaystyle
B_{c}^{\mathds{L}}(v)=\Lambda_{k+1}^{(c)}(i-1)-\Lambda_{k+1}^{(c)}(i)=\mathds{L}_{k+1}^{(c)}(i-1)-\mathds{L}_{k+1}^{(c)}(i)-\big{(}\mathds{L}_{k+1}^{(c+1)}(i-1)-\mathds{L}_{k+1}^{(c+1)}(i)\big{)};$
$\displaystyle
C_{c}^{\mathds{L}}(v)=\Lambda_{k}^{(c)}(i)-\Lambda_{k+1}^{(c)}(i)=\mathds{L}_{k}^{(c)}(i)-\mathds{L}_{k+1}^{(c)}(i)-\big{(}\mathds{L}_{k}^{(c+1)}(i)-\mathds{L}_{k+1}^{(c+1)}(i)\big{)};$
$\displaystyle
D_{c}^{\mathds{L}}(v)=\Lambda_{k}^{(c)}(i-1)-\Lambda_{k}^{(c)}(i)=\mathds{L}_{k}^{(c)}(i-1)-\mathds{L}_{k}^{(c)}(i)-\big{(}\mathds{L}_{k}^{(c+1)}(i-1)-\mathds{L}_{k}^{(c+1)}(i)\big{)},$

where we observe that all four quantities are nonnegative since
$\bm{\Lambda}^{(c)}$ is a line ensemble. This assignment of arrow
configurations is consistent and therefore defines a colored fused path
ensemble $\mathcal{E}^{\mathds{L}}$ associated with the colored line ensemble
$\mathds{L}$.

The following lemma indicates that the assocations from Definition 7.1 and
Definition 7.4 are compatible; we omit its proof, which is a quick
verification using the second and third properties of 7.2.

######  Lemma 7.5.

If $\mathcal{E}^{\mathds{L}}=\mathcal{E}_{\bm{\mu}}$ for some
$(M;\sigma;\bm{\mathrm{R}})$-ascending sequence $\bm{\mu}$ of
$n$-compositions, then $\mathds{L}$ is associated with $\bm{\mu}$ in the sense
of Definition 7.1.

###  7.2. Properties of Random Fused Colored Line Ensembles

In this section we discuss some properties of colored line ensembles
$\mathds{L}_{\bm{\mu}}$ associated (recall Definition 7.1) with an
$(M;\sigma;\bm{\mathrm{R}})$-ascending sequence $\bm{\mu}$ of $n$-compositions
sampled from the measure $\mathbb{P}_{\mathbbm{f}\mathbb{G}}^{\sigma}$ (recall
Definition 6.19). Let us first give notation to this law on colored line
ensembles.

######  Definition 7.6.

Let
$\mathbb{P}_{\operatorname{cL}}^{\sigma}=\mathbb{P}_{\operatorname{cL};n;s;\bm{x};\bm{y};\bm{r};\bm{t}}^{\sigma}$
denote the law of a colored line ensemble $\mathds{L}_{\bm{\mu}}$ associated
with a random $(M;\sigma;\bm{\mathrm{R}})$-ascending sequence $\bm{\mu}$ of
$n$-compositions (as in Definition 7.1) sampled from the measure
$\mathbb{P}_{\mathbbm{f}\mathbb{G};n;s;\bm{x};\bm{y};\bm{r};\bm{t}}^{\sigma}$.

The following result, which is a quick consequence of 6.24, provides under
this setup a matching in law between the top curves of $\mathds{L}$ (under
$\mathbb{P}_{\operatorname{cL}}^{\sigma}$) and the height functions for a
colored stochastic fused vertex model (recall Definition 6.22).

######  Theorem 7.7.

Sample a colored line ensemble $\mathds{L}$ on $\llbracket 0,M+N\rrbracket$
from the measure
$\mathbb{P}_{\operatorname{cL};n;s;\bm{x};\bm{y};\bm{r};\bm{t}}^{\sigma}$, and
sample a random colored fused path ensemble $\mathcal{E}$ under
$\mathbb{P}_{\operatorname{FV};\bm{x};\bm{y};\bm{r};\bm{t}}^{\sigma}$. For
each $c\in\llbracket 1,n\rrbracket$, define the function $H_{c}:\llbracket
0,M+N\rrbracket\rightarrow\mathbb{Z}$ by setting

$\displaystyle H_{c}(k)=\mathfrak{h}_{\geq c}^{\leftarrow}(M,k),\quad\text{if
$k\in\llbracket 0,N\rrbracket$};\qquad H_{c}(k)=\mathfrak{h}_{\geq
c}^{\leftarrow}(M+N-k,N),\quad\text{if $k\in\llbracket N,M+N\rrbracket$},$

where $\mathfrak{h}_{\geq c}^{\leftarrow}$ is the height function associated
with $\mathcal{E}$. Then, the joint law of
$\big{(}\mathds{L}_{1}^{(1)},\mathds{L}_{1}^{(2)},\ldots,\mathds{L}_{1}^{(n)}\big{)}$
is the same as that of $(H_{1},H_{2},\ldots,H_{n})$.

###### Proof.

Since $\mathfrak{h}_{\geq c}^{\leftarrow}(i,j)=\ell_{[c,n]}-\mathfrak{h}_{\geq
c}^{\rightarrow}(i,j)$ holds for any integer $c\in\llbracket 1,n\rrbracket$
and vertex
$(i,j)\in\big{\\{}(M,0),(M,1),\ldots,(M,N),(M,N-1),\ldots,(0,N)\big{\\}}$
along the northeast boundary of $\llbracket 0,M\rrbracket\times\llbracket
0,N\rrbracket$, this theorem follows from Definition 7.1 and 6.24. ∎

The next theorem explains the effect of conditioning on some of the curves in
$\mathds{L}$ (the Gibbs property), if $\bm{\mu}$ is sampled under the
$\mathbb{P}_{\operatorname{cL}}^{\sigma}$ measure, which is given by the
following theorem; its proof is omitted, as it is very similar to that of 4.8.
Below, we recall the vertex weights $W_{x;r,s}$ from Definition 6.3; the
association of a colored line ensemble with an ascending sequence of
$n$-compositions from Definition 7.1; and the notation from Definition 7.4.

######  Theorem 7.8.

Sample $\mathds{L}=\mathds{L}_{\bm{\mu}}$ under
$\mathbb{P}_{\operatorname{cL};n;s;\bm{x};\bm{y};\bm{r};\bm{t}}^{\sigma}$. Fix
integers $j>i\geq 0$ and $u,v\in\llbracket 0,M+N\rrbracket$ with $u<v$; set
$i_{0}=\max\\{i,1\\}$; and condition on the curves $\mathds{L}_{k}^{(c)}(m)$
for all $c\in\llbracket 1,n\rrbracket$ and
$(k,m)\in\big{(}\mathbb{Z}_{>0}\times\llbracket
0,M+N\rrbracket\big{)}\setminus\big{(}\llbracket
i+1,j\rrbracket\times\llbracket u,v-1\rrbracket\big{)}$. For any colored line
ensemble $\bm{\bm{\mathsf{l}}}$ that is $\llbracket
i+1,j\rrbracket\times\llbracket u,v-1\rrbracket$-compatible with $\mathds{L}$,
we have

(7.2) $\displaystyle\begin{aligned}
\mathbb{P}[\mathds{L}=\bm{\bm{\mathsf{l}}}]&=\mathcal{Z}^{-1}\cdot\displaystyle\prod_{k=i_{0}}^{j}\displaystyle\prod_{\begin{subarray}{c}m\in\llbracket
u,v\rrbracket\\\ m\leq
N\end{subarray}}W_{x_{m};r_{m},s}\big{(}\bm{A}^{\bm{\bm{\mathsf{l}}}}(-k,m),\bm{B}^{\bm{\bm{\mathsf{l}}}}(-k,m);\bm{C}^{\bm{\bm{\mathsf{l}}}}(-k,m),\bm{D}^{\bm{\bm{\mathsf{l}}}}(-k,m)\big{)}\\\
&\qquad\times\displaystyle\prod_{k=i_{0}}^{j}\displaystyle\prod_{\begin{subarray}{c}m\in\llbracket
u,v\rrbracket\\\
m>N\end{subarray}}W_{y_{m-N};t_{m-N},s}\big{(}\bm{A}^{\bm{\bm{\mathsf{l}}}}(-k,m),\bm{B}^{\bm{\bm{\mathsf{l}}}}(-k,m);\bm{C}^{\bm{\bm{\mathsf{l}}}}(-k,m),\bm{D}^{\bm{\bm{\mathsf{l}}}}(-k,m)\big{)}.\end{aligned}$

Here, the probability on the left side of (7.2) is with respect to the
conditional law of $\mathds{L}$. Moreover, $\mathcal{Z}$ is a normalizing
constant defined so that the sum of the right side of (7.2), over all colored
line ensembles $\bm{\bm{\mathsf{l}}}$ that are $\llbracket
i+1,j\rrbracket\times\llbracket u,v-1\rrbracket$-compatible with
$\mathds{L}_{\bm{\mu}}$, is equal to $1$.

Let us also describe color merging properties for line ensembles sampled
according to $\mathbb{P}_{\operatorname{cL}}^{\sigma}$, which will be parallel
to those discussed in Section 4.3. To that end, we have the following
proposition generalizing 4.11, where below we recall the definition of
$\vartheta$ from (4.10) (as a function on $\operatorname{Comp}_{n}$), (4.11)
(as a functional on the set of functions $\varsigma:\llbracket
1,N\rrbracket\rightarrow\llbracket 1,n\rrbracket$), and (4.12) (as a function
on sequences of $n$-compositions); we also recall that
$\breve{\sigma}=\vartheta(\sigma)$ from (4.11).

######  Proposition 7.9.

Sample
$\mathds{L}=\big{(}\mathds{L}^{(1)},\mathds{L}^{(2)},\ldots,\mathds{L}^{(n)}\big{)}$
under
$\mathbb{P}_{\operatorname{cL};n;s;\bm{x};\bm{y};\bm{r};\bm{t}}^{\sigma}$.
Then the joint law of the colored line ensemble
$\breve{\mathds{L}}=\big{(}\mathds{L}^{(1)},\mathds{L}^{(3)},\mathds{L}^{(4)},\ldots,\mathds{L}^{(n)}\big{)}$
(with $n-1$ colors) is given by
$\mathbb{P}_{\operatorname{cL};n-1;s;\bm{x};\bm{y};\bm{r};\bm{t}}^{\breve{\sigma}}$.

To establish 7.9, we require the following lemma that is parallel to 4.9.

######  Lemma 7.10.

Fix an integer $k\geq 1$; $(n-1)$-compositions
$\breve{\mu},\breve{\nu},\breve{\kappa}\in\operatorname{Comp}_{n-1}$; and
$n$-compositions $\kappa,\mu\in\operatorname{Comp}_{n}$, such that
$\vartheta(\kappa)=\breve{\kappa}$ and $\vartheta(\mu)=\breve{\mu}$. We have

(7.3) $\displaystyle\displaystyle\sum_{\begin{subarray}{c}\varsigma:\llbracket
1,N\rrbracket\rightarrow\llbracket 1,n\rrbracket\\\
\vartheta(\varsigma)=\breve{\sigma}\end{subarray}}\displaystyle\sum_{\begin{subarray}{c}\nu\in\operatorname{Comp}_{n}\\\
\vartheta(\nu)=\breve{\nu}\end{subarray}}\mathbbm{f}_{\mu/\nu;s}^{\varsigma}(\bm{x};\bm{r})=\mathbbm{f}_{\breve{\mu}/\breve{\nu};s}^{\breve{\sigma}}(\bm{x};\bm{r});\qquad\displaystyle\sum_{\begin{subarray}{c}\nu\in\operatorname{Comp}_{n}\\\
\vartheta(\nu)=\breve{\nu}\end{subarray}}\mathbb{G}_{\nu/\kappa;s}(\bm{y};\bm{t})=\mathbb{G}_{\breve{\nu}/\breve{\kappa};s}(\bm{y};\bm{t}).$

###### Proof.

Combining 4.9 with 6.13 yields the first statement of (7.3), as well as the
second one if $t_{i}^{2}\in q^{\mathbb{Z}_{<0}}$ for each $i\in\llbracket
1,M\rrbracket$. The fact that the second statement in (7.3) holds for
arbitrary $\bm{t}$ then follows from uniqueness of analytic continuation, as
$\mathbb{G}$ is rational in $\bm{t}$ (since the $W$-weights of Definition 6.3
are). ∎

The next lemma is parallel to 4.10. Its proof given 7.10 is entirely analogous
to that of 4.10 given 4.9 and is therefore omitted.

######  Lemma 7.11.

If an $(M;\sigma;\bm{\mathrm{R}})$-ascending sequence of $n$-compositions
$\bm{\mu}$ is sampled from
$\mathbb{P}_{\mathbbm{f}\mathbb{G};n;s;\bm{x};\bm{y}}^{\sigma}$, then the
$(M;\breve{\sigma})$-ascending sequence $\vartheta(\bm{\mu})$ of
$(n-1)$-compositions has law
$\mathbb{P}_{\mathbbm{f}\mathbb{G};n-1;s;\bm{x};\bm{y}}^{\breve{\sigma}}$.

Now we can establish 7.9.

###### Proof of 7.9.

Sample $\bm{\mu}$ under
$\mathbb{P}_{\mathbbm{f}\mathbb{G};n;s;\bm{x};\bm{y};\bm{r};\bm{t}}^{\sigma}$
(recall Definition 6.19). By Definition 7.6, we may identify $\mathds{L}$ as
the colored line ensemble $\mathds{L}_{\bm{\mu}}$ associated with $\bm{\mu}$.
By Definition 7.1, the $n-1$ line ensembles
$\breve{\mathds{L}}=\big{(}\mathds{L}^{(1)},\mathds{L}^{(3)},\ldots,\mathds{L}^{(n)}\big{)}$
in $\mathds{L}=\mathds{L}_{\bm{\mu}}$ constitute the colored line ensemble
associated with $\vartheta(\bm{\mu})$, which has law
$\mathbb{P}_{\mathbbm{f}\mathbb{G};n-1;s;\bm{x};\bm{y};\bm{r};\bm{t}}^{\breve{\sigma}}$
by 7.11. Hence, again by Definition 7.6, $\breve{\mathds{L}}$ has law
$\mathbb{P}_{\operatorname{cL};n-1;s;\bm{x};\bm{y};\bm{r};\bm{t}}^{\breve{\sigma}}$,
thereby establishing the proposition. ∎

######  Remark 7.12.

The above discussion describes the merging of colors $1$ and $2$. As in 4.12,
it is more generally possible to merge several (disjoint) intervals of colors,
which would correspond in 7.9 to omitting line ensembles $\mathds{L}^{(i)}$,
for $i\in\llbracket 2,n\rrbracket$ arbitrary (depending on the corresponding
merged color intervals), in $\mathds{L}$.

##  8\. Line Ensembles for Discrete Time $q$-Boson Models

In this section we specialize the results of Section 7 to the discrete time
$q$-boson model, which involves setting each $\mathrm{R}_{j}$ there equal to
$1$. Throughout this section, we adopt the notation of Section 7 (where here
we do not necessarily assume that (6.8) holds), and set $\mathrm{R}_{j}=1$ for
all $j\in\llbracket 1,n\rrbracket$, so that each $r_{j}=q^{-1/2}$.

###  8.1. Colored Stochastic Higher Spin Vertex and $q$-Boson Models

In this section we describe the $\mathrm{R}=1$ case of the stochastic fused
vertex models from Section 6.1. Recall that we have set each
$\mathrm{R}_{j}=1$, and thus each $r_{j}=q^{-1/2}$.

Under this specialization, it is quickly verified that the
$U_{z;q^{-1/2},s}$-weights (recall Definition 6.1) permit at most one arrow
along any horizontal edge, that is,
$U_{z;q^{-1/2},s}(\bm{A},\bm{B};\bm{C},\bm{D})=0$ unless there exist indices
$b,d\in\llbracket 0,n\rrbracket$ such that $\bm{B}=\bm{e}_{b}$ and
$\bm{D}=\bm{e}_{d}$. Therefore, the associated colored stochastic fused vertex
model (recall Section 6.1) is called the _colored stochastic higher spin
vertex model_ ; it originally appeared in [63, Appendix A]. For any
$\bm{A},\bm{C}\in\mathbb{Z}_{\geq 0}^{n}$ and $b,d\in\llbracket
0,n\rrbracket$, we denote these specialized stochastic weights by

$\displaystyle
U_{z;s}^{\operatorname{hs}}(\bm{A},b;\bm{C},d)=U_{z;q^{-1/2},s}(\bm{A},\bm{e}_{b};\bm{C},\bm{e}_{d}).$

These weights are depicted in the second to last row of Figure 22. We also
denote the associated probability measure on path ensembles (recall Definition
6.22) by
$\mathbb{P}_{\operatorname{hs};\bm{x};\bm{y};\bm{t}}^{\sigma}=\mathbb{P}_{\operatorname{hs};\bm{x};\bm{y};q^{-1/2};\bm{t}}^{\sigma}$.

$i$$j$$i$$i$$i$$j$$i$$i$$\bm{A}$$\bm{A}_{i}^{-}$$\bm{A}_{i}^{+}$$\bm{A}_{ij}^{+-}$$\bm{A}_{ji}^{+-}$$\bm{A}$$\bm{A}$$\bm{A}$$\bm{A}$$\bm{A}$$\bm{A}$$\bm{A}$$1\leq
i\leq n$$1\leq i<j\leq
n$$(\bm{A},0;\bm{A},0)$$\big{(}\bm{A},0;\bm{A}_{i}^{-},i\big{)}$$\big{(}\bm{A},i;\bm{A}_{i}^{+},0\big{)}$$\big{(}\bm{A},i;\bm{A}_{ij}^{+-},j\big{)}$$\big{(}\bm{A},j;\bm{A}_{ji}^{+-},i\big{)}$$(\bm{A},i;\bm{A},i)$$\displaystyle\frac{1+\nu
q^{A_{[1,n]}}}{1+\nu}$$\displaystyle\frac{\nu(1-q^{A_{i}})q^{A_{[i+1,n]}}}{1+\nu}$$\displaystyle\frac{1+\nu
q^{A_{[1,n]}}}{1+\nu}$$\displaystyle\frac{\nu(1-q^{A_{j}})q^{A_{[j+1,n]}}}{1+\nu}$$\displaystyle\frac{\nu(1-q^{A_{i}})q^{A_{[i+1,n]}}}{1+\nu}$$\displaystyle\frac{\nu(1-q^{A_{i}})q^{A_{[i+1,n]}}}{1+\nu}$$\displaystyle\frac{1-s^{2}xq^{A_{[1,n]}}}{1-s^{2}x}$$\displaystyle\frac{s^{2}x(q^{A_{i}}-1)q^{A_{[i+1,n]}}}{1-s^{2}x}$$\displaystyle\frac{1-s^{2}q^{A_{[1,n]}}}{1-s^{2}x}$$\displaystyle\frac{s^{2}x(q^{A_{j}}-1)q^{A_{[j+1,n]}}}{1-s^{2}x}$$\displaystyle\frac{s^{2}(q^{A_{i}}-1)q^{A_{[i+1,n]}}}{1-s^{2}x}$$\displaystyle\frac{s^{2}(q^{A_{i}}-x)q^{A_{[i+1,n]}}}{1-s^{2}x}$
Figure 22.  Depicted in the second to last row are the weights for the colored
stochastic higher spin vertex model, and depicted in the last row are those
for the discrete time $q$-boson model.

Given an $M$-tuple of positive parameters
$\bm{\nu}=(\nu_{1},\nu_{2},\ldots,\nu_{M})$, the (Bernoulli) _discrete time
$q$-boson model_ is the special case of the colored stochastic higher spin
vertex model when

(8.1) $\displaystyle
x_{j}y_{i}^{-1}=1,\quad\text{and}\quad\nu_{i}=-t_{i}^{2},\qquad\text{for each
$(i,j)\in\llbracket 1,M\rrbracket\times\llbracket 1,N\rrbracket$}.$

We denote the associated stochastic weights

$\displaystyle
U_{\nu}^{\operatorname{dqb}}(\bm{A},b;\bm{C},d)=U_{1;\sqrt{-\nu}}^{\operatorname{hs}}(\bm{A},b;\bm{C},d),$

which satisfy

$\displaystyle
U_{\nu}^{\operatorname{dqb}}(\bm{A},i;\bm{A}_{i}^{+},0)=\displaystyle\frac{1+\nu
q^{A_{[1,n]}}}{1+\nu},\quad\text{and}\quad
U_{\nu}^{\operatorname{dqb}}(\bm{A},i;\bm{A}_{ij}^{+-},j)=\displaystyle\frac{\nu}{1+\nu}\cdot(1-q^{A_{i}})q^{A_{[i+1,n]}},$

for any $i,j\in\llbracket 0,n\rrbracket$ and $\bm{A}\in\mathbb{Z}_{\geq
0}^{n}$; see the last row of Figure 22. The colorless ($n=1$) case of this
model was introduced in [17, Definition 1.6]. We denote the associated
probability measure by $\mathbb{P}_{\operatorname{dqb};\bm{\nu}}^{\sigma}$
(which is the specialization of
$\mathbb{P}_{\operatorname{hs};\bm{x};\bm{y};\bm{t}}^{\sigma}$ under (8.1)).

######  Remark 8.1.

Fix parameters $\bm{\alpha}=(\alpha_{1},\alpha_{2},\ldots)$. Suppose we
consider the discrete time $q$-boson model, set
$\nu_{j}=\varepsilon\alpha_{j}$ for each $j\in\llbracket 1,M\rrbracket$, scale
$N$ by $\varepsilon^{-1}$, let $M$ tend to $\infty$, and let $\varepsilon$
tend to $0$. This gives rise to a continuous-time Markov process on
$\mathbb{Z}_{>0}$, in which a particle of color $c\in\llbracket 1,n\rrbracket$
at site $k\in\mathbb{Z}_{>0}$ jumps to the right according to an exponential
clock of rate $\alpha_{k}q^{A_{[i+1,n]}(k)}(1-q^{A_{i}(k)})$, where $A_{j}(k)$
denotes the number of particles of color $j$ at site $k$; see [25, Proposition
12.4.1]. This model is called the _colored $q$-boson model_ or the _colored
$q$-deformed totally asymmetric zero range process_ (TAZRP). The colorless
($n=1$) case of this model was introduced in [80, Equation (2.6)], which under
a change of variables is equivalent to the $q$-TASEP introduced in [16,
Definition 3.3.7].

###  8.2. $q$-Hahn Weights

We eventually seek to degenerate 7.8 to the ($\mathrm{R}=1$) colored
stochastic higher spin vertex model case. Recall that this result involves the
$W$-weights from (6.2) with an arbitrary choice of the parameter $s$, which
are in general a bit intricate. So, in this section we discuss a specific
choice for $s$ that simplifies these weights considerably.151515This is a
colored generalization of what was done in [27, Section 7.3], where a similar
idea was implemented to match the height function of the uncolored stochastic
higher spin vertex model with the length of a partition sampled under the spin
Hall-Littlewood / spin Whittaker measure. This corresponds to setting $s=x$, a
special case that has appeared numerous times in the prior literature [14, 24,
63] and is sometimes known as the $q$-Hann specialization; it is given as the
below lemma.

######  Lemma 8.2.

Adopting the notation of Definition 6.3, we have for any $s,t\in\mathbb{C}$
that

$\displaystyle W_{s;t,s}(\bm{A},\bm{B};\bm{C},\bm{D})$
$\displaystyle=(-st^{-2})^{d}q^{\varphi(\bm{D},\bm{A}-\bm{D})}\cdot\displaystyle\frac{(t^{-2}s^{2};q)_{a-d}(t^{2};q)_{d}}{(s^{2};q)_{a}}\cdot\displaystyle\prod_{i=1}^{n}\displaystyle\frac{(q;q)_{A_{i}}}{(q;q)_{A_{i}-D_{i}}(q;q)_{D_{i}}}$
$\displaystyle\qquad\times\mathbbm{1}_{\bm{A}+\bm{B}=\bm{C}+\bm{D}}\cdot\mathbbm{1}_{\bm{A}\geq\bm{D}}.$

###### Proof.

We assume throughout this proof that $\bm{A}+\bm{B}=\bm{C}+\bm{D}$, as
otherwise $W_{s;t,s}(\bm{A},\bm{B};\bm{C},\bm{D})=0$ by Definition 6.3. Due to
the factor of $(s^{-1}x;q)_{b-p}$ in (6.2), any nonzero summand on the right
side of (6.2) must at $x=s$ satisfy $p=b$. Together with the fact that
$\bm{P}\leq\bm{B}$ in this sum, it follows that the only nonzero summand is
given by $\bm{P}=\bm{B}$; in particular (since $\bm{P}\leq\bm{C}$), we must
have $\bm{B}\leq\bm{C}$ for $W_{s;t,s}(\bm{A},\bm{B};\bm{C},\bm{D})$ to be
nonzero. We assume this in what follows, meaning (as
$\bm{A}+\bm{B}=\bm{C}+\bm{D}$) that $\bm{A}\geq\bm{D}$. Then, (6.2) yields

$\displaystyle W_{s;t,s}(\bm{A},\bm{B};\bm{C},\bm{D})$
$\displaystyle=(-1)^{d}s^{d}t^{2(c-a)}q^{\varphi(\bm{D},\bm{C})}\displaystyle\frac{(t^{2};q)_{d}}{(t^{2};q)_{b}}\cdot
t^{-2b}\displaystyle\frac{(t^{-2}s^{2};q)_{c-b}(t^{2};q)_{b}}{(s^{2};q)_{c+d-b}}$
$\displaystyle\qquad\times
q^{-\varphi(\bm{D},\bm{B})}\displaystyle\prod_{i=1}^{n}\displaystyle\frac{(q;q)_{C_{i}+D_{i}-B_{i}}}{(q;q)_{D_{i}}(q;q)_{C_{i}-B_{i}}}.$

This, together with the fact that $\bm{A}+\bm{B}=\bm{C}+\bm{D}$ (and hence
$a+b=c+d$) yields the lemma. ∎

######  Remark 8.3.

Observe for any $\mu,\kappa\in\operatorname{Comp}_{n}$ that, if $y_{i}=s$ for
each $\in\llbracket 1,n\rrbracket$, then
$\mathbb{G}_{\mu/\kappa;s}(\bm{y};\bm{t})$ (from Definition 6.11) is nonzero
only if $\max\mu-\max\kappa\leq k$. Indeed, if $\max\mu\geq\max\kappa+k+1$,
then any fused path ensemble
$\mathcal{E}\in\mathfrak{P}_{\mathbb{G}}(\mu/\kappa)$ must have at least one
arrow configuration $(\bm{A},\bm{B};\bm{C},\bm{D})$ that does not satisfy
$\bm{B}\leq\bm{C}$ (equivalently, that does not satisfy $\bm{D}\geq\bm{A}$).
Hence, 8.2 implies that $W_{\bm{y};\bm{t},s}(\mathcal{E})=0$ for each
$\mathcal{E}\in\mathfrak{P}_{\mathbb{G}}(\mu/\kappa)$, meaning by (6.7) that
$\mathbb{G}_{\mu/\kappa;s}(\bm{y};\bm{t})=0$.

######  Remark 8.4.

By 8.3, if $y_{i}=s$ for each $i\in\llbracket 1,M\rrbracket$, then the
probability measure $P_{\operatorname{hs};\bm{x};\bm{y};\bm{t}}^{\sigma}$ from
Section 8.1 (see also (6.11)) is supported on
$(M;\sigma;\bm{\mathrm{R}})$-ascending sequences of $n$-compositions
$\bm{\mu}=\big{(}\mu(0),\mu(1),\ldots,\mu(M+N)\big{)}$ satisfying
$\max\mu(N)\leq M$. It follows that this measure is supported on only finitely
many such sequences $\bm{\mu}$, and so by analytic continuation (6.11) defines
a probability measure for any161616It can be verified that the quantity
$\prod_{j=1}^{N}\mathbbm{f}_{\mu(j)/\mu(j-1);s}^{\sigma(j)}(x_{j};q^{-1/2})\cdot\prod_{i=1}^{M}\prod_{j=1}^{N}(1-x_{j}s^{-1})$
appearing in the $\mathrm{R}_{j}=1$ case of (6.11) is a polynomial in $\bm{x}$
whenever $\max\mu(N)\leq M$ (using the fact that
$(1-xs^{-1})\cdot\widehat{W}_{x;q^{-1/2},s}$ is a polynomial in $x$), and so
this measure has no singularities in $\bm{x}$. choice of the parameters
$\bm{x}$ and $\bm{t}$, even if (6.8) is not satisfied. In particular, the
results from Section 6.6 and Section 7 continue to hold when all $y_{i}=s$,
without assuming (6.8).

###  8.3. Colored Line Ensembles

In this section we establish the following proposition describing a colored
line ensemble for the discrete time $q$-Boson model (from Section
8.1).171717Proposition 8.5 below examines this line ensemble on the interval
$\llbracket N,M+N\rrbracket$, which corresponds to the behavior of the
discrete time $q$-boson model along the north boundary of the rectangle
$\llbracket 1,M\rrbracket\times\llbracket 1,N\rrbracket$. One can also
formulate more general statements about this line ensemble on its full domain
$\llbracket 0,M+N\rrbracket$, but we will not do so here. In the below, we
recall the notation from Definition 7.4, and we define the vertex weights (see
8.2)

(8.2)
$\displaystyle\mathcal{W}_{\nu}(\bm{A},\bm{B};\bm{C},\bm{D})=q^{\varphi(\bm{D},\bm{A}-\bm{D})}\nu^{-d}(-\nu;q)_{d}\cdot\displaystyle\prod_{i=1}^{n}\displaystyle\frac{(q;q)_{A_{i}}}{(q;q)_{A_{i}-D_{i}}(q;q)_{D_{i}}}\cdot\mathbbm{1}_{\bm{A}+\bm{B}=\bm{C}+\bm{D}}\cdot\mathbbm{1}_{\bm{A}\geq\bm{D}},$

for any $\nu\in\mathbb{C}$ and $\bm{A},\bm{B},\bm{C},\bm{D}\in\mathbb{Z}_{\geq
0}^{n}$, where we have set $d=|\bm{D}|$.

######  Proposition 8.5.

Fix $\bm{\nu}=(\nu_{1},\nu_{2},\ldots,\nu_{M})\in\mathbb{R}_{>0}^{M}$; sample
a colored path ensemble $\mathcal{E}$ on $\llbracket
1,M\rrbracket\times\llbracket 1,N\rrbracket$ according to the discrete time
$q$-boson model $\mathbb{P}_{\operatorname{dqb};\bm{\nu}}^{\sigma}$; and for
each $c\in\llbracket 1,n\rrbracket$ define $H_{c}:\llbracket
N,M+N\rrbracket\rightarrow\mathbb{Z}$ by setting $H_{c}(k)=\mathfrak{h}_{\geq
c}^{\leftarrow}(M+N-k,N)$ for each $k\in\llbracket N,M+N\rrbracket$, where
$\mathfrak{h}_{\geq c}^{\leftarrow}$ is the height function with respect to
$\mathcal{E}$. There exists a random colored line ensemble
$\mathds{L}=\big{(}\mathds{L}^{(1)},\mathds{L}^{(2)},\ldots,\mathds{L}^{(n)}\big{)}$
on $\llbracket 0,M+N\rrbracket$ satisfying the following properties.

  1. (1)

The joint law of the functions $\big{(}\mathds{L}_{1}^{(1)}|_{\llbracket
N,M+N\rrbracket},\mathds{L}_{1}^{(2)}|_{\llbracket
N,M+N\rrbracket},\ldots,\mathds{L}_{1}^{(n)}|_{\llbracket
N,M+N\rrbracket}\big{)}$ is the same as that of $(H_{1},H_{2},\ldots,H_{n})$.

  2. (2)

We almost surely have $\bm{A}^{\mathds{L}}(-k,i)\geq\bm{D}^{\mathds{L}}(-k,i)$
for each $(-k,i)\in\mathbb{Z}_{\leq 0}\times\llbracket N,M+N\rrbracket$.

  3. (3)

Fix integers $j>i\geq 0$ and $u,v\in\llbracket N+1,M+N\rrbracket$ with $u<v$;
set $i_{0}=\max\\{i,1\\}$; and condition on the curves
$\mathds{L}_{k}^{(c)}(m)$ for all $c\in\llbracket 1,n\rrbracket$ and
$(k,m)\notin\llbracket i+1,j\rrbracket\times\llbracket u,v-1\rrbracket$. For
any colored line ensemble $\bm{\mathsf{l}}$ that is $\llbracket
i+1,j\rrbracket\times\llbracket u,v-1\rrbracket$-compatible with $\mathds{L}$,
we have

(8.3)
$\displaystyle\mathbb{P}[\mathds{L}=\bm{\mathsf{l}}]=\mathcal{Z}^{-1}\cdot\displaystyle\prod_{k=i_{0}}^{j}\displaystyle\prod_{m=u}^{v}\mathcal{W}_{\nu_{m-N}}\big{(}\bm{A}^{\bm{\mathsf{l}}}(-k,m),\bm{B}^{\bm{\mathsf{l}}}(-k,m);\bm{C}^{\bm{\mathsf{l}}}(-k,m),\bm{D}^{\bm{\mathsf{l}}}(-k,m)\big{)}.$

Here, the probability on the left side of (8.3) is with respect to the
conditional law of $\mathds{L}$. Moreover, $\mathcal{Z}$ is a normalizing
constant defined so that the sum of the right side of (8.3), over all colored
line ensembles $\bm{\mathsf{l}}$ that are $\llbracket
i+1,j\rrbracket\times\llbracket u,v-1\rrbracket$-compatible with $\mathds{L}$,
is equal to $1$.

###### Proof.

Let $s>0$ be a small real number; set $x_{j}=s=y_{i}$ for each
$(i,j)\in\llbracket 1,M\rrbracket\times\llbracket 1,N\rrbracket$ (recalling
that $r_{j}=q^{-1/2}$ for each $j\in\llbracket 1,N\rrbracket$); and set
$t_{i}^{2}=-\nu_{i}$ for each $i\in\llbracket 1,M\rrbracket$. Observe from
(8.1) that this specialization sends the stochastic fused vertex model to the
discrete time $q$-boson model. Sample a colored line ensemble
$\mathds{L}=\big{(}\mathds{L}^{(1)},\mathds{L}^{(2)},\ldots,\mathds{L}^{(n)}\big{)}$
from the measure
$\mathbb{P}_{\operatorname{cL};n;s;\bm{x};\bm{y};\bm{r};\bm{t}}^{\sigma}$
(recall Definition 7.6); we will show that the proposition holds for the
limit181818This limit exists since that of
$\mathbb{P}_{\mathbbm{f}\mathbb{G};n;s;\bm{x};\bm{y};\bm{r};\bm{t}}^{\sigma}$
does as $s$ tends to $0$, as it can be quickly verified that the products
$s^{|\mu(N)|}\cdot\prod_{j=1}^{N}\mathbbm{f}_{\mu(j)/\mu(j-1);s}^{\sigma(j)}(x_{j};q^{-1/2})$
and
$s^{-|\mu(N)|}\cdot\prod_{i=1}^{N}\mathbb{G}_{\mu(i-1)/\mu(i);s}(s;t_{i-N})$
appearing in (6.11) remain nonsingular as $s$ tends to $0$ (since the weights
$s^{\mathbbm{1}_{j>0}}\cdot\widehat{W}_{x;q^{-1/2},s}(\bm{A},i;\bm{A}_{ij}^{+-},j)$
and $s^{|\bm{D}|}\cdot W_{s;t,s}(\bm{A},\bm{B};\bm{C},\bm{D})$ do). of
$\mathds{L}$, as $s$ tends to $0$.

The fact that $\mathds{L}$ (for any $s\geq 0$) satisfies the first statement
of the proposition follows from 7.7 (and 8.4, so that we need not assume that
(6.8) holds). That it satisfies the second (also for any $s\geq 0$) follows
from (7.2) (and 8.4), together with the fact by 8.2 that for $m>N$ and
$y_{m-N}=s$ we have

$\displaystyle
W_{y_{m-N};t_{m-N},s}\big{(}\bm{A}^{\bm{\mathsf{l}}}(-k,m),\bm{B}^{\bm{\mathsf{l}}}(-k,m);\bm{C}^{\bm{\mathsf{l}}}(-k,m),\bm{D}^{\bm{\mathsf{l}}}(-k,m)\big{)}=0,$

unless $\bm{A}^{\bm{\mathsf{l}}}(-k,m)\geq\bm{D}^{\bm{\mathsf{l}}}(-k,m)$.

To verify the third, observe by 7.8 (and 8.4) that, for any $s\geq 0$,

(8.4) $\displaystyle\begin{aligned}
\mathbb{P}[\mathds{L}=\bm{\mathsf{l}}]&=\mathcal{Z}_{0}^{-1}\cdot\displaystyle\prod_{k=i_{0}}^{j}\displaystyle\prod_{m=u}^{v}s^{|\bm{D}^{\bm{\mathsf{l}}}(-k,m)|}\\\
&\quad\times\displaystyle\prod_{k=i_{0}}^{j}\displaystyle\prod_{m=u}^{v}s^{-|\bm{D}^{\bm{\mathsf{l}}}(-k,m)|}\cdot
W_{s;t_{m-N};s}\big{(}\bm{A}^{\bm{\mathsf{l}}}(-k,m),\bm{B}^{\bm{\mathsf{l}}}(-k,m);\bm{C}^{\bm{\mathsf{l}}}(-k,m),\bm{D}^{\bm{\mathsf{l}}}(-k,m)\big{)},\end{aligned}$

for some normalization constant $\mathcal{Z}_{0}>0$. Next, we have

$\displaystyle\displaystyle\sum_{m=u}^{v}\big{|}\bm{D}^{\bm{\mathsf{l}}}(-k,m)\big{|}=\displaystyle\sum_{m=u}^{v}\big{(}\bm{\mathsf{l}}_{k}^{(1)}(m-1)-\bm{\mathsf{l}}_{k}^{(1)}(m)\big{)}=\mathsf{l}_{k}^{(1)}(u-1)-\mathsf{l}_{k}^{(1)}(v)=\mathds{L}_{k}^{(1)}(u-1)-\mathds{L}_{k}^{(1)}(v),$

where the first equality holds by Definition 7.4, the second by performing the
sum, and the third by the fact that $\bm{\mathsf{l}}$ is $\llbracket
u,v-1\rrbracket\times\llbracket i+1,j\rrbracket$-compatible with $\mathds{L}$.
This quantity is fixed by the conditioning, so the first product on the right
side of (8.4) can be incorporated into the normalization constant
$\mathcal{Z}_{0}$. This gives

$\displaystyle\mathbb{P}[\mathds{L}=\bm{\mathsf{l}}]=\mathcal{Z}_{1}^{-1}\cdot\displaystyle\prod_{k=i_{0}}^{j}\displaystyle\prod_{m=u}^{v}s^{-|\bm{D}^{\bm{\mathsf{l}}}(-k,m)|}\cdot
W_{s;t_{m-N};s}\big{(}\bm{A}^{\bm{\mathsf{l}}}(-k,m),\bm{B}^{\bm{\mathsf{l}}}(-k,m);\bm{C}^{\bm{\mathsf{l}}}(-k,m),\bm{D}^{\bm{\mathsf{l}}}(-k,m)\big{)},$

for some normalization constant $\mathcal{Z}_{1}>0$. Letting $s$ tend to $0$,
together with the fact by 8.2 and (8.2) (and as $t_{i}^{2}=-\nu_{i}$) that

$\displaystyle\lim_{s\rightarrow 0}s^{-|\bm{D}|}\cdot
W_{s;t_{i},s}(\bm{A},\bm{B};\bm{C},\bm{D})=\mathcal{W}_{\nu_{i}}(\bm{A},\bm{B};\bm{C},\bm{D}),$

yields the third part of the proposition. ∎

##  Appendix A Degeneration to the Log-Gamma Polymer

In this section we explain how to recover the log-gamma polymer (introduced in
[81]) from the colored stochastic fused vertex model (recall Section 6.1);
this involves a complementation procedure (of the type alluded to in 6.5), an
analytic continuation, and limit degeneration. We begin by describing the
complementation procedure and evaluating the associated complemented weights
in Section A.1. We then analyze (limits of) anaytic continuations of these
weights in Section A.2 and Section A.3, and analyze the behavior of them as
$q$ tends to $1$ in Section A.4, Section A.5, and Section A.6. We conclude in
Section A.7 by explaining convergence of the vertex model with these
degenerated weights to the log-gamma polymer (and also describing the reason
for choosing this specialization in A.25). In what follows, given a real
number $\theta>0$, we say that $X$ is a $\mathrm{Gamma}(\theta)$ random
variable if $\mathbb{P}[x\leq X\leq x+dx]=\Gamma(\theta)^{-1}\cdot
x^{\theta-1}e^{-x}dx$ for any $x\in\mathbb{R}$. Throughout this section,
$n\geq 1$ is an integer, and we assume that $q\in(0,1)$.

###  A.1. Complemented Fused Weights

In this section we implement (a case of) the complementation procedure
mentioned in 6.5, which will be necessary to degenerate the fused stochastic
vertex model of Section 6.1 to the log-gamma polymer. To that end, we require
some additional notation. Let $\mathrm{L}\in\mathbb{Z}_{\geq 0}$ be an
integer, and let

(A.1)
$\displaystyle\overline{\bm{X}}=(X_{1},X_{2},\ldots,X_{n-1})\in\mathbb{Z}_{\geq
0}^{n-1};\qquad\text{for any $\bm{X}\in\mathbb{Z}_{\geq 0}^{n}$, and
set}\quad\overline{x}=|\overline{\bm{X}}|.$

For any $\bm{B},\bm{D}\in\mathbb{Z}_{\geq 0}^{n}$ such that
$|\bm{B}|\leq\mathrm{L}$ and $\bm{D}\leq\mathrm{L}$, we set

(A.2)
$\displaystyle\mathsf{B}_{n}=\mathrm{L}-B_{n};\qquad\mathsf{b}=\mathrm{L}-|\bm{B}|;\qquad\mathsf{D}_{n}=\mathrm{L}-D_{n};\qquad\mathsf{d}=\mathrm{L}-|\bm{D}|.$

In this way, we have $\bm{B}=(\overline{\bm{B}},\mathrm{L}-\mathsf{B}_{n})$
and $\bm{D}=(\overline{\bm{D}},\mathrm{L}-\mathsf{D}_{n})$, so that

(A.3)
$\displaystyle\mathsf{b}=\mathsf{B}_{n}-\overline{b};\qquad\mathsf{d}=\mathsf{D}_{n}-\overline{d}.$

We will evaluate the weight $U_{z;r,s}(\bm{A},\bm{B};\bm{C},\bm{D})$ when
$r^{2}=q^{-\mathrm{L}}$ (which is the reason we imposed the conditions
$|\bm{B}|,|\bm{D}|\leq\mathrm{L}$ above), $s^{2}=q^{-\mathfrak{M}}$, and
$z=q^{\mathfrak{N}-\mathrm{L}+1}$ (for some complex numbers
$\mathfrak{M},\mathfrak{N}\in\mathbb{C}$). This weight will happen to be
rational in $q^{\mathrm{L}}$, so let us define it when the integer
$\mathrm{L}$ is replaced by an arbitrary complex number $\mathfrak{L}$. In
what follows, for any integers $m,k\geq 0$, and complex numbers
$z\in\mathbb{C}$, $a_{1},a_{2},\ldots,a_{k}\in\mathbb{C}$, and
$b_{1},b_{2},\ldots,b_{k}\in\mathbb{C}$, we denote the terminating basic
hypergeometric series by

(A.6)
$\displaystyle{{}_{k+1}\varphi_{k}}\bigg{(}\begin{array}[]{ccc}q^{-m};a_{1},a_{2},\ldots,a_{k}\\\
b_{1},b_{2},\ldots,b_{k}\end{array}\bigg{|}q,z\bigg{)}=\displaystyle\sum_{i=0}^{m}z^{i}\displaystyle\frac{(q^{-m};q)_{i}}{(q;q)_{i}}\cdot\displaystyle\prod_{j=1}^{k}\displaystyle\frac{(a_{j};q)_{i}}{(b_{j};q)_{i}}.$

######  Definition A.1.

Let $\mathfrak{L},\mathfrak{M},\mathfrak{N}\in\mathbb{C}$ be complex numbers;
$\mathsf{B}_{n},\mathsf{D}_{n}\in\mathbb{Z}_{\geq 0}$ be integers;
$\overline{\bm{B}},\overline{\bm{D}}\in\mathbb{Z}_{\geq 0}^{n}$ be
$(n-1)$-tuples, such that $\overline{b}=|\bm{\overline{B}}|\leq\mathsf{B}_{n}$
and $\overline{d}=|\overline{\bm{D}}|\leq\mathsf{D}_{n}$; and
$\bm{A},\bm{C}\in\mathbb{Z}_{\geq 0}^{n}$ be $n$-tuples. Setting $a=|\bm{A}|$
and $c=|\bm{C}|$; letting $\mathsf{b},\mathsf{d}\in\mathbb{Z}_{\geq 0}$ be as
in (A.3), and recalling (A.1), define the weight

$\displaystyle\mathcal{U}_{q^{\mathfrak{L}};q^{\mathfrak{M}};q^{\mathfrak{N}}}\big{(}\bm{A},(\overline{\bm{B}},\mathsf{B}_{n});\bm{C},(\overline{\bm{D}},\mathsf{D}_{n})\big{)}$
$\displaystyle\quad=(-1)^{C_{n}}q^{\varphi(\overline{\bm{D}},\overline{\bm{C}})+\binom{C_{n}+1}{2}+\overline{c}\mathsf{d}+\mathfrak{L}C_{n}}\cdot\mathbbm{1}_{\overline{\bm{A}}+\overline{\bm{B}}=\overline{\bm{C}}+\overline{\bm{D}}}\cdot\mathbbm{1}_{\mathsf{b}-a=\mathsf{d}-c}$
$\displaystyle\qquad\times\displaystyle\frac{(q;q)_{\mathsf{b}}}{(q;q)_{\mathsf{d}}}\displaystyle\frac{(q^{\mathfrak{M}-\mathfrak{N}-c};q)_{\mathsf{d}}}{(q;q)_{C_{n}}}\displaystyle\frac{(q^{\mathsf{D}_{n}-C_{n}-\mathfrak{L}};q)_{C_{n}}}{(q^{-\mathfrak{N}};q)_{\mathsf{b}}}\displaystyle\frac{(q^{-\mathfrak{N}};q)_{\infty}}{(q^{\mathfrak{L}-\mathfrak{N}};q)_{\infty}}\displaystyle\frac{(q^{\mathfrak{L}+\mathfrak{M}-\mathfrak{N}};q)_{\infty}}{(q^{\mathfrak{M}-\mathfrak{N}};q)_{\infty}}\cdot\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{B_{j}}}{(q;q)_{D_{j}}}$
$\displaystyle\qquad\times\displaystyle\sum_{\overline{\bm{P}}\leq\overline{\bm{B}},\overline{\bm{C}}}(-1)^{\overline{p}}q^{\varphi(\overline{\bm{B}}-\overline{\bm{D}}-\overline{\bm{P}},\overline{\bm{P}})}\displaystyle\frac{(q^{-\mathfrak{N}-1};q)_{\overline{p}}(q^{\mathfrak{M}-\mathfrak{N}+\mathsf{d}-c};q)_{\overline{p}}}{(q^{\mathfrak{M}-\mathfrak{N}-c};q)_{\overline{p}}(q^{\mathsf{b}-\mathfrak{N}};q)_{\overline{p}}}\cdot\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{C_{j}+D_{j}-P_{j}}}{(q;q)_{C_{j}-P_{j}}(q;q)_{P_{j}}(q;q)_{B_{j}-P_{j}}}$
$\displaystyle\qquad\qquad\quad\times
q^{\overline{p}(\mathsf{b}-\mathsf{d}+1)+\binom{\overline{p}}{2}}\cdot{{}_{4}\varphi_{3}}\bigg{(}\begin{array}[]{cc}q^{-C_{n}};q^{\mathsf{B}_{n}-\mathfrak{L}},q^{\overline{p}-\mathfrak{N}+1},q^{\mathfrak{M}-\mathfrak{N}+\mathsf{d}-c+\overline{p}}\\\
q^{\mathfrak{M}-\mathfrak{N}-c+\overline{p}},q^{\mathsf{b}-\mathfrak{N}+\overline{p}},q^{\mathsf{D}_{n}-C_{n}-\mathfrak{L}}\end{array}\bigg{|}q,q\bigg{)}.$

######  Remark A.2.

The appearance of the ${{}_{4}\varphi_{3}}$ basic hypergeometric series in the
$\mathcal{U}$-weights defined above is a common phenomenon for fused vertex
weights in the colorless $n=1$ case; see, for example, [65, Equation (1.1)]
and [39, Theorem 3.15].

The following lemma provides an expression for the $U_{z}$-weight at
$z=q^{\mathfrak{N}-\mathrm{L}+1}$ in terms of $\mathcal{U}$.

######  Lemma A.3.

Fix an integer $\mathrm{L}\geq 1$ and complex number
$\mathfrak{M},\mathfrak{N}\in\mathbb{C}$. Let
$\bm{A},\bm{B},\bm{C},\bm{D}\in\mathbb{Z}_{\geq 0}^{n}$, let $a=|\bm{A}|$ and
$c=|\bm{C}|$, and adopt the notation in (A.1) and (A.2). For any
$\mathfrak{N}\in\mathbb{C}$, we have

$\displaystyle
U_{q^{\mathfrak{N}-\mathrm{L}+1};q^{-\mathrm{L}/2},q^{-\mathfrak{M}/2}}(\bm{A},\bm{B};\bm{C},\bm{D})=\mathcal{U}_{q^{\mathrm{L}};q^{\mathfrak{M}};q^{\mathfrak{N}}}\big{(}\bm{A},(\overline{\bm{B}},\mathsf{B}_{n});\bm{C},(\overline{\bm{D}},\mathsf{D}_{n})\big{)}.$

###### Proof.

Let us assume throughout this proof that $\bm{A}+\bm{B}=\bm{C}+\bm{D}$, or
equivalently that
$\overline{\bm{A}}+\overline{\bm{B}}=\overline{\bm{C}}+\overline{\bm{D}}$ and
$\mathsf{b}-a=\mathrm{L}-|\bm{A}|-|\bm{B}|=\mathrm{L}-|\bm{C}|-|\bm{D}|=\mathsf{d}-c$;
otherwise, (6.1) implies that
$U_{q^{\mathfrak{N}-\mathrm{L}+1};q^{-\mathrm{L}/2},q^{-\mathfrak{M}/2}}(\bm{A},\bm{B};\bm{C},\bm{D})=0$,
which matches with
$\mathcal{U}_{q^{\mathfrak{L}};q^{\mathfrak{M}};q^{\mathfrak{N}}}(\bm{A},\bm{B};\bm{C},\bm{D})$,
by Definition A.1. Next, inserting (A.2) into (6.1) (and using the facts that
$a-c=|\bm{A}|-|\bm{C}|=|\bm{D}|-|\bm{B}|=\mathsf{b}-\mathsf{d}$ and that
$\varphi(\bm{B},\bm{X})=\varphi(\overline{\bm{B}},\bm{X})$ and
$\varphi(\bm{D},\bm{X})=\varphi(\overline{\bm{D}},\bm{X})$ for any
$\bm{X}\in\mathbb{Z}_{\geq 0}^{n}$), we obtain

(A.7) $\displaystyle\begin{aligned}
&U_{q^{\mathfrak{N}-\mathrm{L}+1};q^{-\mathrm{L}/2},q^{-\mathfrak{M}/2}}(\bm{A},\bm{B};\bm{C},\bm{D})\\\
&\quad=q^{(\mathfrak{N}+1)(\mathsf{b}-\mathsf{d})-(\mathrm{L}-\mathsf{d})\mathfrak{M}+\varphi(\overline{\bm{D}},\bm{C})}\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathrm{L}-\mathsf{d}}}{(q^{-\mathrm{L}};q)_{\mathrm{L}-\mathsf{b}}}\displaystyle\frac{(q;q)_{\mathrm{L}-\mathsf{B}_{n}}}{(q;q)_{\mathrm{L}-\mathsf{D}_{n}}}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{B_{j}}}{(q;q)_{D_{j}}}\\\
&\qquad\times\displaystyle\sum_{p=0}^{\min\\{\mathrm{L}-\mathsf{b},c\\}}q^{(\mathfrak{N}+1)p}\displaystyle\frac{(q^{\mathfrak{N}-\mathfrak{M}+1};q)_{c-p}}{(q^{\mathfrak{N}-\mathrm{L}-\mathfrak{M}+1};q)_{\mathrm{L}+c-\mathsf{d}-p}}(q^{-\mathfrak{N}-1};q)_{p}(q^{\mathfrak{N}-\mathrm{L}+1};q)_{\mathrm{L}-\mathsf{b}-p}\\\
&\qquad\times\displaystyle\sum_{\begin{subarray}{c}\bm{P}\leq\bm{B},\bm{C}\\\
|\bm{P}|=p\end{subarray}}q^{\varphi(\overline{\bm{B}}-\overline{\bm{D}}-\overline{\bm{P}},\bm{P})}\displaystyle\frac{(q;q)_{\mathrm{L}+C_{n}-\mathsf{D}_{n}-P_{n}}}{(q;q)_{C_{n}-P_{n}}(q;q)_{P_{n}}(q;q)_{\mathrm{L}-\mathsf{B}_{n}-P_{n}}}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{C_{j}+D_{j}-P_{j}}}{(q;q)_{C_{j}-P_{j}}(q;q)_{P_{j}}(q;q)_{B_{j}-P_{j}}}.\end{aligned}$

We will next separate the parameter $\mathrm{L}$ (as well as $p$ and $P_{n}$)
from the other ones in the subscripts of the $q$-Pochhammer symbols. To that
end, observe that

(A.8)
$\displaystyle\displaystyle\frac{(q;q)_{\mathrm{L}-\mathsf{B}_{n}}}{(q;q)_{\mathrm{L}-\mathsf{D}_{n}}}\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathrm{L}-\mathsf{d}}}{(q^{-\mathrm{L}};q)_{\mathrm{L}-\mathsf{b}}}=(-1)^{\mathsf{b}+\mathsf{B}_{n}-\mathsf{d}-\mathsf{D}_{n}}q^{\binom{\mathsf{B}_{n}}{2}-\binom{\mathsf{D}_{n}}{2}+\mathrm{L}(\mathsf{D}_{n}-\mathsf{B}_{n})+\binom{\mathsf{d}+1}{2}-\binom{\mathsf{b}+1}{2}}\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathsf{D}_{n}}}{(q^{-\mathrm{L}};q)_{\mathsf{B}_{n}}}\displaystyle\frac{(q;q)_{\mathsf{b}}}{(q;q)_{\mathsf{d}}},$

due to the identities

$\displaystyle\displaystyle\frac{(q;q)_{\mathrm{L}-\mathsf{B}_{n}}}{(q;q)_{\mathrm{L}-\mathsf{D}_{n}}}=\displaystyle\frac{(q^{\mathrm{L}};q^{-1})_{\mathsf{D}_{n}}}{(q^{\mathrm{L}};q^{-1})_{\mathsf{B}_{n}}}=(-1)^{\mathsf{B}_{n}-\mathsf{D}_{n}}q^{\binom{\mathsf{B}_{n}}{2}-\binom{\mathsf{D}_{n}}{2}+\mathrm{L}(\mathsf{D}_{n}-\mathsf{B}_{n})}\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathsf{D}_{n}}}{(q^{-\mathrm{L}};q)_{\mathsf{B}_{n}}};$
$\displaystyle\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathrm{L}-\mathsf{d}}}{(q^{-\mathrm{L}};q)_{\mathrm{L}-\mathsf{b}}}=\displaystyle\frac{(q^{-\mathsf{b}};q)_{\mathsf{b}}}{(q^{-\mathsf{d}};q)_{\mathsf{d}}}=(-1)^{\mathsf{b}-\mathsf{d}}q^{\binom{\mathsf{d}+1}{2}-\binom{\mathsf{b}+1}{2}}\displaystyle\frac{(q;q)_{\mathsf{b}}}{(q;q)_{\mathsf{d}}},$

where in the first we used the fact that
$(u;q^{-1})_{k}=(-u)^{k}q^{-\binom{k}{2}}(u^{-1};q)_{k}$ for any
$k\in\mathbb{Z}_{\geq 0}$ and $u\in\mathbb{C}$. We further have that

(A.9) $\displaystyle\begin{aligned}
\displaystyle\frac{(q^{\mathfrak{N}-\mathfrak{M}+1};q)_{c-p}}{(q^{\mathfrak{N}-\mathrm{L}-\mathfrak{M}+1};q)_{\mathrm{L}+c-\mathsf{d}-p}}(q^{\mathfrak{N}-\mathrm{L}+1};q)_{\mathrm{L}-\mathsf{b}-p}&=\displaystyle\frac{(q^{\mathfrak{N}-\mathfrak{M}+1};q)_{c}}{(q^{\mathfrak{N}-\mathfrak{M}+c};q^{-1})_{p}}\displaystyle\frac{(q^{\mathfrak{N}-\mathrm{L}+1};q)_{\mathrm{L}}}{(q^{\mathfrak{N}};q^{-1})_{\mathsf{b}+p}}\displaystyle\frac{(q^{\mathfrak{N}-\mathfrak{M}};q^{-1})_{\mathsf{d}-c-p}}{(q^{\mathfrak{N}-\mathfrak{M}-\mathrm{L}+1};q)_{\mathrm{L}}};\\\
\displaystyle\frac{(q;q)_{\mathrm{L}+C_{n}-\mathsf{D}_{n}-P_{n}}}{(q;q)_{C_{n}-P_{n}}(q;q)_{\mathrm{L}-\mathsf{B}_{n}-P_{n}}}&=\displaystyle\frac{(q^{\mathrm{L}};q^{-1})_{\mathsf{B}_{n}+P_{n}}(q^{C_{n}};q^{-1})_{P_{n}}}{(q;q)_{C_{n}}(q^{\mathrm{L}};q^{-1})_{\mathsf{D}_{n}-C_{n}+P_{n}}},\end{aligned}$

due the identities

$\displaystyle(q^{\mathfrak{N}-\mathrm{L}+1};q)_{\mathrm{L}-\mathsf{b}-p}=\displaystyle\frac{(q^{\mathfrak{N}-\mathrm{L}+1};q)_{\mathrm{L}}}{(q^{\mathfrak{N}};q^{-1})_{\mathsf{b}+p}};\qquad(q^{\mathfrak{N}-\mathfrak{M}+1};q)_{c-p}=\displaystyle\frac{(q^{\mathfrak{N}-\mathfrak{M}+1};q)_{c}}{(q^{\mathfrak{N}-\mathfrak{M}+c};q^{-1})_{p}};$
$\displaystyle(q^{\mathfrak{N}-\mathfrak{M}-\mathrm{L}+1};q)_{\mathrm{L}+c-\mathsf{d}-p}=\displaystyle\frac{(q^{\mathfrak{N}-\mathfrak{M}-\mathrm{L}+1};q)_{\mathrm{L}}}{(q^{\mathfrak{N}-\mathfrak{M}};q^{-1})_{\mathsf{d}-c+p}};\qquad(q;q)_{\mathrm{L}-\mathsf{B}_{n}-P_{n}}=\displaystyle\frac{(q;q)_{\mathrm{L}}}{(q^{\mathrm{L}};q^{-1})_{\mathsf{B}_{n}+P_{n}}};$
$\displaystyle(q;q)_{\mathrm{L}+C_{n}-\mathsf{D}_{n}-P_{n}}=\displaystyle\frac{(q;q)_{\mathrm{L}}}{(q^{\mathrm{L}};q^{-1})_{\mathsf{D}_{n}-C_{n}+P_{n}}};\qquad(q;q)_{C_{n}-P_{n}}=\displaystyle\frac{(q;q)_{C_{n}}}{(q^{C_{n}};q^{-1})_{P_{n}}}.$

Inserting (A.8) and (A.9) into (A.7), we obtain

(A.10) $\displaystyle\begin{aligned}
&U_{q^{\mathfrak{N}-\mathrm{L}+1};q^{-\mathrm{L}/2},q^{-\mathfrak{M}/2}}(\bm{A},\bm{B};\bm{C},\bm{D})\\\
&\quad=(-1)^{\mathsf{b}+\mathsf{B}_{n}-\mathsf{d}-\mathsf{D}_{n}}q^{(\mathfrak{N}+1)(b-\mathsf{d})-(\mathrm{L}-\mathsf{d})\mathfrak{M}+\varphi(\overline{\bm{D}},\bm{C})}q^{\binom{\mathsf{B}_{n}}{2}-\binom{\mathsf{D}_{n}}{2}+\mathrm{L}(\mathsf{D}_{n}-\mathsf{B}_{n})}q^{\binom{\mathsf{d}+1}{2}-\binom{\mathsf{b}+1}{2}}\\\
&\qquad\times\displaystyle\frac{(q;q)_{\mathsf{b}}}{(q;q)_{\mathsf{d}}}\displaystyle\frac{(q^{\mathfrak{N}-\mathfrak{M}+1};q)_{c}}{(q;q)_{C_{n}}}\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathsf{D}_{n}}}{(q^{-\mathrm{L}};q)_{\mathsf{B}_{n}}}\displaystyle\frac{(q^{\mathfrak{N}-\mathrm{L}+1};q)_{\mathrm{L}}}{(q^{\mathfrak{N}-\mathfrak{M}-\mathrm{L}+1};q)_{\mathrm{L}}}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{B_{j}}}{(q;q)_{D_{j}}}\\\
&\qquad\times\displaystyle\sum_{p=0}^{\min\\{b,c\\}}q^{(\mathfrak{N}+1)p}\displaystyle\frac{(q^{-\mathfrak{N}-1};q)_{p}(q^{\mathfrak{N}-\mathfrak{M}};q^{-1})_{\mathsf{d}-c+p}}{(q^{\mathfrak{N}-\mathfrak{M}+c};q^{-1})_{p}(q^{\mathfrak{N}};q^{-1})_{\mathsf{b}+p}}\\\
&\qquad\quad\times\displaystyle\sum_{\begin{subarray}{c}\bm{P}\leq\bm{B},\bm{C}\\\
|\bm{P}|=p\end{subarray}}q^{\varphi(\overline{\bm{B}}-\overline{\bm{D}}-\overline{\bm{P}},\bm{P})}\displaystyle\frac{(q^{\mathrm{L}};q^{-1})_{\mathsf{B}_{n}+P_{n}}(q^{C_{n}};q^{-1})_{P_{n}}}{(q;q)_{P_{n}}(q^{\mathrm{L}};q^{-1})_{\mathsf{D}_{n}-C_{n}+P_{n}}}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{C_{j}+D_{j}-P_{j}}}{(q;q)_{C_{j}-P_{j}}(q;q)_{P_{j}}(q;q)_{B_{j}-P_{j}}}.\end{aligned}$

Next observe, as $(u;q)_{k}=(-u)^{k}q^{\binom{k}{2}}(q^{1-k}u^{-1};q)_{k}$ for
any $k\in\mathbb{Z}_{\geq 0}$ and $u\in\mathbb{C}$, that

(A.11)
$\displaystyle\displaystyle\frac{(q^{\mathfrak{N}-\mathrm{L}+1};q)_{\mathrm{L}}}{(q^{\mathfrak{N}-\mathrm{L}-\mathfrak{M}+1};q)_{\mathrm{L}}}=q^{\mathrm{L}\mathfrak{M}}\displaystyle\frac{(q^{-\mathfrak{N}};q)_{\mathrm{L}}}{(q^{\mathfrak{M}-\mathfrak{N}};q)_{\mathrm{L}}};\qquad(q^{\mathfrak{N}-\mathfrak{M}+1};q)_{c}=(-1)^{c}q^{(\mathfrak{N}-\mathfrak{M})c+\binom{c+1}{2}}(q^{\mathfrak{M}-\mathfrak{N}-c};q)_{c}.$

We further convert the quantities of the form $(u;q^{-1})_{k}$ on the right
side of (A.10) into ones of the form $(u^{\prime};q)_{k}$. To do so, we
repeatedly use the identity
$(u;q^{-1})_{k}=(-u)^{k}q^{-\binom{k}{2}}(u^{-1};q)_{k}$ for any
$k\in\mathbb{Z}_{\geq 0}$ and $u\in\mathbb{C}$, which gives

(A.12) $\displaystyle\begin{aligned}
\displaystyle\frac{(q^{\mathfrak{N}-\mathfrak{M}};q^{-1})_{\mathsf{d}-c+p}}{(q^{\mathrm{L}};q^{-1})_{\mathsf{D}_{n}-C_{n}+P_{n}}}&=(-1)^{\mathsf{d}-c+p-\mathsf{D}_{n}+C_{n}-P_{n}}q^{(\mathfrak{N}-\mathfrak{M})(\mathsf{d}-c+p)-\mathrm{L}(\mathsf{D}_{n}-C_{n}+P_{n})}q^{\binom{\mathsf{D}_{n}-C_{n}+P_{n}}{2}-\binom{\mathsf{d}-c+p}{2}}\\\
&\qquad\times\displaystyle\frac{(q^{\mathfrak{M}-\mathfrak{N}};q)_{\mathsf{d}-c+p}}{(q^{-\mathrm{L}};q)_{\mathsf{D}_{n}-C_{n}+P_{n}}};\end{aligned}$

and

(A.13) $\displaystyle\begin{aligned}
&\displaystyle\frac{(q^{\mathrm{L}};q^{-1})_{\mathsf{B}_{n}+P_{n}}}{(q^{\mathfrak{N}};q^{-1})_{\mathsf{b}+p}}=(-1)^{\mathsf{B}_{n}+P_{n}-b-p}q^{\mathrm{L}(\mathsf{B}_{n}+P_{n})-\mathfrak{N}(\mathsf{b}+p)}q^{\binom{\mathsf{b}+p}{2}-\binom{\mathsf{B}_{n}+P_{n}}{2}}\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathsf{B}_{n}+P_{n}}}{(q^{-\mathfrak{N}};q)_{\mathsf{b}+p}};\\\
&\displaystyle\frac{(q^{C_{n}};q^{-1})_{P_{n}}}{(q^{\mathfrak{N}-\mathfrak{M}+c};q^{-1})_{p}}=(-1)^{p+P_{n}}q^{P_{n}C_{n}+(\mathfrak{M}-\mathfrak{N}-c)p}q^{\binom{p}{2}-\binom{P_{n}}{2}}\displaystyle\frac{(q^{-C_{n}};q)_{P_{n}}}{(q^{\mathfrak{M}-\mathfrak{N}-c};q)_{p}}.\end{aligned}$

Inserting (A.11), (A.12), and (A.13) into (A.10) gives

$\displaystyle
U_{q^{\mathfrak{N}-\mathrm{L}+1};q^{-\mathrm{L}/2},q^{-\mathfrak{M}/2}}(\bm{A},\bm{B};\bm{C},\bm{D})$
$\displaystyle\quad=(-1)^{\mathsf{b}+\mathsf{B}_{n}-\mathsf{d}-\mathsf{D}_{n}+c}q^{(\mathfrak{N}+1)(\mathsf{b}-\mathsf{d})+\mathsf{d}\mathfrak{M}+\varphi(\overline{\bm{D}},\bm{C})}q^{\binom{\mathsf{B}_{n}}{2}-\binom{\mathsf{D}_{n}}{2}+\mathrm{L}(\mathsf{D}_{n}-\mathsf{B}_{n})}q^{\binom{\mathsf{d}+1}{2}-\binom{\mathsf{b}+1}{2}+c(\mathfrak{N}-\mathfrak{M})+\binom{c+1}{2}}$
$\displaystyle\qquad\times\displaystyle\frac{(q;q)_{\mathsf{b}}}{(q;q)_{\mathsf{d}}}\displaystyle\frac{(q^{\mathfrak{M}-\mathfrak{N}-c};q)_{c}}{(q;q)_{C_{n}}}\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathsf{D}_{n}}}{(q^{-\mathrm{L}};q)_{\mathsf{B}_{n}}}\displaystyle\frac{(q^{-\mathfrak{N}};q)_{\mathrm{L}}}{(q^{\mathfrak{M}-\mathfrak{N}};q)_{\mathrm{L}}}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{B_{j}}}{(q;q)_{D_{j}}}$
$\displaystyle\qquad\times\displaystyle\sum_{p=0}^{\min\\{\mathrm{L}-\mathsf{b},c\\}}\displaystyle\sum_{\bm{P}\leq\bm{B},\bm{C}}(-1)^{\mathsf{d}-c+p-\mathsf{D}_{n}-C_{n}+P_{n}}q^{(\mathfrak{N}+1)p}q^{(\mathfrak{N}-\mathfrak{M})(\mathsf{d}-c+p)-\mathrm{L}(\mathsf{D}_{n}-C_{n}+P_{n})}$
$\displaystyle\qquad\qquad\times
q^{\binom{\mathsf{D}_{n}-C_{n}+P_{n}}{2}-\binom{\mathsf{d}-c+p}{2}}(-1)^{\mathsf{B}_{n}+P_{n}-\mathsf{b}-p}q^{\mathrm{L}(\mathsf{B}_{n}+P_{n})-\mathfrak{N}(\mathsf{b}+p)}q^{\binom{\mathsf{b}+p}{2}-\binom{\mathsf{B}_{n}+P_{n}}{2}}$
$\displaystyle\qquad\qquad\times\displaystyle\frac{(q^{-\mathfrak{N}-1};q)_{p}(q^{\mathfrak{M}-\mathfrak{N}};q)_{\mathsf{d}-c+p}}{(q^{\mathfrak{M}-\mathfrak{N}-c};q)_{p}(q^{-\mathfrak{N}};q)_{\mathsf{b}+p}}(-1)^{p+P_{n}}q^{P_{n}C_{n}+p(\mathfrak{M}-\mathfrak{N}-c)}q^{\binom{p}{2}-\binom{P_{n}}{2}}q^{\varphi(\overline{\bm{B}}-\overline{\bm{D}}-\overline{\bm{P}},\bm{P})}$
$\displaystyle\qquad\qquad\times\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathsf{B}_{n}+P_{n}}(q^{-C_{n}};q)_{P_{n}}}{(q;q)_{P_{n}}(q^{-\mathrm{L}};q)_{\mathsf{D}_{n}-C_{n}+P_{n}}}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{C_{j}+D_{j}-P_{j}}}{(q;q)_{C_{j}-P_{j}}(q;q)_{P_{j}}(q;q)_{B_{j}-P_{j}}}.$

Before proceeding, we next simplify the powers of $-1$ and $q$ appearing on
the right side. Doing so directly yields

$\displaystyle
U_{q^{\mathfrak{N}-\mathrm{L}+1};q^{-\mathrm{L}/2},q^{-\mathfrak{M}/2}}(\bm{A},\bm{B};\bm{C},\bm{D})$
$\displaystyle\quad=(-1)^{C_{n}}q^{\varphi(\overline{\bm{D}},\bm{C})}q^{\mathrm{L}C_{n}+\binom{\mathsf{B}_{n}}{2}-\binom{\mathsf{D}_{n}}{2}}q^{\binom{\mathsf{d}+1}{2}-\binom{\mathsf{b}+1}{2}+\mathsf{b}-\mathsf{d}+\binom{c+1}{2}}$
$\displaystyle\qquad\times\displaystyle\frac{(q;q)_{\mathsf{b}}}{(q;q)_{\mathsf{d}}}\displaystyle\frac{(q^{\mathfrak{M}-\mathfrak{N}-c};q)_{c}}{(q;q)_{C_{n}}}\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathsf{D}_{n}}}{(q^{-\mathrm{L}};q)_{\mathsf{B}_{n}}}\displaystyle\frac{(q^{-\mathfrak{N}};q)_{\mathrm{L}}}{(q^{\mathfrak{M}-\mathfrak{N}};q)_{\mathrm{L}}}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{B_{j}}}{(q;q)_{D_{j}}}$
$\displaystyle\qquad\times\displaystyle\sum_{p=0}^{\min\\{\mathrm{L}-\mathsf{b},c\\}}\displaystyle\sum_{\bm{P}\leq\bm{B},\bm{C}}(-1)^{p+P_{n}}q^{\binom{\mathsf{D}_{n}-C_{n}+P_{n}}{2}+P_{n}C_{n}-\binom{P_{n}}{2}-\binom{\mathsf{d}-c+p}{2}+\binom{p}{2}+p-pc}q^{\binom{\mathsf{b}+p}{2}-\binom{\mathsf{B}_{n}+P_{n}}{2}}$
$\displaystyle\qquad\qquad\qquad\qquad\quad\times
q^{\varphi(\overline{\bm{B}}-\overline{\bm{D}}-\overline{\bm{P}},\bm{P})}\displaystyle\frac{(q^{-\mathfrak{N}-1};q)_{p}(q^{\mathfrak{M}-\mathfrak{N}};q)_{\mathsf{d}-c+p}}{(q^{\mathfrak{M}-\mathfrak{N}-c};q)_{p}(q^{-\mathfrak{N}};q)_{\mathsf{b}+p}}\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathsf{B}_{n}+P_{n}}(q^{-C_{n}};q)_{P_{n}}}{(q;q)_{P_{n}}(q^{-\mathrm{L}};q)_{\mathsf{D}_{n}-C_{n}+P_{n}}}$
$\displaystyle\qquad\qquad\qquad\qquad\quad\times\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{C_{j}+D_{j}-P_{j}}}{(q;q)_{C_{j}-P_{j}}(q;q)_{P_{j}}(q;q)_{B_{j}-P_{j}}}.$

Let us continue to simplify the power of $q$ above. Since

$\displaystyle\binom{\mathsf{D}_{n}-C_{n}+P_{n}}{2}+P_{n}C_{n}-\binom{P_{n}}{2}=\binom{\mathsf{D}_{n}-C_{n}}{2}+P_{n}\mathsf{D}_{n};$
$\displaystyle\binom{\mathsf{B}_{n}+P_{n}}{2}=\binom{\mathsf{B}_{n}}{2}+\binom{P_{n}}{2}+\mathsf{B}_{n}P_{n};$
$\displaystyle\binom{\mathsf{d}-c+p}{2}-\binom{p}{2}+pc=\binom{\mathsf{d}-c}{2}+p\mathsf{d};\qquad\binom{\mathsf{b}+p}{2}=\binom{\mathsf{b}+1}{2}-\mathsf{b}+\binom{p}{2}+\mathsf{b}p,$

it follows that

(A.14) $\displaystyle\begin{aligned}
&U_{q^{\mathfrak{N}-\mathrm{L}+1};q^{-\mathrm{L}/2},q^{-\mathfrak{M}/2}}(\bm{A},\bm{B};\bm{C},\bm{D})\\\
&\quad=(-1)^{C_{n}}q^{\varphi(\overline{\bm{D}},\bm{C})}q^{\binom{\mathsf{D}_{n}-C_{n}}{2}-\binom{\mathsf{d}-c}{2}+\mathrm{L}C_{n}-\binom{\mathsf{D}_{n}}{2}}q^{\binom{\mathsf{d}+1}{2}-\mathsf{d}+\binom{c+1}{2}}\\\
&\qquad\times\displaystyle\frac{(q;q)_{\mathsf{b}}}{(q;q)_{\mathsf{d}}}\displaystyle\frac{(q^{\mathfrak{M}-\mathfrak{N}-c};q)_{c}}{(q;q)_{C_{n}}}\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathsf{D}_{n}}}{(q^{-\mathrm{L}};q)_{\mathsf{B}_{n}}}\displaystyle\frac{(q^{-\mathfrak{N}};q)_{\mathrm{L}}}{(q^{\mathfrak{M}-\mathfrak{N}};q)_{\mathrm{L}}}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{B_{j}}}{(q;q)_{D_{j}}}\\\
&\qquad\times\displaystyle\sum_{p=0}^{\min\\{\mathrm{L}-\mathsf{b},c\\}}\displaystyle\sum_{\bm{P}\leq\bm{B},\bm{C}}(-1)^{p+P_{n}}q^{P_{n}(\mathsf{D}_{n}-\mathsf{B}_{n})+p(\mathsf{b}-\mathsf{d}+1)+\binom{p}{2}-\binom{P_{n}}{2}}\\\
&\qquad\qquad\qquad\qquad\quad\times
q^{\varphi(\overline{\bm{B}}-\overline{\bm{D}}-\overline{\bm{P}},\bm{P})}\displaystyle\frac{(q^{-\mathfrak{N}-1};q)_{p}(q^{\mathfrak{M}-\mathfrak{N}};q)_{\mathsf{d}-c+p}}{(q^{\mathfrak{M}-\mathfrak{N}-c};q)_{p}(q^{-\mathfrak{N}};q)_{\mathsf{b}+p}}\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathsf{B}_{n}+P_{n}}(q^{-C_{n}};q)_{P_{n}}}{(q;q)_{P_{n}}(q^{-\mathrm{L}};q)_{\mathsf{D}_{n}-C_{n}+P_{n}}}\\\
&\qquad\qquad\qquad\qquad\quad\times\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{C_{j}+D_{j}-P_{j}}}{(q;q)_{C_{j}-P_{j}}(q;q)_{P_{j}}(q;q)_{B_{j}-P_{j}}}.\end{aligned}$

We additionally have by (A.3) and the equality $p=\overline{p}+P_{n}$ that

$\displaystyle\binom{\mathsf{d}+1}{2}-\mathsf{d}=\binom{\mathsf{d}}{2};\qquad\varphi(\overline{\bm{B}}-\overline{\bm{D}}-\overline{\bm{P}},\bm{P})=\varphi(\overline{\bm{B}}+\overline{\bm{D}}-\overline{\bm{P}},\overline{\bm{P}})+(\overline{b}-\overline{d}-\overline{p})P_{n};$
$\displaystyle\binom{p}{2}-\binom{P_{n}}{2}-\overline{p}P_{n}=\binom{\overline{p}}{2};\qquad
p(\mathsf{b}-\mathsf{d}+1)=\overline{p}(\mathsf{b}-\mathsf{d}+1)+P_{n}(\mathsf{b}-\mathsf{d})+P_{n};$
$\displaystyle
P_{n}(\mathsf{b}-\mathsf{d}+\mathsf{D}_{n}-\mathsf{B}_{n}+\overline{b}-\overline{d})=0;\qquad\binom{\mathsf{D}_{n}-C_{n}}{2}-\binom{\mathsf{D}_{n}}{2}=\binom{C_{n}+1}{2}-C_{n}\mathsf{D}_{n};$
$\displaystyle\varphi(\overline{\bm{D}},\bm{C})=\varphi(\overline{\bm{D}},\overline{\bm{C}})+C_{n}\overline{d};\qquad\binom{c+1}{2}+\binom{\mathsf{d}}{2}-\binom{\mathsf{d}-c}{2}=c\mathsf{d}=\overline{c}\mathsf{d}+C_{n}(\mathsf{D}_{n}-\overline{d}).$

Together with (A.3) and (A.14), these give

$\displaystyle
U_{q^{\mathfrak{N}-\mathrm{L}+1};q^{-\mathrm{L}/2},q^{-\mathfrak{M}/2}}(\bm{A},\bm{B};\bm{C},\bm{D})$
$\displaystyle=(-1)^{C_{n}}q^{\varphi(\overline{\bm{D}},\overline{\bm{C}})+\binom{C_{n}+1}{2}+\overline{c}\mathsf{d}+\mathrm{L}C_{n}}$
$\displaystyle\qquad\times\displaystyle\frac{(q;q)_{\mathsf{b}}}{(q;q)_{\mathsf{d}}}\displaystyle\frac{(q^{\mathfrak{M}-\mathfrak{N}-c};q)_{c}}{(q;q)_{C_{n}}}\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathsf{D}_{n}}}{(q^{-\mathrm{L}};q)_{\mathsf{B}_{n}}}\displaystyle\frac{(q^{-\mathfrak{N}};q)_{\mathrm{L}}}{(q^{\mathfrak{M}-\mathfrak{N}};q)_{\mathrm{L}}}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{B_{j}}}{(q;q)_{D_{j}}}$
$\displaystyle\qquad\times\displaystyle\sum_{p=0}^{\min\\{\mathrm{L}-\mathsf{b},c\\}}\displaystyle\sum_{\bm{P}\leq\bm{B},\bm{C}}(-1)^{\overline{p}}q^{\overline{p}(\mathsf{b}-\mathsf{d}+1)+P_{n}+\binom{\overline{p}}{2}}q^{\varphi(\overline{\bm{B}}-\overline{\bm{D}}-\overline{\bm{P}},\overline{\bm{P}})}$
$\displaystyle\qquad\qquad\times\displaystyle\frac{(q^{\mathfrak{M}-\mathfrak{N}};q)_{\mathsf{d}-c+p}}{(q^{\mathfrak{M}-\mathfrak{N}-c};q)_{p}}\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathsf{B}_{n}+P_{n}}(q^{-C_{n}};q)_{P_{n}}}{(q;q)_{P_{n}}(q^{-\mathrm{L}};q)_{\mathsf{D}_{n}-C_{n}+P_{n}}}$
$\displaystyle\qquad\qquad\times\displaystyle\frac{(q^{-\mathfrak{N}-1};q)_{p}}{(q^{-\mathfrak{N}};q)_{\mathsf{b}+p}}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{C_{j}+D_{j}-P_{j}}}{(q;q)_{C_{j}-P_{j}}(q;q)_{P_{j}}(q;q)_{B_{j}-P_{j}}}.$

Next, observe by (A.3) and the equality $p=\overline{p}+P_{n}$ that

$\displaystyle(q^{\mathfrak{M}-\mathfrak{N}-c};q)_{c}(q^{\mathfrak{M}-\mathfrak{N}};q)_{\mathsf{d}-c+p}=(q^{\mathfrak{M}-\mathfrak{N}-c};q)_{\mathsf{d}}(q^{\mathfrak{M}-\mathfrak{N}+\mathsf{d}-c};q)_{\overline{p}}(q^{\mathfrak{M}-\mathfrak{N}+\mathsf{d}-c+\overline{p}};q)_{P_{n}};$
$\displaystyle(q^{\mathfrak{M}-\mathfrak{N}-c};q)_{p}=(q^{\mathfrak{M}-\mathfrak{N}-c};q)_{\overline{p}}(q^{\mathfrak{M}-\mathfrak{N}-c+\overline{p}};q)_{P_{n}};\quad\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathsf{D}_{n}}}{(q^{-\mathrm{L}};q)_{\mathsf{D}_{n}-C_{n}+P_{n}}}=\displaystyle\frac{(q^{\mathsf{D}_{n}-C_{n}-\mathrm{L}};q)_{C_{n}}}{(q^{\mathsf{D}_{n}-C_{n}-\mathrm{L}};q)_{P_{n}}};$
$\displaystyle\displaystyle\frac{(q^{-\mathrm{L}};q)_{\mathsf{B}_{n}+P_{n}}}{(q^{-\mathrm{L}};q)_{\mathsf{B}_{n}}}=(q^{\mathsf{B}_{n}-\mathrm{L}};q)_{P_{n}};\quad\displaystyle\frac{(q^{-\mathfrak{N}-1};q)_{p}}{(q^{-\mathfrak{N}};q)_{\mathsf{b}+p}}=\displaystyle\frac{(q^{-\mathfrak{N}-1};q)_{\overline{p}}(q^{\overline{p}-\mathfrak{N}-1};q)_{P_{n}}}{(q^{-\mathfrak{N}};q)_{\mathsf{b}}(q^{\mathsf{b}-\mathfrak{N}};q)_{\overline{p}}(q^{\mathsf{b}-\mathfrak{N}+\overline{p}};q)_{P_{n}}},$

from which it follows that

$\displaystyle
U_{q^{\mathfrak{N}-\mathrm{L}+1};q^{-\mathrm{L}/2},q^{-\mathfrak{M}/2}}(\bm{A},\bm{B};\bm{C},\bm{D})$
$\displaystyle\quad=(-1)^{C_{n}}q^{\varphi(\overline{\bm{D}},\overline{\bm{C}})+\binom{C_{n}+1}{2}+\overline{c}\mathsf{d}+\mathrm{L}C_{n}}\displaystyle\frac{(q;q)_{\mathsf{b}}}{(q;q)_{\mathsf{d}}}\displaystyle\frac{(q^{\mathfrak{M}-\mathfrak{N}-c};q)_{\mathsf{d}}}{(q;q)_{C_{n}}}\displaystyle\frac{(q^{\mathsf{D}_{n}-C_{n}-\mathrm{L}};q)_{C_{n}}}{(q^{-\mathfrak{N}};q)_{\mathsf{b}}}\displaystyle\frac{(q^{-\mathfrak{N}};q)_{\mathrm{L}}}{(q^{\mathfrak{M}-\mathfrak{N}};q)_{\mathrm{L}}}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{B_{j}}}{(q;q)_{D_{j}}}$
$\displaystyle\qquad\times\displaystyle\sum_{\overline{\bm{P}}\leq\overline{\bm{B}},\overline{\bm{C}}}(-1)^{\overline{p}}q^{\varphi(\overline{\bm{B}}-\overline{\bm{D}}-\overline{\bm{P}},\overline{\bm{P}})}\displaystyle\frac{(q^{-\mathfrak{N}-1};q)_{\overline{p}}(q^{\mathfrak{M}-\mathfrak{N}+\mathsf{d}-c};q)_{\overline{p}}}{(q^{\mathfrak{M}-\mathfrak{N}-c};q)_{\overline{p}}(q^{\mathsf{b}-\mathfrak{N}};q)_{\overline{p}}}\displaystyle\prod_{j=1}^{n-1}\displaystyle\frac{(q;q)_{C_{j}+D_{j}-P_{j}}}{(q;q)_{C_{j}-P_{j}}(q;q)_{P_{j}}(q;q)_{B_{j}-P_{j}}}$
$\displaystyle\qquad\quad\times
q^{\overline{p}(\mathsf{b}-\mathsf{d}+1)+\binom{\overline{p}}{2}}\displaystyle\sum_{P_{n}=0}^{C_{n}}q^{P_{n}}\displaystyle\frac{(q^{\overline{p}-\mathfrak{N}-1};q)_{P_{n}}}{(q^{\mathfrak{M}-\mathfrak{N}-c+\overline{p}};q)_{P_{n}}}\displaystyle\frac{(q^{\mathfrak{M}-\mathfrak{N}+\mathsf{d}-c+\overline{p}};q)_{P_{n}}}{(q^{\mathsf{b}-\mathfrak{N}+\overline{p}};q)_{P_{n}}}\displaystyle\frac{(q^{\mathsf{B}_{n}-\mathrm{L}};q)_{P_{n}}(q^{-C_{n}};q)_{P_{n}}}{(q;q)_{P_{n}}(q^{\mathsf{D}_{n}-C_{n}-\mathrm{L}};q)_{P_{n}}},$

where we used the fact that any summands on the right side with
$P_{n}>B_{n}=\mathrm{L}-\mathsf{B}_{n}$ are equal to $0$, due to the factor of
$(q^{\mathsf{B}_{n}-\mathrm{L}};q)_{P_{n}}$. This, with
$(q^{-\mathfrak{N}};q)_{\mathrm{L}}=(q^{-\mathfrak{N}};q)_{\infty}(q^{\mathrm{L}-\mathfrak{N}};q)_{\infty}^{-1}$,
$(q^{\mathfrak{M}-\mathfrak{N}};q)_{\mathrm{L}}^{-1}=(q^{\mathrm{L}+\mathfrak{M}-\mathfrak{N}};q)_{\infty}(q^{\mathfrak{M}-\mathfrak{N}};q)_{\infty}^{-1}$,
and (A.6), yields the lemma. ∎

###  A.2. Degenerations of the Fused Complemented Weights

We next proceed to take limits of the weights from Definition A.1 that will
eventually lead us to the log-gamma polymer. The first is to let
$q^{\mathfrak{M}}$ and $q^{\mathfrak{N}}$ tend to infinity, in such a way that
($\mathfrak{L}$ and) $q^{\mathfrak{M}-\mathfrak{N}}=q^{-\mathfrak{L}}\gamma$
(for some constant $\gamma\in\mathbb{C}$) remains fixed.

######  Lemma A.4.

Adopting the notation of A.3, we have for any complex number
$\gamma\in\mathbb{C}$ that

(A.15) $\displaystyle\begin{aligned}