# Comparative Analyses of the Type D ASEP: Stochastic Fusion and Crystal Bases

Erik Brodsky1, Eva Engel2, Connor Panish3, Lillian Stolberg4

1Undergraduate, Michigan State University, Department of Mathematics
2Undergraduate, Princeton University
3Undergraduate, University of Florida, Department of Mathematics
4Undergraduate, University of Rochester, Department of Mathematics

(July 30, 2024)

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posted at the Texas A&M University Mathematics REU Website. We thank the
National Science Foundation’s Office for Strategic Initiatives on ‘STEM Access
for Persons with Disabilities’ for funding the creation of the accessible
version.

##  1 Abstract

The Type D asymmetric simple exclusion process (ASEP) is a particle system
involving two classes of particles that can be viewed from both a
probabilistic and an algebraic perspective [KLLPZ20]. From a probabilistic
perspective, we perform stochastic fusion on the Type D ASEP and analyze the
outcome on generator matrices, limits of drift speed, stationary
distributions, and Markov self-duality. From an algebraic perspective, we
construct a fused Type D ASEP system from a Casimir element of
$\mathcal{U}_{q}(\mathfrak{so}_{6})$, using crystal bases to analyze and
manipulate various representations of $\mathcal{U}_{q}(\mathfrak{so}_{6})$. We
conclude that both approaches produce different processes and therefore the
previous method of [Kua19], which analyzed the usual ASEP, does not generalize
to all finite-dimensional simple Lie algebras.

##  2 Introduction

[Lig76] is the first paper considering two-class ASEP. Later research by
[CGRS14, BS15, BS15a, CGRS16, BS16, Kua16, Kua17, Kua18, KLLPZ20, RLY23] shows
that generators of multi-class interacting particle systems can be constructed
using a central element from various Drinfeld-Jimbo quantum subgroups of
$\mathcal{U}_{q}(\mathfrak{gl}_{n})$ [Dri85, Jim85]. In [Kua19], the
probabilistic procedure of stochastic fusion is developed and it is shown to
produce the same process as central elements on normal ASEP. We focus on an
interacting particle system, labeled the Type D ASEP [KLLPZ20], that is a
generalization of Spitzer’s ASEP [Spi70]. The Type D ASEP involves two classes
of particles jumping on a lattice, where two particles of different classes
can exist at a site but two particles of the same class cannot. The jump rate
of an individual particle depends on its class and configuration in relation
to other particles.

We question whether constructing a generator according to the probabilistic
method of stochastic fusion defined in [Kua19] will result in the same
generator as the algebraic method, as studied in [CGRS14, CRV14, CGRS16,
Kua16, Kua17, Kua18, Kua21], of applying a ground state transformation on a
Casimir element in the second tensor power of an irreducible representation of
$\mathcal{U}_{q}(\mathfrak{so}_{6})$. We find that, unlike in Type $A$ Lie
algebras, this is not the case for all Type $D$ Lie algebras; in fact, the
probabilistic generator allows for nine states at each lattice site while the
algebraic generator allows for fourteen. We therefore explore various
characteristics of the different processes generated using these separate
approaches. The probabilistic characteristics include limits of component
generator matrices, stationary distributions, spectral gaps, and Markov self-
duality. In terms of algebraic characteristics, we study the algebraic
structure of various representations needed to construct the generator,
creating crystal bases and Young tableaux to decompose
$\mathcal{U}_{q}(\mathfrak{so}_{6})$-modules into irreducible representations
as well as weight spaces, then reflecting on the impacts these decompositions
have on the eigenvalues and overall structure of the constructed Markov
generator.

The paper proceeds by the following outline: algebraic and probabilistic
notation is introduced in the remainder of Section 2, probabilistic and
algebraic results are stated in Section 3 and Section 4, respectively, and the
respective proofs are discussed in Section 5 and Section 6. We list a variety
of matrices and methods of calculating them in the Appendix.

Acknowledgements. This research was conducted during a mathematics REU at
Texas A&M University, funded by the National Science Foundation under REU Site
Grant DMS-2150094 and the National Science Foundation’s Office for Strategic
Initiatives on ‘STEM Access for Persons with Disabilities’. The authors
express their gratitude to their mentor, Dr. Jeffrey Kuan, and teaching
assistant, Kennedi Hatcher, for their involvement in the project.

###  2.1 Probability Notation

####  2.1.1 Type D ASEP

The Type D ASEP is a Markov process in which there are two classes of
particles jumping on a lattice. Two particles of the same class cannot share a
lattice site, but two particles of different classes can. The Type D ASEP is
determined by parameters $(q,n,\delta=0)$. The variable $n$ measures drift
speed, with larger values of $n$ indicating more rapid drifts in the Type D
ASEP. The variable $q$ represents drift direction, with rightwards drift for
$0<q<1$ and leftwards drift for $q>1$. Note that $q$ cannot equal $1$. The
variable $\delta$ represents the interaction between the two particle classes.
The jump rate of a particle in the Type D ASEP is determined by the particle’s
class and configuration on the lattice.

We use $\Gamma$ to denote lattice sites of the Type D ASEP. For each $\Gamma$
site, we use $0$ to denote an empty site (contains no particles), $1$ to
denote a particle of class 1 at a site, $2$ to denote a particle of class 2,
and $3$ to denote a particle of class 1 and a particle of class 2.
Furthermore, we use parentheses to denote a $\Gamma$ state in the Type D ASEP.
For example, we let $(3,1,0,2)$ denote the four $\Gamma$ state particle
configuration in the top line of Figure 2.

We now consider the Type D ASEP over two $\Gamma$ lattice sites with
$\delta=0$. See [RLY23] for a rigorous definition and [KLLPZ20] for a picture
of the generator of Type D ASEP on two $\Gamma$ lattice sites. Note that there
are sixteen $(a,b)$ possible states since $0\leq a,b\leq 3$. We use
$L_{p}^{(2)}$ to denote the generator for Type D ASEP on two $\Gamma$ sites,
which is a direct sum of

$L_{1}=\begin{bmatrix}*&\frac{\left(-q^{1-n}+q^{n-1}\right)^{2}}{q^{2}}&-q^{2n-4}+q^{2n-2}+\frac{2}{q^{2}}\par\\\
q^{2n}-q^{2n-2}+2&*&-q^{2-2n}+2+q^{-2n}&\left(-q^{1-n}+q^{n-1}\right)^{2}\par\\\
q^{2}\left(-q^{1-n}+q^{n-1}\right)^{2}&2q^{2}+q^{2-2n}-q^{4-2n}&*&2q^{2}+q^{2-2n}-q^{4-2n}\par\\\
q^{2n}-q^{2n-2}+2&\left(-q^{1-n}+q^{n-1}\right)^{2}&-q^{2-2n}+2+q^{-2n}&*\end{bmatrix}$

corresponding to the communicating classes $\\{(3,0),(2,1),(0,3),(1,2)\\}$,
four 2 by 2 blocks

$L_{2}=\begin{bmatrix}*&\frac{q^{1-2n}+q^{2n-1}}{q}\\\
q\left(q^{1-2n}+q^{2n-1}\right)&*\end{bmatrix}$

corresponding to the communicating classes
$\\{(1,0),(0,1)\\},\\{(2,0),(0,2)\\},\\{(3,1),(1,3)\\},\\{(3,2),(2,3)\\}$, and
four $1$ by $1$ blocks of zeros corresponding to the communicating classes
$\\{(0,0)\\},\\{(1,1)\\},\\{(2,2)\\},\\{(3,3)\\}$. Finally we express
$L_{p}^{(2)}$,

$L_{p}^{(2)}=L_{1}\oplus\bigoplus_{i=1}^{4}L_{2}\oplus\bigoplus_{i=1}^{4}[0]$

with respect to the ordered basis

$\Omega=\Bigl{(}(3,0),(2,1),(0,3),(1,2),(1,0),(0,1),(2,0),(0,2),(3,1),(1,3),(3,2),(2,3),(0,0),(1,1),(2,2),(3,3)\Bigl{)}.$

Futhermore, we expand the definition of Type D ASEP to $K$ lattice sites where
the generator matrix is given by

$L=L^{1,2}+L^{2,3}+...+L^{K-1,K}$

We let $L^{x,x+1}$ denote the matrix the matrix on lattice sites $x$ and
$x+1$. For this paper, we almost always restrict to the case of Type D ASEP on
4 lattice sites. Therefore we define,

$L_{p}\coloneqq L^{1,2}+L^{2,3}+L^{3,4}$

to represent the generator of Type D ASEP on four $\Gamma$ sites. The
generator $L_{p}$ is a 256 by 256 matrix, $256=4^{4}$. Furthermore, we define
$L_{m}$ to be the generator where only the two middle particles interact, ie.
$L_{m}\coloneqq L^{2,3}$.

####  2.1.2 Stochastic Fusion

Taking a lattice with an even number of sites, we define stochastic fusion as
the merging of each set of two consecutive lattice sites, such that the
particles in each original site of the set are stacked at the new site. Note
that two particles of the same species can exist at one new site, and that the
total number of sites involved in the process is decreased by one half through
stochastic fusion. See [Kua19] for a thorough treatment of stochastic fusion.
We proceed to define notation for the fused Type D ASEP, using $\gamma$ to
denote lattice sites and states that have undergone the process of fusion, and
$\Gamma$ to denote lattice sites and states that have not. For $\gamma$
lattice sites, we use $11$ to denote two particles of class 1 at a site, $22$
to denote two particles of class 2, $3$ to denote a particle of class 1 and a
particle of class 2, $33$ to denote two particles of class 1 and two particles
of class 2, $31$ to denote two particles of class 1 and one particle of class
2, and $32$ to denote one particle of class 1 and two particles of class 2. We
will use angle brackets to denote $\gamma$ states. See Figure 1 for a
visualization of this notation. As another example of $\gamma$-state notation,
note that $\langle 31,2\rangle$ is the $\gamma$ state of $(3,1,0,2)$ and is
depicted in the bottom line of Figure 2.

Figure 1: The fusion states for one $\gamma$ site from left to right $\langle
0\rangle,\langle 1\rangle,\langle 2\rangle,\langle 3\rangle,\langle
11\rangle,\langle 22\rangle,\langle 31\rangle,\langle 32\rangle,\langle
33\rangle$

Figure 2: Depiction of fusion mapping $(3,1,0,2)\rightarrow\langle
31,2\rangle$ and fission mapping $\langle 31,2\rangle\rightarrow(3,1,0,2)$.

Let $S$ be the state space for the four-site Type D ASEP, in which there are
$256=4^{4}$ possible states. Rigorously,

$S=\\{(a,b,c,d)\mid a,b,c,d\in\\{0,1,2,3\\}\\}$

Fusing our lattice, we reach the state space $\hat{S}$ of two $\gamma$ lattice
sites, in which there are $81=9^{2}$ possible states. Rigorously,

$\hat{S}=\\{\langle x,y\rangle\mid x,y\in\\{0,1,2,3,11,22,31,32,33\\}\\}$

Our generator, fusion, and fission matrices depend on the ordering of states,
so we order states in the Type D ASEP over two $\Gamma$ lattice sites as
$\Omega$ and we order states in the Type D ASEP over one $\gamma$ lattice site
as

$\chi=\Bigl{(}\langle 0\rangle,\langle 1\rangle,\langle 2\rangle,\langle
11\rangle,\langle 12\rangle,\langle 22\rangle,\langle 31\rangle,\langle
32\rangle,\langle 33\rangle\Bigl{)}$

A visualization of this ordering and the possible states over one $\gamma$
lattice site can be seen in Figure 1. See section 7.4 for the ordering of
$\Gamma$ and $\gamma$ states over four $\Gamma$ lattice sites and two $\gamma$
lattice sites, respectively.

We now formally define the stochastic fusion map.

######  Definition 1.

We define $\phi:\Omega\rightarrow\chi$ as the stochastic fusion map for the
Type D ASEP on two $\Gamma$ lattice sites, where $x_{i}\in\\{0,1,2,3\\}$
describes the particles at $\Gamma$ lattice site $i$:

$\phi(x_{1},x_{2})\begin{cases}\langle 0\rangle&x_{1},x_{2}=0\\\ \langle
x_{1}\rangle&x_{1}\neq 0,x_{2}=0\\\ \langle x_{2}\rangle&x_{1}=0,x_{2}\neq
0\\\ \langle x_{1}x_{2}\rangle&x_{1},x_{2}\neq 0,x_{1}\geq x_{2}\\\ \langle
x_{2}x_{1}\rangle&x_{1},x_{2}\neq 0,x_{1}\leq x_{2}\end{cases}$

This definition is implied as a specific case in [Kua19] when there are only
class 1 and class 2 particles. Notice that this map corresponds to stacking
particles where order does not matter. To perform stochastic fusion on four
$\Gamma$ lattice sites of the Type D ASEP the stochastic fusion map is
$\phi\otimes\phi$, and so on for additional sites of stochastic fusion.

####  2.1.3 Transition and Generator Matrices

We now turn to the transition matrices of the Type D ASEP. Let $P_{t}$ be the
transition matrix and $L_{p}$ be the generator matrix for the Type D ASEP on
four $\Gamma$ sites defined previously. Note that $L_{m}$, the generator
matrix for the Type D ASEP on four $\Gamma$ lattice sites where only the two
middle $\Gamma$ lattice sites interact, is defined in section 2.1.1. Fusing
down to two $\gamma$ lattice sites, $Q_{t}$ is the fused transition matrix and
$L_{Q}$ is the generator matrix for the Type D ASEP on two $\gamma$ lattice
sites. Summarizing this process, we let $\Lambda:\hat{S}\rightarrow S$ be the
matrix of the fission map from the Type D ASEP on two $\gamma$ lattice sites
to Type D ASEP on four $\Gamma$ sites, and $\Phi:S\rightarrow\hat{S}$ be the
matrix of the stochastic fusion map from the Type D ASEP on four $\Gamma$
lattice sites to two $\gamma$ sites. See Figure 2 for an example of $\Lambda$
and $\Phi$ mappings.

We also define notation for the two-$\Gamma$ site Type D ASEP. Let
$\Lambda^{(2)}$ be the matrix of the fission map from the Type D ASEP on two
$\Gamma$ lattice sites to one $\gamma$ site, and $\Phi^{(2)}$ be the
corresponding matrix of the stochastic fusion map.

###  2.2 Algebraic Notation

We must first define $\mathcal{U}_{q}(\mathfrak{so}_{6})$ as well as its
fundamental representation in order to construct more complex
$\mathcal{U}_{q}(\mathfrak{so}_{6})$-modules. We introduce crystal base theory
as needed in Section 6.

######  Definition 2.

The special orthogonal Lie group $SO_{6}$ is the multiplicative group with
elements of the form

$\displaystyle SO_{6}=\left\\{X\in M_{6\times 6}(\mathbb{C}):XX^{T}=I,\det
X=1\right\\}$

######  Definition 3.

The special orthogonal Lie algebra $\mathfrak{so}_{6}$ is the Lie algebra with
elements of the form

$\displaystyle\mathfrak{so}_{6}=\left\\{\begin{bmatrix}A&C\\\ -C^{T}&B\\\
\end{bmatrix}:A,B,C\in M_{3\times 3}(\mathbb{C}),A=-A^{T},B=-B^{T}\right\\}$

######  Definition 4.

The Universal Enveloping Algebra of $\mathfrak{so}_{6}$, denoted
$\mathcal{U}(\mathfrak{so}_{6})$, is generated by
$\\{E_{1},E_{2},E_{3},F_{1},F_{2},F_{3},H_{1},H_{2},H_{3}\\}$ and the
following relations:

$\displaystyle[E_{i},F_{i}]=H_{i},\text{\hskip 28.45274pt}1\leq i\leq 3$

and

$\displaystyle E_{l}^{2}E_{j}+E_{j}E_{l}^{2}=2E_{l}E_{j}E_{l};\text{\hskip
7.11317pt}F_{l}^{2}F_{j}+F_{j}F_{l}^{2}=2F_{l}F_{j}F_{l}$

for $(l,j)\in\\{(1,2),(2,1),(1,3),(3,1)\\}$.

######  Definition 5.

The $q$-deformed quantum group $\mathcal{U}_{q}(\mathfrak{so}_{6})$, as
described by [Dri85] and [Jim85], is generated by
$\\{E_{1},E_{2},E_{3},F_{1},F_{2},F_{3},q^{H_{1}},q^{H_{2}},q^{H_{3}}\\}$ and
the relations:

$\displaystyle[E_{i},F_{i}]$
$\displaystyle=\frac{q^{H_{i}}-q^{-H_{i}}}{q-q^{-1}}$ $\displaystyle
q^{H_{i}}E_{j}$ $\displaystyle=q^{\alpha_{i}\cdot\alpha_{j}}E_{j}q^{H_{i}}$
$\displaystyle q^{H_{i}}F_{j}$
$\displaystyle=q^{-\alpha_{i}\cdot\alpha_{j}}F_{j}q^{H_{i}}$

for $1\leq i,j\leq 3$ and

$\displaystyle
E_{l}^{2}E_{k}+E_{k}E_{l}^{2}=(q+q^{-1})E_{l}E_{k}E_{l};\text{\hskip
7.11317pt}F_{l}^{2}F_{k}+F_{k}F_{l}^{2}=(q+q^{-1})F_{l}F_{k}F_{l}$

for $(l,k)\in\\{(1,2),(2,1),(1,3),(3,1)\\}$.

For a matrix $H$ in the Cartan subalgebra $\mathfrak{h}$ of
$\mathfrak{so}_{6}$, let $L_{i}\in\mathfrak{h}^{*}$ map $H$ to $H_{i,i}$. Then
we define the simple roots of $\mathcal{U}_{q}(\mathfrak{so}_{6})$ to be

$\Pi=\\{\alpha_{1}=L_{1}-L_{2},\alpha_{2}=L_{2}-L_{2},\alpha_{3}=L_{2}+L_{3}\\}$

and the fundamental weights to be

$\left\\{\omega_{1}=L_{1},\omega_{2}=\frac{1}{2}(L_{1}+L_{2}-L_{3}),\omega_{3}=\frac{1}{2}(L_{1}+L_{2}+L_{3})\right\\}.$

######  Definition 6.

We define the coproduct $\Delta$, counit $\epsilon$, and antipode $S$ of
$\mathcal{U}_{q}(\mathfrak{so}_{6})$ to be

$\displaystyle\Delta(E_{i})=E_{i}\otimes 1+q^{H_{i}}\otimes E_{i},\text{\hskip
32.43611pt}\epsilon($ $\displaystyle E_{i})=0,\text{\hskip
33.28966pt}S(E_{i})=-E_{i}q^{H_{i}}$ $\displaystyle\Delta(F_{i})=1\otimes
F_{i}+F_{i}\otimes q^{-H_{i}},\text{\hskip 28.45274pt}\epsilon($
$\displaystyle F_{i})=0,\text{\hskip 34.14322pt}S(F_{i})=-q^{-H_{i}}F_{i}$
$\displaystyle\Delta(q^{H_{i}})=q^{H_{i}}\otimes q^{H_{i}},\text{\hskip
70.27834pt}\epsilon($ $\displaystyle q^{H_{i}})=1,\text{\hskip
28.45274pt}S(q^{H_{i}})=q^{-H_{i}}.$

Equipping $\mathcal{U}_{q}(\mathfrak{so}_{6})$ with $\Delta$, $\epsilon$, and
$S$ makes the quantum group a Hopf algebra.

######  Definition 7.

A fundamental representation of $\mathcal{U}_{q}(\mathfrak{so}_{6})$ is
defined to be the subset of $M_{6\times 6}(\mathbb{R}[q,q^{-1}])$ as follows:

| $E_{i}$ | $F_{i}$ | $q^{H_{i}}$
---|---|---|---
$i=1$ | $E_{1,2}-E_{5,4}$ | $E_{2,1}-E_{4,5}$ | $qE_{1,1}+q^{-1}E_{2,2}+E_{3,3}+q^{-1}E_{4,4}+qE_{5,5}+E_{6,6}$
$i=2$ | $E_{2,3}-E_{6,5}$ | $E_{3,2}-E_{5,6}$ | $E_{1,1}+qE_{2,2}+q^{-1}E_{3,3}+E_{4,4}+q^{-1}E_{5,5}+qE_{6,6}$
$i=3$ | $E_{2,6}-E_{3,5}$ | $E_{6,2}-E_{5,3}$ | $E_{1,1}+qE_{2,2}+qE_{3,3}+E_{4,4}+q^{-1}E_{5,5}+q^{-1}E_{6,6}$

$E_{i,j}$ indicates the $6\times 6$ zero matrix with a $1$ in the $(i,j)^{th}$
entry, and $q^{-H_{i}}$ is defined as the multiplicative inverse of
$q^{H_{i}}$. We denote $V$ to be $\mathbb{R}^{6}\otimes\mathbb{R}^{6}$.

##  3 Probabilistic Results

We begin by defining the generator for stochastic fusion on the Type D ASEP,
noting the block diagonal form of this matrix. We then examine the unique
characteristics of the generator’s component blocks (communicating classes),
particularly their limits and spectral gaps as drift speeds are taken to
infinity and their stationary distributions in relation to drift speed. Fixing
drift speed to two, we explore two approaches to finding the matrix of Markov
self-duality for our system. The matrix of Markov self-duality will be
expressed in terms of the eigenvectors of the block-diagonalized fusion
generator, allowing for the dual and spectral gap to be found through the same
eigenvalue-eigenvector calculation.

###  3.1 Producing Stochastic Fusion Matrix

In this section we perform stochastic fusion on Type D ASEP for four $\Gamma$
lattice sites following the procedure developed in Section 3 of [Kua19]. To
begin, we define

$Q_{t}\coloneqq\Lambda P_{t}\Phi$

Taking the derivative on both sides at zero, we express the same equation in
terms of generators.

$L_{Q}=\Lambda L_{p}\Phi$

The goal of this section is to explicitly calculate $L_{Q}$, the generator
matrix for two $\gamma$ sites. This process reduces the $256\times 256$ matrix
$L_{p}$ to the fused $81\times 81$ matrix $L_{Q}$.

We construct $\Lambda^{(2)}$ based on the reversibility measures given in
Proposition 1.3 of [KLLPZ20] and we define
$\Lambda\coloneqq\Lambda^{(2)}\otimes\Lambda^{(2)}$ in lemma 4. Similarly, we
construct $\Phi^{(2)}$ based on the map $\phi$ from two $\Gamma$ lattice sites
to a single $\gamma$ lattice site, and we define
$\Phi\coloneqq\Phi^{(2)}\otimes\Phi^{(2)}$ in lemma 5.

The generator matrix $L_{m}$ corresponds to only the two middle of the four
$\Gamma$ lattice sites interacting, and can be expressed as a permutation of
$Id_{16\times 16}\otimes L_{p}^{(2)}$ as shown in lemma 6, this stands in
contrast to $L_{p}$ where all four $\Gamma$ sites can interact. We show that
only considering when the two middle of the four $\Gamma$ sites interact will
lead to the same stochastically fused process $L_{Q}$.

######  Theorem 1.

The generator for Type D ASEP on two $\gamma$ sites denoted $L_{Q}$ is equal
to $\Lambda L_{m}\Phi$, where $L_{m}$ is the generator for Type D ASEP on four
$\Gamma$ sites where only the middle two $\Gamma$ sites interact.

See a proof of this theorem in section 5.1.

Next, we want to understand the communicating classes of $L_{Q}$. We show that
$L_{Q}$ can be broken up into twenty-five communicating classes, where several
of the generators on these twenty-five communicating classes are the same and
every communicating class with the same amount of states has the same
generator. We define $L^{Q}_{D}$ as a permutation of $L_{Q}$ that is a block
diagonal matrix, where the ordering is based on the communicating classes.

######  Proposition 1.

Let
$L^{D}_{Q}=\mathcal{L}_{9}\oplus\bigoplus_{i=1}^{4}\mathcal{L}_{6}\oplus\bigoplus_{i=1}^{4}\mathcal{L}_{4}\oplus\bigoplus_{i=1}^{4}\mathcal{L}_{3}\oplus\bigoplus_{i=1}^{8}\mathcal{L}_{2}\oplus\bigoplus_{i=1}^{4}[0]$,
where $\mathcal{L}_{9}$ is the generator for the 9-state communicating class,
$\mathcal{L}_{6}$ is the generator for each 6-state communicating class, and
so on for $\mathcal{L}_{4},\mathcal{L}_{3}$ and $\mathcal{L}_{2}$. There
exists a permutation matrix $C$ such that $L_{Q}$ = $CL^{D}_{Q}C^{-1}$.

The permutation of states from $L_{Q}$ to the block diagonal $L^{D}_{Q}$ which
defines the permutation matrix $C$ is listed in section 7.4. Below are the
four three-state communicating classes of $L_{Q}$:

$\\{\langle 1,1\rangle,\langle 0,11\rangle,\langle 11,0\rangle\\},\\{\langle
2,2\rangle,\langle 0,22\rangle,\langle 22,0\rangle\\},\\{\langle
31,31\rangle,\langle 11,33\rangle,\langle 33,11\rangle\\},\\{\langle
32,32\rangle,\langle 22,33\rangle,\langle 33,22\rangle\\}$

.

$\mathcal{L}_{3}=\
\begin{bmatrix}*&\frac{q^{2}+q^{4n}}{(q^{6}+2q^{4}+q^{2})q^{2n}}&\frac{q^{6}+q^{4n+4}}{(q^{4}+2q^{2}+1)q^{2n}}\\\
\frac{q^{2}+q^{4n}}{q^{2n}}&*&0\\\
\frac{q^{2}+q^{4n}}{q^{2n+2}}&0&*\end{bmatrix}$

The remaining
$\mathcal{L}_{9},\mathcal{L}_{6},\mathcal{L}_{4},\mathcal{L}_{2}$ matrices are
outlined in section 7.3.

###  3.2 Taking $n$ to Infinity and Spectral Gaps

Recall that $n$ affects drift speed, with larger values of $n$ causing faster
drifting in the Type D ASEP. Also recall that Type D ASEP particles drift to
the right for $0<q<1$ and to the left for $q>1$. We are interested in how the
drift speed of the Type D ASEP affects the drift speed of the stochastically
fused process and its relaxation times.

Faster drift speeds in the Type D ASEP correspond to faster drift speeds in
the fused Type D ASEP process. Recall that $L_{m}$ is the generator for the
Type D ASEP on four lattice sites where only the two middle $\Gamma$ sites
interact, and note that $L_{m}$ depends on $n$. Thus, increasing values of $n$
will indicate faster drift speeds for the Type D ASEP on four $\Gamma$ lattice
sites where only the two middle $\Gamma$ sites interact. We see that $\Lambda$
and $\Phi$ do not depend on $n$, so faster drift speeds in $L_{m}$ will lead
to faster drift speeds in $L_{Q}$, the fusion matrix. This statement is proved
rigorously below.

######  Lemma 1.

We have the following limits:
$\lim_{n\to\infty}L_{Q}=\lim_{n\to\infty}(\Lambda
L_{m}\Phi)=\Lambda(\lim_{n\to\infty}L_{m})\Phi$. In other words, the limit as
$n\to\infty$ is commutative on $L_{Q}$.

######  Proposition 2.

The following are finite: $\lim_{n\to\infty}(q^{-2n}L_{m})$ for $q>1$ and
$\lim_{n\to\infty}(q^{2n}L_{m})$ for $0<q<1$.

######  Theorem 2.

The following are finite: $\lim_{n\to\infty}(q^{-2n}L_{Q})$ for $q>1$, and
$\lim_{n\to\infty}(q^{2n}L_{Q})$ for $0<q<1$.

See a proof of this theorem in section 5.2.

The previous theorem proves the intuitive statement that, as the drift speed
of the Type D ASEP on four $\Gamma$ lattice sites increases, so does the drift
speed of the stochastically fused process.

Continuing our study of limits as $n$ approaches infinity, we now consider the
limits of spectral gaps. The spectral gap of a Markov process is the absolute
value of the second-largest (ie. least negative) eigenvalue of the generator
matrix. Note that eigenvalues of generator matrices are less than or equal to
zero. From our calculations, the spectral gaps corresponding to generators
$\mathcal{L}_{2}$ and $\mathcal{L}_{3}$ simplify to reasonable expressions,
while the spectral gaps for $\mathcal{L}_{4},\mathcal{L}_{6}$, and
$\mathcal{L}_{9}$ do not simplify to expressions of a reasonable length.

######  Proposition 3.

The spectral gaps for $\mathcal{L}_{2}$ and $\mathcal{L}_{3}$ communicating
classes are denoted in the form $\lvert\lambda_{\mathcal{L}_{i}}\rvert$ and
are:

$\lvert\lambda_{\mathcal{L}_{2}}\rvert=\frac{(q^{4n-2}+1)(q^{4}+1)}{q^{2n}(q^{2}+1)}$
$\displaystyle\lvert\lambda_{\mathcal{L}_{3}}\rvert$
$\displaystyle=\bigg{|}-\frac{(q^{4n-2}+1)(2q^{4}+q^{2}+2)}{2q^{2n}(q^{2}+1)}+\frac{(q^{2}+q^{4n})}{q^{2n}(q^{2}+1)}\bigg{|}=\frac{(q^{4}+1)(q^{2}+q^{4n})}{q^{2n+2}(q^{2}+1).}$

For $q>1$ and $0<q<1$, notice that the spectral gaps
$|\lambda_{\mathcal{L}_{2}}|$ and $|\lambda_{\mathcal{L}_{3}}|$ go to infinity
as $n$ goes to infinity, meaning the relaxation times
$\frac{1}{\lambda_{\mathcal{L}_{2}}}$ and
$\frac{1}{\lambda_{\mathcal{L}_{3}}}$ go to zero. This implies that as the
drift speed of the Type D ASEP goes to infinity, the time to convergence to
the stationary distribution approaches zero for the 2 and 3-state
communicating classes. The authors suspect the same is true for the 4, 6, and
9-state communicating classes.

Note that, for $i\in\\{2,3\\}$, the eigenvalues of $q^{\pm 2n}\mathcal{L}_{i}$
are those of $\mathcal{L}_{i}$ multiplied by $q^{\pm 2n}$. Thus, the spectral
gaps of $q^{\pm 2n}\mathcal{L}_{2}$ and $q^{\pm 2n}\mathcal{L}_{2}$ are
$|q^{\pm 2n}\lambda_{\mathcal{L}_{2}}|$ and $|q^{\pm
2n}\lambda_{\mathcal{L}_{3}}|$, respectively. Note that the following spectral
gaps and relaxation times are the same for the 2 and 3-state communicating
classes. The limit as $n$ goes to infinity of the spectral gap of
$q^{-2n}\mathcal{L}_{i}$ for $q>1$ is 1. The limit as $n$ goes to infinity of
the spectral gap of $q^{2n}\mathcal{L}_{i}$ for $0<q<1$ is
$\frac{q^{4}+1}{q^{2}+1}$. Thus, the relaxation times for both generators for
$q>1$ are 1, and the relaxation times for $0<q<1$ are
$\frac{q^{2}+1}{q^{4}+1}$. This result suggests that the time to convergence
to the stationary distribution increases (relative to the time to convergence
in the previous paragraph) as $n$ goes to infinity for the 2 and 3-state
communicating classes. The authors suspect the same is true for the 4, 6, and
9-state communicating classes.

######  Remark 1.

As $n$ goes to infinity, some of the jump rates in the (time-rescaled) fused
Type D ASEP converge to $0$, which can be seen by taking the limit as
$n\to\infty$ of entries in the fusion generator. Thus, as $n$ approaches
infinity, the fused Type D ASEP (where $\delta=0$ and time is rescaled by a
factor of $q^{\pm 2n}$ for respective values of $q$) degenerates to the usual
ASEP. In this degeneration, we would expect the 2-state communicating class
and the 3-state communicating class to have the same relaxation time, which is
consistent with the calculations in the previous paragraph.

Due to the difficulty of displaying the eigenvalues for general $q$ and $n$,
we note that for $n=2$, the eigenvalues simplify nicely and are used to find a
dual for $L_{Q}$ when $n=2$ in section 3.4.2.

######  Lemma 2.

The eigenvalues for
$\mathcal{L}_{9},\mathcal{L}_{6},\mathcal{L}_{4},\mathcal{L}_{3}$, and
$\mathcal{L}_{2}$ when $n=2$ are the eigenvalues of $L_{Q}^{D}$ for $n=2$.

###  3.3 Stationary Distributions

Take Markov process $X_{t}$ with generator matrix $L$ and stationary
distribution $\pi$. Then $\pi L=0$. In other words, $\pi$ is a left
eigenvector of $L$. We use this result to obtain Proposition 4, which depends
on the communicating classes listed in Section 7.2.

######  Proposition 4.

For the 9-state communicating class, the stationary distribution is the
normalization of

$\Big{[}q^{4}(q^{2}+1)^{4},1,q^{16},q^{2}(q^{2}+1)^{2},q^{10}(q^{2}+1)^{2},q^{2}(q^{2}+1)^{2},q^{10}(q^{2}+1)^{2},q^{8},q^{8}\Big{]}.$

For the 6-state communicating class, the stationary distribution is the
normalization of

$\Big{[}q^{2}(q^{2}+1)^{2},q^{6}(q^{2}+1)^{2},1,q^{12},q^{4},q^{8}\Big{]}.$

For the 4-state communicating class, the stationary distribution is the
normalization of

$\Big{[}q^{8},q^{4},q^{4},1\Big{]}.$

For the 3-state communicating class, the stationary distribution is the
normalization of

$\Big{[}q^{2}(q^{2}+1)^{2},1,q^{8}\Big{]}.$

For the 2-state communicating class, the stationary distribution is the
normalization of

$\Big{[}1,q^{4}\Big{]}.$

for all $q>0$ where $q\neq 1$.

######  Remark 2.

Our advisor, Dr. Jeffrey Kuan, noticed that the stationary distributions we
found resemble q-exchangeable measures [GO09, GO11, GO10], but we do not delve
into the details of this relationship in our paper.

In order to find the stationary distributions, we found the left eigenvectors
corresponding to zero for $n=2$ (drift speed 2), and then verified that the
same eigenvectors work for general $n$. We then normalized the vectors to
obtain our final stationary distributions. This emphasizes that, although the
jump rates of particles in the Type D ASEP do depend on $n$, the stationary
distributions do not.

###  3.4 Markov Duality

######  Definition 8.

Let $X_{t}$ be a time-homogeneous Markov process on a discrete state space
$\mathcal{X}$. Label the generator matrix of the Markov process as $L_{X}$.
Let $D$ be a matrix with rows and columns indexed by $\mathcal{X}$. Then
$X_{t}$ is self-dual with respect to the matrix $D$ if $L_{X}D=DL_{X}^{T}$
[DF90].

####  3.4.1 Duality with Quantum q-Krawtchouk Polynomials

######  Remark 3.

Our mentor, Dr. Jeffrey Kuan, previously dropped the term “quantum” from
“quantum q-Krawtchouk polynomials” [Kua24a]. We have used the correct term in
our paper.

There are several papers on quantum q-Krawtchouk polynomials, see [KS96,
CFG21, FKZ24, Zho21, Gro18] for example. Furthermore, in Theorem 3.1 of
[BBKLUZ23], the authors propose a self-duality function for the unfused Type D
ASEP with parameters $(q,n=2,\delta=0)$ and $(q,n=3,\delta=0)$. We question
whether their self-duality function works for $L_{Q}^{D}$, the generator of
the Type D ASEP over two $\gamma$ lattice sites with ordered state space
$\mathcal{X}_{2}$ and parameters $(q,n=2,\delta=0)$, when approached from a
probability perspective.

Using the notation of [BBKLUZ23] with minimal changes for clarity, we denote
the self-duality function given by [BBKLUZ23] as
$D_{\alpha_{1},\alpha_{2}}^{(L)}(\eta,\xi)=D_{\alpha_{1}}^{(L)}(\eta_{1},\xi_{1})\cdot
D_{\alpha_{2}}^{(L)}(\eta_{2},\xi_{2})$ where
$D_{\alpha_{i}}^{(L)}(\eta_{i},\xi_{i})$ is taken over particle class
$i\in\\{1,2\\}$. We take variables $\alpha_{1},\alpha_{2}\in(0,q^{4})$. Using
the parameters $L=2$ and $n=2$, we find a potential self-duality function
(given by Theorem 3.1 in [BBKLUZ23]) for this 2-site Type D ASEP, where $D$ is
the matrix of this proposed self-duality function. For each
$\mathcal{X}_{2}[i]=\eta$ and $\mathcal{X}_{2}[j]=\xi$, the $(i,j)$th entry of
$D$ is given by $D_{\alpha_{1},\alpha_{2}}^{(2)}(\eta,\xi)$.

However, using the matrix $D$ that we computed, $L_{Q}^{D}D[1,2]\neq
D(L_{Q}^{D})^{T}[1,2]$, so $L_{Q}^{D}D\neq D(L_{Q}^{D})^{T}$ and thus $D$ is
not the matrix of the self-duality function of the Type D ASEP with two
$\gamma$ lattice sites. In our case, constructing our particle system from a
probabilistic rather than an algebraic process [BBKLUZ23], resulted in our
particle systems having different duality functions.

####  3.4.2 Other Avenues to Duality

Since the aforementioned duality function does not generate a valid dual for
the stochastically fused Type D ASEP, another logical approach is to find an
algebraic connection between the dual of the Type D ASEP generator on two
$\Gamma$ lattice sites and the dual of the stochastically fused process on two
$\gamma$ lattice sites. Unfortunately, we were not able to find a direct
algebraic relation between a dual of $L_{p}^{(2)}$ and a dual of $L_{Q}$.

Due to the eigenvalues of $L_{Q}$ not simplifying to reasonable expressions,
we now restrict to the case where $n=2$ and find a dual based on the
eigenvectors. We construct a non-trivial dual for $L^{D}_{Q}$, a permutation
of $L_{Q}$ in block diagonal form defined in proposition 1, for $n=2$. Note
that the eigenvalues for the communicating classes of $L_{Q}^{D}$ when $n=2$
were already given in lemma 2.

Recall the structure of the block diagonal generator in proposition 1:

$L^{D}_{Q}=\mathcal{L}_{9}\oplus\bigoplus_{i=1}^{4}\mathcal{L}_{6}\oplus\bigoplus_{i=1}^{4}\mathcal{L}_{4}\oplus\bigoplus_{i=1}^{4}\mathcal{L}_{3}\oplus\bigoplus_{i=1}^{8}\mathcal{L}_{2}\oplus\bigoplus_{i=1}^{4}[0].$

Since each communicating class is diagonalizable, we diagonalize $L^{D}_{Q}$
block by block and express $L_{Q}^{D}$ in a diagonalized form.

######  Lemma 3.

  *   * a.

The block generators
$\mathcal{L}_{9},\mathcal{L}_{6},\mathcal{L}_{4},\mathcal{L}_{3},\mathcal{L}_{2}$
are diagonalizable for $n=2$. Let $\mathcal{P}_{i}$ denote the matrix of the
right eigenvectors for the diagonalization of $\mathcal{L}_{i}$, and let
$\mathcal{A}_{i}$ be the diagonal matrix of eigenvalues listed in lemma 2 such
that $\mathcal{L}_{i}=\mathcal{P}_{i}\mathcal{A}_{i}\mathcal{P}^{-1}_{i}$ for
$i\in\\{2,3,4,6,9\\}$.

  * b.

Define
$\mathcal{P}=\mathcal{P}_{9}\oplus\bigoplus_{i=1}^{4}\mathcal{P}_{6}\oplus\bigoplus_{i=1}^{4}\mathcal{P}_{4}\oplus\bigoplus_{i=1}^{4}\mathcal{P}_{3}\oplus\bigoplus_{i=1}^{8}\mathcal{P}_{2}$,
$\mathcal{A}=\mathcal{A}_{9}\oplus\bigoplus_{i=1}^{4}\mathcal{A}_{6}\oplus\bigoplus_{i=1}^{4}\mathcal{A}_{4}\oplus\bigoplus_{i=1}^{4}\mathcal{A}_{3}\oplus\bigoplus_{i=1}^{8}\mathcal{A}_{2}$
and $\mathcal{Z}=\bigoplus_{i=1}^{4}[0]$.

  * c.

The block diagonal matrix
$L_{Q}^{D}=\mathcal{P}\mathcal{A}\mathcal{P}^{-1}\oplus\mathcal{Z}$.

######  Theorem 3.

The matrix $\mathcal{D}=\mathcal{P}\mathcal{P}^{T}\oplus\mathcal{Z}$ is a non-
trivial dual for $L_{Q}^{D}$.

See a proof of this theorem in section 5.4.

Since $\mathcal{P}$ has a block diagonal form, it follows that
$\mathcal{P}\mathcal{P}^{T}$ has a block diagonal form,

$\mathcal{D}=\mathcal{P}(\mathcal{P})^{T}\oplus\mathcal{Z}=\mathcal{P}_{9}(\mathcal{P}_{9})^{T}\oplus\bigoplus_{i=1}^{4}\mathcal{P}_{6}(\mathcal{P}_{6})^{T}\oplus\bigoplus_{i=1}^{4}\mathcal{P}_{4}(\mathcal{P}_{4})^{T}\oplus\bigoplus_{i=1}^{4}\mathcal{P}_{3}(\mathcal{P}_{3})^{T}\oplus\bigoplus_{i=1}^{8}\mathcal{P}_{2}(\mathcal{P}_{2})^{T}\oplus\mathcal{Z}.$

Note that $\mathcal{D}$ is a non-trivial dual because $\mathcal{D}$ is not a
constant matrix. For example

$\displaystyle\mathcal{P}_{4}(\mathcal{P}_{4})^{T}=$
$\displaystyle\begin{bmatrix}1+\frac{1}{q^{8}}+\frac{1}{q^{16}}&1-\frac{q^{4}-1}{q^{8}}-\frac{1}{q^{12}}&1-\frac{1}{q^{12}}&1-\frac{1}{q^{4}}+\frac{1}{q^{8}}\\\
1-\frac{q^{4}-1}{q^{8}}-\frac{1}{q^{12}}&2+\frac{\left(q^{4}-1\right)^{2}}{q^{8}}+\frac{1}{q^{8}}&\frac{1}{q^{8}}&1+\frac{q^{4}-1}{q^{4}}-\frac{1}{q^{4}}\\\
1-\frac{1}{q^{12}}&\frac{1}{q^{8}}&2+\frac{1}{q^{8}}&1-\frac{1}{q^{4}}\\\
1-\frac{1}{q^{4}}+\frac{1}{q^{8}}&1+\frac{q^{4}-1}{q^{4}}-\frac{1}{q^{4}}&1-\frac{1}{q^{4}}&3\end{bmatrix}.$

##  4 Algebraic Results

Previous research used the symmetries of Type $A$, $C$, and $D$ quantum groups
to construct asymmetric interacting particle systems on various lattice sites
[CGRS14, BS15a, BS15, CGRS16, BS16, Kua16, Kua17, Kua18, KLLPZ20, RLY23]. The
main algebraic focus of this project was to investigate how this construction
differs for the Type $D$ Lie algebra $\mathfrak{so}_{6}$. We are also
interested in whether this method produced the same Markov generator as the
probabilistic approach did. We found the following key result, with the second
research question answered in Proposition 9. Our definition of a ground state
transformation can be found in Definition 10.

######  Theorem 4.

A Markov generator is produced by performing a ground state transformation on
the representation $W\otimes W$ of [KLLPZ20]’s Casimir element; the explicit
generator can be found in Section 7.8.

To prove this, we need to first construct the aforementioned representation of
$\mathcal{U}_{q}(\mathfrak{so}_{6})$. We therefore prove the following two
propositions.

######  Proposition 5.

Define $W$ to be the 20-dimensional subspace of
$\mathbb{R}^{6}\otimes\mathbb{R}^{6}$ satisfying

$\mathrm{Sym}_{q}^{2}(\mathbb{R}^{6})=W\oplus\mathrm{span}\\{q^{-2}e_{1}\otimes
e_{4}+q^{2}e_{4}\otimes e_{1}+q^{-1}e_{2}\otimes e_{5}+qe_{5}\otimes
e_{2}+e_{3}\otimes e_{6}+e_{6}\otimes e_{3}\\}$

with $\\{e_{i}|i=1,...,6\\}$ standard basis vectors of $\mathbb{R}^{6}$. Then,
$W$ is an irreducible representation of $\mathcal{U}_{q}(\mathfrak{so}_{6})$.

######  Proposition 6.

As a representation of $\mathcal{U}_{q}(\mathfrak{so}_{6})$, $W\otimes W$
decomposes into a direct sum of irreducible representations and thus a direct
sum of weight spaces, listed respectively as follows:

$W\otimes W\cong V(4L_{1})\oplus V(3L_{1}+L_{2})\oplus V(2L_{1}+2L_{2})\oplus
V(2L_{1})\oplus V(L_{1}+L_{2})\oplus V(0)$

Let $\langle i,j,k\rangle$ denote the weight space $W\otimes W[i,j,k]$. Then

$\displaystyle W\otimes W\cong$ $\displaystyle\langle
0,0,0\rangle\oplus\langle 1,1,0\rangle\oplus\langle 1,0,1\rangle\oplus\langle
0,1,1\rangle\oplus\langle-1,1,0\rangle\oplus\langle-1,0,1\rangle\oplus\langle
0,-1,1\rangle\oplus\langle 1,-1,0\rangle$ $\displaystyle\oplus\langle
1,0,-1\rangle\oplus\langle
0,1,-1\rangle\oplus\langle-1,-1,0\rangle\oplus\langle-1,0,-1\rangle\oplus\langle
0,-1,-1\rangle\oplus\langle 2,0,0\rangle\oplus\langle 0,2,0\rangle$
$\displaystyle\oplus\langle
0,0,2\rangle\oplus\langle-2,0,0\rangle\oplus\langle 0,-2,0\rangle\oplus\langle
0,0,-2\rangle\oplus\langle 2,1,1\rangle\oplus\langle 1,2,1\rangle\oplus\langle
1,1,2\rangle\oplus\langle 2,1,-1\rangle$ $\displaystyle\oplus\langle
2,-1,1\rangle\oplus\langle
1,2,-1\rangle\oplus\langle-1,2,1\rangle\oplus\langle
1,-1,2\rangle\oplus\langle-1,1,2\rangle\oplus\langle
2,-1,-1\rangle\oplus\langle-1,2,-1\rangle$
$\displaystyle\oplus\langle-1,-1,2\rangle\oplus\langle-2,1,1\rangle\oplus\langle
1,-2,1\rangle\oplus\langle
1,-2,1\rangle\oplus\langle-2,1,-1\rangle\oplus\langle-2,-1,1\rangle\oplus\langle
1,-2,-1\rangle$ $\displaystyle\oplus\langle-1,-2,1\rangle\oplus\langle
1,-1,-2\rangle\oplus\langle-1,1,-2\rangle\oplus\langle-2,-1,-1\rangle\oplus\langle-1,-2,-1\rangle\oplus\langle-1,-1,-2\rangle$
$\displaystyle\oplus\langle 2,2,0\rangle\oplus\langle
2,0,2\rangle\oplus\langle
0,2,2\rangle\oplus\langle-2,2,0\rangle\oplus\langle-2,0,2\rangle\oplus\langle
0,-2,2\rangle\oplus\langle 2,-2,0\rangle\oplus\langle 2,0,-2\rangle$
$\displaystyle\oplus\langle
0,2,-2\rangle\oplus\langle-2,-2,0\rangle\oplus\langle-2,0,-2\rangle\oplus\langle
0,-2,-2\rangle\oplus\langle 3,1,0\rangle\oplus\langle
3,0,1\rangle\oplus\langle 0,3,1\rangle$ $\displaystyle\oplus\langle
1,3,0\rangle\oplus\langle 1,0,3\rangle\oplus\langle 0,1,3\rangle\oplus\langle
3,-1,0\rangle\oplus\langle 3,0,-1\rangle\oplus\langle
0,3,-1\rangle\oplus\langle-1,3,0\rangle\oplus\langle-1,0,3\rangle$
$\displaystyle\oplus\langle
0,-1,3\rangle\oplus\langle-3,1,0\rangle\oplus\langle-3,0,1\rangle\oplus\langle
0,-3,1\rangle\oplus\langle 1,-3,0\rangle\oplus\langle
1,0,-3\rangle\oplus\langle 0,1,-3\rangle$
$\displaystyle\oplus\langle-3,-1,0\rangle\oplus\langle-3,0,-1\rangle\oplus\langle
0,-3,-1\rangle\oplus\langle-1,-3,0\rangle\oplus\langle-1,0,-3\rangle\oplus\langle
0,-1,-3\rangle$ $\displaystyle\oplus\langle 4,0,0\rangle\oplus\langle
0,4,0\rangle\oplus\langle 0,0,4\rangle\oplus\langle-4,0,0\rangle\oplus\langle
0,-4,0\rangle\oplus\langle 0,0,-4\rangle.$

Once we characterized $W\otimes W$, we were able to extract enough information
to construct a $400\times 400$ representation of [KLLPZ20]’s Casimir element
without significant computation, which brings us to our next proposition.

######  Proposition 7.

We can write $\pi_{W\otimes W}(C)$ as a block matrix with one $22\times 22$
block, twelve $12\times 12$ blocks, six $8\times 8$ blocks, twenty-four
$4\times 4$ blocks, twelve $3\times 3$ blocks, twenty-four $2\times 2$ blocks,
and six $1\times 1$ blocks. Denoting $v_{i}$ to be the $i^{th}$ basis vector
of $W$, the blocked matrix is with respect to the ordered basis in Section
7.7.

Finally, we must perform a ground state transformation so that each row of
$\pi_{W\otimes W}(C)$ sums to $0$. We introduce and tweak the method in
Section 7.9 to compute this transformation.

######  Proposition 8.

Applying variations of

$\langle u_{i}\otimes
u_{j}|\Delta(F_{1})^{k_{1}}\Delta(F_{2})^{k_{2}}\Delta(F_{3})^{k_{3}}|e_{1}\otimes
e_{1}\rangle$

will generate zero values for the lowest weight vector of the
$\mathcal{U}_{q}(\mathfrak{so}_{6})$-modules $V$, $W$, and $W\otimes W$.

Using the previous four propositions, we compute a $196\times 196$ fused Type
D ASEP system generator from [KLLPZ20]’s central element represented in
$W\otimes W$, which can be found in Section 7.8. This proves Theorem 4, and
also has probabilistic significance: as an interacting particle system, the
process allows for $14$ different states at each site, i.e., all states except
four particles of the same class. This differs from the probabilistically-
constructed generator, which does not allow for three particles of the same
class on a lattice site.

Finally, we compare our result to the fused generator matrix $L_{Q}$ defined
in Section 3 and written explicitly in Section 7.3.

######  Proposition 9.

Regardless of which ground state transformation is applied to $\pi_{W\otimes
W}(C)$, the resulting Markov generator will not match the probabilistically-
generated matrix $L_{Q}$ in Section 7.3.

##  5 Probabilistic Proofs

###  5.1 Producing Stochastic Fusion Matrix

To begin, consider the function $\phi$. The left side of Figure 2 depicts
$\phi(3,1)=\langle 31\rangle$ and the right side depicts $\phi(0,2)=\langle
2\rangle$. Note that the function $\phi$ is surjective.

In order to construct $\Lambda^{(2)}$, we recall from [Kua19], that
$\Lambda^{(2)}$ is dependent on the reversible measures of the process
generated by $L_{p}^{(2)}$, in other words Type D ASEP on two $\Gamma$ sites.
We define the function $G^{2}$ below which is expressed fully in Proposition
1.3 of [KLLPZ20] and gives the reversible measures for the Type D ASEP on two
$\Gamma$ lattice sites.

Any particle configuration on two lattice sites can be represented as a
function $\eta:\\{1,2\\}\rightarrow\\{0,1,2,3\\}$, a function from a lattice
site to the particle configuration on that lattice site. Furthermore,
$A_{1}(\eta)\subset\\{1,2\\}$ is the set of lattice sites that class 1
particles occupy and similarly for $A_{2}(\eta)$. Finally, below is the
function from Proposition 1.3 of [KLLPZ20],

$G^{2}(\eta)=\prod_{x\in A_{1}(\eta)}q^{-2x}\prod_{x\in A_{2}(\eta)}q^{-2x}$

For simplicity, we represent the function $\eta$ as an ordered pair
$(\eta_{1},\eta_{2})$, where $\eta(1)=\eta_{1}$ and $\eta(2)=\eta_{2}$. As an
example for the reader to check their understanding with the new notation,

$G^{2}(2,1)=\prod_{x\in\\{2\\}}q^{-2x}\prod_{x\in\\{1\\}}q^{-2x}=q^{-6}$

######  Lemma 4.

We construct and express $\Lambda^{(2)}$, and define the fission map
$\Lambda\coloneqq\Lambda^{(2)}\otimes\Lambda^{(2)}$.

###### Proof.

Let the rows of $\Lambda^{(2)}$ be indexed by $x\in\chi$ and the columns be
indexed $\omega\in\Omega$. We define

$(\Lambda^{(2)})_{x,\omega}=\begin{cases}G^{2}(\omega)&\phi(\omega)=x\\\
0&\phi(\omega)\neq x\end{cases}$

with the additional step that the rows of $\Lambda^{(2)}$ are normalized to
sum to 1. $\Lambda^{(2)}$ is below.

$\setcounter{MaxMatrixCols}{16}\begin{bmatrix}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1\\\
0&0&0&0&\frac{q^{2}}{q^{2}+1}&\frac{1}{q^{2}+1}&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&\frac{q^{2}}{q^{2}+1}&\frac{1}{q^{2}+1}&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0\\\
\frac{q^{4}}{(q^{2}+1)^{2}}&\frac{q^{2}}{(q^{2}+1)^{2}}&\frac{1}{(q^{2}+1)^{2}}&\frac{q^{2}}{(q^{2}+1)^{2}}&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0\\\
0&0&0&0&0&0&0&0&\frac{q^{2}}{q^{2}+1}&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&\frac{q^{2}}{q^{2}+1}&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1\end{bmatrix}$

∎

######  Lemma 5.

We construct and express $\Phi^{(2)}$, and define the fusion map
$\Phi\coloneqq\Phi^{(2)}\otimes\Phi^{(2)}$.

###### Proof.

Let the rows of $\Phi^{(2)}$ be indexed by $\omega\in\Omega$ and the columns
be indexed by $x\in\chi$. We define

$(\Phi^{(2)})_{\omega,x}=\begin{cases}1&\phi(\omega)=x\\\ 0&\phi(\omega)\neq
x\end{cases}$

Note that $\Phi^{(2)}$ is a deterministic map and $\Phi^{(2)}$ is the matrix
representation of $\phi$. $\Phi^{(2)}$ is below.

$\begin{bmatrix}0&0&0&0&1&0&0&0&0\\\ 0&0&0&0&1&0&0&0&0\\\ 0&0&0&0&1&0&0&0&0\\\
0&0&0&0&1&0&0&0&0\\\ 0&1&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0&0\\\
0&0&1&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0&0\\\ 0&0&0&0&0&0&1&0&0\\\
0&0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&0&1&0\\\
1&0&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0&0\\\
0&0&0&0&0&1&0&0&1\\\ \end{bmatrix}$

∎

We now prove that $L_{m}$, the generator matrix for the Type D ASEP on four
$\gamma$ lattice sites where only the middle two $\Gamma$ states interact, can
be permuted into $\text{Id}_{16\times 16}\otimes L^{(2)}_{p}$.

######  Lemma 6.

There exists a permutation matrix $J$ such that $L_{m}=J(\text{Id}_{16\times
16}\otimes L_{p}^{(2)})J^{-1}$. In other words, the generator matrix $L_{m}$
can be reordered into $\text{Id}_{16\times 16}\otimes L_{p}^{(2)}$.

###### Proof.

For a short proof by existence, consider the diagonal ordering of states of
four $\Gamma$ lattice sites listed in section 7.4, we permute $L_{m}$
according to these states and get $\text{Id}_{16\times 16}\otimes
L_{p}^{(2)}$.

For a more intuitive proof, consider that any state in the Type D ASEP on four
$\Gamma$ lattice sites can be represented as $(x_{1},x_{2},x_{3},x_{4})$ where
$x_{1},x_{2},x_{3},x_{4}\in\\{0,1,2,3\\}$. Letting only the middle $\Gamma$
sites interact corresponds to $x_{1}$ and $x_{4}$ being held constant, while
the interaction of $x_{2}$ and $x_{3}$ is governed by $L_{p}^{(2)}$. Now
consider the bijective function
$h:(x_{1},x_{2},x_{3},x_{4})\mapsto\Big{(}(x_{1},x_{4}),(x_{2},x_{3})\Big{)}$.
Since $(x_{1},x_{4})$ is held constant while $(x_{2},x_{3})$ interact, it
follows that $L_{m}$ can be permuted into $\bigoplus_{i=1}^{16}L_{p}^{(2)}$,
since there are 16 possibilities for $(x_{1},x_{4})$. Finally, it follows that
$\bigoplus_{i=1}^{16}L_{p}^{(2)}=\text{Id}_{16\times 16}\otimes L_{p}^{(2)}$.
This proves that $L_{m}$ can be permuted into $\text{Id}_{16\times 16}\otimes
L_{p}^{(2)}$. ∎

Finally we prove theorem 1, and therefore that the fused process only depends
on the middle interacting particles. See 1

###### Proof.

We can express $L_{p}=L^{1,2}+L^{2,3}+L^{3,4}$. Note that $L_{m}\coloneqq
L^{2,3}$, as stated in section 2.1. Now we show that,

$L_{Q}\coloneqq\Lambda L_{p}\Phi=\Lambda(L^{1,2}+L_{m}+L^{3,4})\Phi$ $=\Lambda
L^{1,2}\Phi+\Lambda L_{m}\Phi+\Lambda L^{3,4}\Phi$
$=(\Lambda^{(2)}\otimes\Lambda^{(2)})L^{1,2}(\Phi^{(2)}\otimes\Phi^{(2)})+\Lambda
L_{m}\Phi+(\Lambda^{(2)}\otimes\Lambda^{(2)})L^{3,4}(\Phi^{(2)}\otimes\Phi^{(2)})$
$=0+\Lambda L_{m}\Phi+0$ $=\Lambda L_{m}\Phi$

The second to last step follows because $L^{1,2}$ and $L^{3,4}$ are the
generators acting on the sites that get fused down by $\Phi$ and fissioned by
$\Lambda$. The fused state of the Type D ASEP on two $\Gamma$ lattice sites
which are being fused cannot change because the stochastically fused state is
completely dependent on the number of type 1 and type 2 particles on those two
$\Gamma$ lattice sites. This cannot change when only those two sites interact.

∎

See 1

###### Proof.

Permute $L_{Q}$ according to the diagonal ordering for two $\gamma$ given in
section 7.4. This is a permutation of $L_{Q}$ according to its communicating
classes. ∎

###  5.2 Taking n to Infinity

See 1

###### Proof.

We remind the reader that $\Lambda$ and $\Phi$ do not depend on n. Therefore,

$\lim_{n\to\infty}L_{Q}=\lim_{n\to\infty}(\Lambda
L_{m}\Phi)=\Lambda(\lim_{n\to\infty}L_{m})\Phi$

∎

See 2

###### Proof.

Take $q>1$. Observing $L^{(2)}_{p}$, we see that multiplying it by $q^{-2n}$
then taking the limit as $n\to\infty$ is finite. The same applies to
multiplying $L^{(2)}_{p}$ by $q^{2n}$ for $0<q<1$. Finally, use lemma 6 and
notice that

$q^{\pm 2n}L_{m}=q^{\pm 2n}J(\text{Id}_{16\times 16}\otimes
L^{(2)}_{p})J^{-1}=J(\text{Id}_{16\times 16}\otimes(q^{\pm
2n}L^{(2)}_{p}))J^{-1}$

for respective boundaries on $q$, by the properties of the Kronecker product
and standard matrix multiplication. Noting that $J$ and $J^{-1}$ do not depend
on n, we take the limit as $n\to\infty$ of both sides and see that
$\lim_{n\to\infty}(q^{\pm 2n}L_{m})=\lim_{n\to\infty}(J(\text{Id}_{16\times
16}\otimes(q^{\pm 2n}L^{(2)}_{p}))J^{-1})=J(\text{Id}_{16\times
16}\otimes(\lim_{n\to\infty}q^{\pm 2n}L^{(2)}_{p}))J^{-1}$, where the right
(and therefore left) side of the equation is finite since
$\lim_{n\to\infty}(q^{\pm 2n}L^{(2)}_{p})$ is finite for respective boundaries
on $q$. This implies that the rate matrix grows on the order of $q^{2n}$
irrespective of drift direction. ∎

See 2

###### Proof.

Note that Proposition 2 suggests that multiplying $q^{\pm 2n}L_{m}$ (for
respective boundaries on $q$) by any constant preserves the finite limit,
since the constant can be moved outside the limit. Note that

$q^{\pm 2n}L_{Q}=q^{\pm 2n}\Lambda L_{m}\Phi=\Lambda q^{\pm 2n}L_{m}\Phi$

by the commutativity of scalar multiplication. Taking the limit of both sides
of the equation, we have

$\lim_{n\to\infty}(q^{\pm 2n}L_{Q})=\lim_{n\to\infty}(\Lambda q^{\pm
2n}L_{m}\Phi)=\Lambda\lim_{n\to\infty}(q^{\pm 2n}L_{m})\Phi$

by Lemma 1. We know that $\lim_{n\to\infty}(q^{\pm 2n}L_{m})$ is finite for
respective values of $q$, so $\lim_{n\to\infty}q^{\pm 2n}L_{Q}$ must also be
finite for such values of $q$. ∎

See 2

###### Proof.

We list the eigenvalues for the generator $\mathcal{L}_{9}$ of the 9-state
communicating class:

$0,\frac{-q^{8}-q^{4}-1}{q^{4}},\,\frac{-q^{8}+q^{6}-2q^{4}+q^{2}-1}{q^{4}},\,\frac{-q^{12}-4q^{10}-2q^{8}-6q^{6}-2q^{4}-4q^{2}-1}{q^{8}+2q^{6}+q^{4}},\,$
$\frac{-q^{12}-4q^{10}+q^{8}-8q^{6}+q^{4}-4q^{2}-1}{q^{8}+2q^{6}+q^{4}}$

The eigenvalues for the fusion generator $\mathcal{L}_{6}$ of the 6-state
communicating class:

$0,\frac{-q^{8}-q^{4}-1}{q^{4}},\,\frac{-q^{8}+q^{6}-2q^{4}+q^{2}-1}{q^{4}},\,$
$\frac{-q^{12}-4q^{10}-2q^{8}-6q^{6}-2q^{4}-4q^{2}-1}{q^{8}+2q^{6}+q^{4}},\,\frac{-q^{12}-4q^{10}+q^{8}-8q^{6}+q^{4}-4q^{2}-1}{q^{8}+2q^{6}+q^{4}}$

The eigenvalues for the fusion generator $\mathcal{L}_{4}$ of the 4-state
communicating class:

$0,\frac{-q^{8}+q^{6}-2q^{4}+q^{2}-1}{q^{4}},\,\frac{-q^{12}-4q^{10}+q^{8}-8q^{6}+q^{4}-4q^{2}-1}{q^{8}+2q^{6}+q^{4}}$

The eigenvalues for the fusion generator $\mathcal{L}_{3}$ of the 3-state
communicating class:

$0,\frac{-q^{8}-q^{4}-1}{q^{4}},\,\frac{-q^{8}+q^{6}-2q^{4}+q^{2}-1}{q^{4}}$

The eigenvalues for the fusion generator $\mathcal{L}_{2}$ of the 2-state
communicating class:

$0,\frac{-q^{8}+q^{6}-2q^{4}+q^{2}-1}{q^{4}}$

∎

###  5.3 Stationary Distribution

See 4

###### Proof.

For communicating class $i$ with generator matrix $\mathcal{L}_{i}$, the
stationary distribution for the class is represented by the normalized left
eigenvector $\pi$ corresponding to the eigenvalue zero for the generator
matrix, ie. $\pi L_{i}=0$. Through standard computation, one can see that a
left eigenvector exists for each communicating class and is equal to the one
listed in the proposition, making the normalization of that vector the
stationary distribution for the corresponding communicating class. ∎

###  5.4 Markov Duality

See 3

###### Proof.

To prove part a, we show that for each communicating class $\mathcal{L}_{i}$,
the matrix of right eigenvectors is invertible for $q>0$ where $q\neq 1$ (all
possible values $q$ can take), and therefore that all the eigenvectors are
linearly independent, which implies diagonalizability for $\mathcal{L}_{i}$.
Below we show $\mathcal{P}_{2}$ is invertible. We list all the other cases in
section 7.6.

$\mathcal{P}_{2}=\begin{bmatrix}1&-q^{4}\\\ 1&1\end{bmatrix}$

It follows that $\det(\mathcal{P}_{2})=q^{4}+1\neq 0$ for $q>0$. Therefore,
$\mathcal{P}_{2}$ is invertible for $q>0$, and therefore $\mathcal{L}_{2}$ is
diagonalizable for $q>0$.

To prove part c, we show that $L_{Q}^{D}$ is the direct sum of the
diagonalization over all of the communicating classes.

$\mathcal{P}\mathcal{A}\mathcal{P}^{-1}\oplus\mathcal{Z}$
$=\Big{(}\mathcal{P}_{9}\oplus\bigoplus_{i=1}^{4}\mathcal{P}_{6}\oplus\bigoplus_{i=1}^{4}\mathcal{P}_{4}\oplus\bigoplus_{i=1}^{4}\mathcal{P}_{3}\oplus\bigoplus_{i=1}^{8}\mathcal{P}_{2}\Big{)}\Big{(}\mathcal{A}_{9}\oplus\bigoplus_{i=1}^{4}\mathcal{A}_{6}\oplus\bigoplus_{i=1}^{4}\mathcal{A}_{4}\oplus\bigoplus_{i=1}^{4}\mathcal{A}_{3}\oplus\bigoplus_{i=1}^{8}\mathcal{A}_{2}\Big{)}$
$\Big{(}\mathcal{P}^{-1}_{9}\oplus\bigoplus_{i=1}^{4}\mathcal{P}^{-1}_{6}\oplus\bigoplus_{i=1}^{4}\mathcal{P}^{-1}_{4}\oplus\bigoplus_{i=1}^{4}\mathcal{P}^{-1}_{3}\oplus\bigoplus_{i=1}^{8}\mathcal{P}^{-1}_{2}\Big{)}\oplus\mathcal{Z}$
$=\mathcal{P}_{9}\mathcal{A}_{9}\mathcal{P}^{-1}_{9}\oplus\bigoplus_{i=1}^{4}\mathcal{P}_{6}\mathcal{A}_{6}\mathcal{P}^{-1}_{6}\oplus\bigoplus_{i=1}^{4}\mathcal{P}_{4}\mathcal{A}_{4}\mathcal{P}^{-1}_{4}\oplus\bigoplus_{i=1}^{4}\mathcal{P}_{3}\mathcal{A}_{3}\mathcal{P}^{-1}_{3}\oplus\bigoplus_{i=1}^{8}\mathcal{P}_{2}\mathcal{A}_{2}\mathcal{P}^{-1}_{2}\oplus\mathcal{Z}$
$=\mathcal{L}_{9}\oplus\bigoplus_{i=1}^{4}\mathcal{L}_{6}\oplus\bigoplus_{i=1}^{4}\mathcal{L}_{4}\oplus\bigoplus_{i=1}^{4}\mathcal{L}_{3}\oplus\bigoplus_{i=1}^{8}\mathcal{L}_{2}\oplus\mathcal{Z}$
$=L_{Q}^{D}.$

∎

See 3

###### Proof.

Our goal is to prove that $L_{Q}^{D}\mathcal{D}=\mathcal{D}(L_{Q}^{D})^{T}$,
and we use lemma 3.
Consider,

$L_{Q}^{D}\mathcal{D}=(\mathcal{P}\mathcal{A}\mathcal{P}^{-1}\oplus\mathcal{Z})(\mathcal{P}\mathcal{P}^{T}\oplus\mathcal{Z})$
$=(\mathcal{P}\mathcal{A}\mathcal{P}^{-1}\mathcal{P}\mathcal{P}^{T})\oplus\mathcal{Z}\mathcal{Z}=\mathcal{P}\mathcal{A}\mathcal{P}^{T}\oplus\mathcal{Z}$
$=\mathcal{P}(\mathcal{P}^{T}(\mathcal{P}^{T})^{-1})\mathcal{A}\mathcal{P}^{T}\oplus\mathcal{Z}$
$=(\mathcal{P}\mathcal{P}^{T})((\mathcal{P}^{-1})^{T}\mathcal{A}\mathcal{P}^{T})\oplus\mathcal{Z}$
$=(\mathcal{P}\mathcal{P}^{T})(\mathcal{P}\mathcal{A}\mathcal{P}^{-1})^{T}\oplus\mathcal{Z}$
$=(\mathcal{P}\mathcal{P}^{T}\oplus\mathcal{Z})((\mathcal{P}\mathcal{A}\mathcal{P}^{-1})^{T}\oplus\mathcal{Z})$
$=(\mathcal{P}\mathcal{P}^{T}\oplus\mathcal{Z})((\mathcal{P}\mathcal{A}\mathcal{P}^{-1})\oplus\mathcal{Z})^{T}$
$=\mathcal{D}(L_{Q}^{D})^{T}.$

Note that

$\mathcal{D}=\mathcal{P}\mathcal{P}^{T}\oplus\mathcal{Z}=\mathcal{P}_{9}(\mathcal{P}_{9})^{T}\oplus\bigoplus_{i=1}^{4}\mathcal{P}_{6}(\mathcal{P}_{6})^{T}\oplus\bigoplus_{i=1}^{4}\mathcal{P}_{4}(\mathcal{P}_{4})^{T}\oplus\bigoplus_{i=1}^{4}\mathcal{P}_{3}(\mathcal{P}_{3})^{T}\oplus\bigoplus_{i=1}^{8}\mathcal{P}_{2}(\mathcal{P}_{2})^{T}\oplus\mathcal{Z}$

and that

$\mathcal{P}_{2}(\mathcal{P}_{2})^{T}=\begin{bmatrix}q^{8}+1&1-q^{4}\\\
1-q^{4}&2\end{bmatrix}.$

contains non-trivial values, completing the proof. ∎

Note that the remaining entries of $\mathcal{D}$ can be easily calculated from
section 7.6, and that $\mathcal{P}_{4}(\mathcal{P}_{4})^{T}$ is displayed in
section 3.4.2.

##  6 Algebraic Proofs

###  6.1 $W$ is an irreducible representation

We begin by proving Proposition 5. First, we must first establish a few
definitions.

######  Definition 9.

We define $P$ to be the weight lattice, with $\lambda\in P$ called a weight. A
vector $v_{\lambda}$ is called a highest weight vector if
$E_{1}v_{\lambda}=E_{2}v_{\lambda}=E_{3}v_{\lambda}=0$. Then, a
$\mathcal{U}_{q}(\mathfrak{so}_{6})$-module $M$ is a highest weight module if
$M=\mathcal{U}_{q}(\mathfrak{so}_{6})v_{\lambda}$, and will be denoted
$M=V(\lambda)$.

Since $\mathcal{U}_{q}(\mathfrak{so}_{6})$ can be triangularly decomposed into
its three subalgebras generated by $\\{E_{i}\\}$, $\\{q^{H_{i}}\\}$, and
$\\{F_{i}\\}$ (sometimes known as the Cartan decomposition), $M$ can be
computed by multiplying compositions of $F_{i}$s with a highest weight vector
$v_{\lambda}$ as described in [HK02]. Note that, in the fundamental
representation of $\mathcal{U}_{q}(\mathfrak{so}_{6})$, each $E_{i}$ has an
empty first column and thus $e_{1}\in\mathbb{R}^{6}$ is a highest weight
vector. Denote $\lambda=\lambda_{1}L_{1}+\lambda_{2}L_{2}+\lambda_{3}L_{3}$.
Since $e_{1}$ has weight $L_{1}$, we can use the following equation from
[FH04] to determine the dimension of $V(L_{1})$:

$\displaystyle\text{dim}(V(\lambda))=\frac{1}{12}\prod_{1\leq i<j\leq
3}(\lambda_{i}-\lambda_{j}+j-i)(\lambda_{i}+\lambda_{j}+6-i-j).$ (1)

Therefore $V(L_{1})$ is a 6-dimensional
$\mathcal{U}_{q}(\mathfrak{so}_{6})$-module, and $V(L_{1})$ is thus the entire
fundamental representation.

We will prove multiple statements partially using the theory of crystal bases
as stated in [HK02]. We use the correspondence between a weight module
$V(\lambda)$ and its crystal base
$(\mathcal{L}(\lambda),\mathcal{B}(\lambda))$, formed of the crystal lattice
and crystal basis respectively. We also use the tensor product rule, Theorem
4.4.1 on pg 83 of [HK02].

######  Lemma 7.

$V(L_{1})\otimes V(L_{1})\cong V(2L_{1})\oplus V(L_{1}+L_{2})\oplus V(0)$.

###### Proof.

First, define $1=e_{1}$, $2=e_{2}$, $3=e_{3}$, $\bar{1}=e_{4}$,
$\bar{2}=-e_{5}$, $\bar{3}=e_{6}$. Then, note that $V(L_{1})$ has crystal base
as shown in Figure 3, with arrow index $i$ corresponding to Kashiwara operator
$\tilde{f_{i}}$.

Figure 3: Crystal graph of fundamental representation
$\mathcal{U}_{q}(\mathfrak{so}_{6})$

Then, by the tensor product rule, we can draw the crystal graph of $V\otimes
V$ as shown in Figure 5.

Figure 4: Crystal graph of $V(2L_{1})$ in the representation $V\otimes V$
Figure 5: Crystal graph of $\mathcal{U}_{q}(\mathfrak{so}_{6})$ in the
representation $V\otimes V$

Thus, by reading off the disjoint connected components in the crystal graph of
Figure 5, we see that

$V(L_{1})\otimes V(L_{1})\cong V(2L_{1})\oplus V(L_{1}+L_{2})\oplus V(0).$

Note that, by applying the corresponding $F_{i}$, we can trace a path of
arrows to each vertex in $W$ and obtain the corresponding basis vector of $W$.
This process is used to generate the compositions of $F_{i}$s shown in Lemma
8. ∎

Now, we move on to constructing $V(2L_{1})$ explicitly.

######  Lemma 8.

The $\mathcal{U}_{q}(\mathfrak{so}_{6})$-module $V(2L_{1})$ is 20-dimensional
with basis

$\displaystyle\begin{split}\\{e_{1}\otimes e_{1}&,\text{\hskip
5.69046pt}e_{1}\otimes e_{2}+q^{-1}e_{2}\otimes e_{1},\text{\hskip
5.69046pt}e_{1}\otimes e_{3}+q^{-1}e_{3}\otimes e_{1},\\\ &-e_{1}\otimes
e_{4}+q^{-1}e_{2}\otimes e_{5}-q^{-2}e_{4}\otimes e_{1}+q^{-1}e_{5}\otimes
e_{2},\text{\hskip 5.69046pt}e_{1}\otimes e_{5}+q^{-1}e_{5}\otimes e_{1},\\\
&-e_{1}\otimes e_{6}-q^{-1}e_{6}\otimes e_{1},\text{\hskip
5.69046pt}(q+q^{-1})e_{2}\otimes e_{2},\text{\hskip 5.69046pt}e_{2}\otimes
e_{3}+q^{-1}e_{3}\otimes e_{2},\\\ &-e_{2}\otimes e_{4}-q^{-1}e_{4}\otimes
e_{2},\text{\hskip 5.69046pt}e_{2}\otimes e_{5}-q^{-1}e_{3}\otimes
e_{6}+q^{-2}e_{5}\otimes e_{2}-(q^{2}+1)e_{6}\otimes e_{3},\\\ &-e_{2}\otimes
e_{6}-q^{{}_{1}}e_{6}\otimes e_{2},(q^{-1}+q)e_{3}\otimes e_{3},\\\
&-(q^{2}+2+q^{-2})e_{3}\otimes e_{4}-(q+2q^{-1}+q^{-3})e_{4}\otimes e_{3},\\\
&(q^{2}+2+q^{-2})e_{3}\otimes e_{5}+(q+2q^{-1}+q^{-3})e_{5}\otimes
e_{3},\text{\hskip 5.69046pt}q^{-2}e_{3}\otimes e_{6}-q^{-1}e_{6}\otimes
e_{3},\\\ &(q^{4}+4q^{2}+6+4q^{-2}+q^{-4})e_{4}\otimes e_{4},\\\
&-(q^{2}+3+3q^{-2}+q^{-4})e_{4}\otimes
e_{5}-(q^{3}+3q+3q^{-1}+q^{-3})e_{5}\otimes e_{4}\\\ &(q^{-2}+1)e_{4}\otimes
e_{6}+(q+q^{-1})e_{6}\otimes e_{4},(q^{2}+2+q^{-2})e_{5}\otimes e_{5},\\\
&-(q^{-2}+1)e_{5}\otimes e_{6}-(q^{-1}+q)e_{6}\otimes
e_{5},(q^{-1}+q)e_{6}\otimes e_{6}\\}\end{split}$ (2)

###### Proof.

Define $\pi_{V}(F_{i})$ to be the projection of $\Delta(F_{i})$ into
$V=\mathbb{R}^{6}\otimes\mathbb{R}^{6}$. The tensor product assumes the role
of the Kronecker product in this 36-dimensional representation $V$ of
$\mathcal{U}_{q}(\mathfrak{so}_{6})$. Note that $\\{\pi_{V}(E_{i})\\}$
annihilates $e_{1}\otimes e_{1}$, making $e_{1}\otimes e_{1}$ a highest weight
vector with highest weight $2L_{1}$. To generate $V(2L_{1})$, we use Lemma 7
to obtain the correct compositions of $F_{i}$s applied to $e_{1}\otimes e_{1}$
to form each basis vector: we apply $F_{i}$s corresponding to the arrows in
the path from $e_{1}\otimes e_{1}$ to each $e_{i}\otimes e_{j}$ contained in
the upper right-hand cycle as displayed Figure 5. A few examples are done
below, and the rest are left to the reader. The notation $F_{ijk}$ is used to
shorten $F_{i}F_{j}F_{k}$.

$\displaystyle\pi_{V}(F_{1})(e_{1}\otimes e_{1})$ $\displaystyle=(1\otimes
F_{1})(e_{1}\otimes e_{1})+(F_{1}\otimes q^{-H_{1}})(e_{1}\otimes e_{1})$
$\displaystyle=e_{1}\otimes e_{2}+q^{-1}e_{2}\otimes e_{1}$
$\displaystyle\pi_{V}(F_{11})\coloneqq\pi_{V}(F_{1}F_{1})(e_{1}\otimes e_{1})$
$\displaystyle=(q+q^{-1})e_{2}\otimes e_{2}$
$\displaystyle\pi_{V}(F_{11332211})(e_{1}\otimes e_{1})$
$\displaystyle=(q^{4}+4q^{2}+6+4q^{-2}+q^{-4})e_{4}\otimes e_{4}$

After complete evaluation, we obtain the basis shown in Equation 2 for
$V(2L_{1})$. Note that the number of basis vectors generated agrees with the
dimension formula in Equation 1. ∎

Using Lemma 7 and Lemma 8, we are now ready to prove Proposition 5. See 5

###### Proof.

Note that the basis vectors of $V(2L_{1})$ span $W$ and are linearly
independent. Since finite-dimensional highest weight modules of
$\mathcal{U}_{q}(\mathfrak{so}_{6})$ are irreducible representations, $W$ is
an irreducible representation. ∎

As a sanity check, if $C$ is a central element of
$\mathcal{U}_{q}(\mathfrak{so}_{6})$, we will see that $\pi_{V}(C)|_{W}$ is a
constant times $\mathrm{Id}_{20\times 20}$.

######  Example 1.

In 2020, Kuan, Landry, Lin, Park, and Zhou [KLLPZ20] found the following
central element of $\mathcal{U}_{q}(\mathfrak{so}_{6})$ (note that the
fundamental representation of $\mathfrak{so}_{2n}$ presented in that paper is
slightly incorrect, leading to some sign errors in its central elements
[Kua24]. The $E_{3}$ used in that paper is $-E_{3}$ under our notation). We
write $F_{ij}$ to shorten $F_{i}F_{j}$ and $r$ to represent $q+q^{-1}$.

$\displaystyle C$
$\displaystyle=q^{-4-2H_{1}-H_{2}-H_{3}}+q^{-2-H_{2}-H_{3}}+q^{H_{2}-H_{3}}+q^{H_{3}-H_{2}}+q^{2+H_{2}+H_{3}}$
$\displaystyle+q^{4+2H_{1}+H_{2}+H_{3}}+\frac{r^{2}}{q^{3}}F_{1}q^{-H_{1}-H_{2}-H_{3}}E_{1}+\frac{r^{2}}{q}F_{2}q^{-H_{3}}E_{2}$
$\displaystyle+\frac{r^{2}}{q}F_{3}q^{-H_{2}}E_{3}+r^{2}qF_{2}q^{H_{3}}E_{2}+r^{2}qF_{3}q^{H_{2}}E_{3}+r^{2}q^{3}F_{1}q^{H_{1}+H_{2}+H_{3}}E_{1}$
$\displaystyle+\frac{r^{2}}{q^{3}}(qF_{12}-F_{21})q^{-H_{1}-H_{3}}(qE_{21}-E_{12})$
$\displaystyle+\frac{r^{2}}{q^{3}}(qF_{13}-F_{31})q^{-H_{1}-H_{2}}(qE_{31}-E_{13})$
$\displaystyle+r^{2}q(qF_{21}-F_{12})q^{H_{1}+H_{3}}(qE_{12}-E_{21})$
$\displaystyle+r^{2}q(qF_{31}-F_{13})q^{H_{1}+H_{2}}(qE_{13}-E_{31})$
$\displaystyle+\frac{r^{2}}{q^{3}}(q^{2}F_{123}-qF_{213}-qF_{312}+F_{231})q^{-H_{1}}(q^{2}E_{231}-qE_{312}-qE_{213}+E_{123})$
$\displaystyle+\frac{r^{2}}{q}(q^{2}F_{231}-qF_{312}-qF_{213}+F_{123})q^{H_{1}}(q^{2}E_{123}-qE_{213}-qE_{312}+E_{231})$
$\displaystyle+\frac{r^{4}}{q^{2}}((q^{2}+1)F_{1231}-qF_{1312}-qF_{2131})((q^{2}+1)E_{1231}-qE_{1312}-qE_{2131})$
$\displaystyle+r^{4}F_{2}F_{3}E_{2}E_{3}$

By some tedious $36\times 36$ matrix computations, one can find $\pi_{V}(C)$.
Then, construct the change of basis matrix $Y$ with first 20 columns being the
above-constructed basis of $W$ and the last 16 columns extending to a basis of
$V$. Then, the matrix $Y^{-1}\pi_{V}(C)Y$ has an upper left $20\times 20$
block of $(q^{8}+q^{2}+2+q^{-2}+q^{-8})\mathrm{Id}_{20\times
20}=\pi_{V}(C)|_{W}$.

###  6.2 Decompositions of $W\otimes W$

In this section, we analyze the structure of $W\otimes W$. This will allow us
to deduce properties of $\pi_{W\otimes W}(C)$ in Section 6.3.

####  6.2.1 Decomposition of $W\otimes W$ into irreducible representations

In order to decompose $W\otimes W$ into a direct sum of irreducible
representations, we would like to use some analog of the tensor product rule
as done in Lemma 7. However, since our current crystal graph of $W$ is three-
dimensional, creating the crystal graph of $W\otimes W$ is very difficult.
Therefore, we introduce Young tableaux to be able to visualize $W\otimes W$,
and use the corresponding tensor product rule to decompose $W\otimes W$. An
interested reader may reference [HK02] chapters 7 and 8 for all appropriate
background.

The chain of correspondence is as follows: $\mathcal{B}(2L_{1})$ describes
$W$, as seen in Lemma 7, and

$\mathcal{B}\left(\vbox{\hbox{\text{ \leavevmode\hbox to29.25pt{\vbox
to15.03pt{\pgfpicture\makeatletter\hbox{\hskip 0.4pt\lower-7.51318pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{{}} {}{}{{}} {}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-7.11319pt}\pgfsys@lineto{28.45276pt}{-7.11319pt}\pgfsys@lineto{28.45276pt}{7.11319pt}\pgfsys@lineto{0.0pt}{7.11319pt}\pgfsys@lineto{0.0pt}{-7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{-7.11319pt}\pgfsys@lineto{14.22638pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}
}}}\right)\coloneqq\mathcal{B}(\mathcal{Y})$

describes $\mathcal{B}(2L_{1})$. Thus, the decomposition of
$\mathcal{B}(\mathcal{Y})\otimes\mathcal{B}(\mathcal{Y})$ dictates the
structure of $W\otimes W$. This brings us to Lemma 9.

######  Lemma 9.

$W\otimes W$ decomposes as follows into irreducible representations:

$W\otimes W\cong V(4L_{1})\oplus V(3L_{1}+L_{2})\oplus V(2L_{1}+2L_{2})\oplus
V(2L_{1})\oplus V(L_{1}+L_{2})\oplus V(0).$

As expected, the sum of the dimensions of these irreducible representations is
$400=\mathrm{dim}(W\otimes W)$.

###### Proof.

As can be read off of Figure 5,

$\mathcal{B}(2L_{1})=\left\\{\vbox{\hbox{\text{ \leavevmode\hbox
to31.03pt{\vbox to18.23pt{\pgfpicture\makeatletter\hbox{\hskip
0.98105pt\lower-9.51695pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{{}} {}{}{{}} {}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-7.11319pt}\pgfsys@lineto{28.45276pt}{-7.11319pt}\pgfsys@lineto{28.45276pt}{7.11319pt}\pgfsys@lineto{0.0pt}{7.11319pt}\pgfsys@lineto{0.0pt}{-7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{-7.11319pt}\pgfsys@lineto{14.22638pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{ }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.47025pt}{-3.57549pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$a$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{ }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{19.19374pt}{-3.47221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}
}}}\mathrel{}\middle|\mathrel{}b\succeq a\right\\}$

We choose the English notation and convention of reading Young diagrams down
columns from right to left. By the tensor product rule for Young diagrams
(Theorem 8.6.6, pg 206 of [HK02]),

$\mathcal{B}(\mathcal{Y})\otimes\mathcal{B}(\mathcal{Y})\cong\bigoplus_{b_{1}\otimes
b_{2}\in W}\mathcal{B}(\mathcal{Y}[b_{1},b_{2}]).$

To compute the right hand side, note that:

$\displaystyle\mathcal{B}(\mathcal{Y}[v_{1},v_{1}])$
$\displaystyle=\mathcal{B}\left(\vbox{\hbox{\text{ \leavevmode\hbox
to57.71pt{\vbox to15.03pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-7.51318pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{{}} {}{}{{}} {}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-7.11319pt}\pgfsys@lineto{56.90552pt}{-7.11319pt}\pgfsys@lineto{56.90552pt}{7.11319pt}\pgfsys@lineto{0.0pt}{7.11319pt}\pgfsys@lineto{0.0pt}{-7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{-7.11319pt}\pgfsys@lineto{14.22638pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{-7.11319pt}\pgfsys@lineto{28.45276pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{42.67914pt}{-7.11319pt}\pgfsys@lineto{42.67914pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }}}\right)$
$\displaystyle\mathcal{B}(\mathcal{Y}[v_{2},v_{1}])$
$\displaystyle=\mathcal{B}\left(\vbox{\hbox{\text{ \leavevmode\hbox
to43.48pt{\vbox to29.25pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-21.73956pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{{}} {}{}{{}} {}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-7.11319pt}\pgfsys@lineto{42.67914pt}{-7.11319pt}\pgfsys@lineto{42.67914pt}{7.11319pt}\pgfsys@lineto{0.0pt}{7.11319pt}\pgfsys@lineto{0.0pt}{-7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{-7.11319pt}\pgfsys@lineto{14.22638pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{28.45276pt}{-7.11319pt}\pgfsys@lineto{28.45276pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-21.33957pt}\pgfsys@lineto{14.22638pt}{-21.33957pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{7.11319pt}\pgfsys@lineto{0.0pt}{-21.33957pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{7.11319pt}\pgfsys@lineto{14.22638pt}{-21.33957pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }}}\right)$
$\displaystyle\mathcal{B}(\mathcal{Y}[v_{2},v_{2}])$
$\displaystyle=\mathcal{B}\left(\vbox{\hbox{\text{ \leavevmode\hbox
to29.25pt{\vbox to29.25pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-21.73956pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{{}} {}{}{{}} {}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{7.11319pt}\pgfsys@lineto{0.0pt}{-21.33957pt}\pgfsys@lineto{28.45276pt}{-21.33957pt}\pgfsys@lineto{28.45276pt}{7.11319pt}\pgfsys@lineto{0.0pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-7.11319pt}\pgfsys@lineto{28.45276pt}{-7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{7.11319pt}\pgfsys@lineto{14.22638pt}{-21.33957pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }}}\right)$
$\displaystyle\mathcal{B}(\mathcal{Y}[v_{\bar{1}},v_{1}])$
$\displaystyle=\mathcal{B}\left(\vbox{\hbox{\text{ \leavevmode\hbox
to29.25pt{\vbox to15.03pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-7.51318pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{{}} {}{}{{}} {}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-7.11319pt}\pgfsys@lineto{28.45276pt}{-7.11319pt}\pgfsys@lineto{28.45276pt}{7.11319pt}\pgfsys@lineto{0.0pt}{7.11319pt}\pgfsys@lineto{0.0pt}{-7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{14.22638pt}{-7.11319pt}\pgfsys@lineto{14.22638pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }}}\right)$
$\displaystyle\mathcal{B}(\mathcal{Y}[v_{\bar{1}},v_{2}])$
$\displaystyle=\mathcal{B}\left(\vbox{\hbox{\text{ \leavevmode\hbox
to15.03pt{\vbox to29.25pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-21.73956pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{{}} {}{}{{}} {}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{7.11319pt}\pgfsys@lineto{0.0pt}{-21.33957pt}\pgfsys@lineto{14.22638pt}{-21.33957pt}\pgfsys@lineto{14.22638pt}{7.11319pt}\pgfsys@lineto{0.0pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-7.11319pt}\pgfsys@lineto{14.22638pt}{-7.11319pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }}}\right)$
$\displaystyle\mathcal{B}(\mathcal{Y}[v_{\bar{1}},v_{\bar{1}}])$
$\displaystyle=\mathcal{B}(\varnothing).$

Thus, we have that

$\mathcal{B}(\mathcal{Y})\otimes\mathcal{B}(\mathcal{Y})\cong\mathcal{B}\left(\vbox{\hbox{\text{
\leavevmode\hbox to57.71pt{\vbox
to15.03pt{\pgfpicture\makeatletter\hbox{\hskip 0.4pt\lower-7.51318pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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}}}\right)\oplus\mathcal{B}\left(\vbox{\hbox{\text{ \leavevmode\hbox
to29.25pt{\vbox to29.25pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}
}}}\right)\oplus\mathcal{B}\left(\vbox{\hbox{\text{ \leavevmode\hbox
to29.25pt{\vbox to15.03pt{\pgfpicture\makeatletter\hbox{\hskip
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}}}\right)\oplus\mathcal{B}\left(\vbox{\hbox{\text{ \leavevmode\hbox
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}}}\right)\oplus\mathcal{B}(\varnothing)$

which implies that

$W\otimes W\cong V(4L_{1})\oplus V(3L_{1}+L_{2})\oplus V(2L_{1}+2L_{2})\oplus
V(2L_{1})\oplus V(L_{1}+L_{2})\oplus V(0).$

As for the equality of dimensions, note that

$\displaystyle\mathrm{dim}(W\otimes W)$ $\displaystyle=400$
$\displaystyle=105+175+84+20+15+1$
$\displaystyle=\sum_{\lambda}\mathrm{dim}V(\lambda)$

using the dimension formula Equation 1. The order of dimensions in the above
sum is the same order in which the irreducible representations are listed in
the decomposition above. ∎

We proceed to investigating our next trait of $W\otimes W$.

####  6.2.2 Decomposition of $W\otimes W$ into weight spaces

By definition, each highest-weight $\mathcal{U}_{q}(\mathfrak{so}_{6})$-module
has a weight space decomposition. Therefore, we can express $W\otimes W$ as a
direct sum of weight spaces using the Young tableaux found in Lemma 9. We are
most interested in the dimension of each weight space, which will help deduce
the shape of $\pi_{W\otimes W}(C)$.

######  Lemma 10.

$W\otimes W$ admits a decomposition into eighty-five weight spaces, one with
dimension $22$, twelve with dimension $12$, six with dimension $8$, twenty-
four with dimension $4$, twelve with dimension $3$, twenty-four with dimension
$2$, and six with dimension $1$.

###### Proof.

For ease of notation, define

$\displaystyle L_{-i}\coloneqq-L_{i}$

For $1\leq i\leq 3$. Recall that the weights associated with the standard
basis vectors of the fundamental representation $\mathbb{R}^{6}$ of
$\mathfrak{so}_{6}$ are as follows:

$e_{1}$ | $L_{1}$
---|---
$e_{2}$ | $L_{2}$
$e_{3}$ | $L_{3}$
$e_{4}$ | $L_{-1}$
$e_{5}$ | $L_{-2}$
$e_{6}$ | $L_{-3}$

It is a property of weights that if $v$ and $v^{\prime}$ are vectors in a
representation $V$ with respective weights $\lambda$ and $\lambda^{\prime}$,
then $v\otimes v^{\prime}$ is a vector in $V\otimes V$ with weight
$\lambda+\lambda^{\prime}$. Thus, we can observe from the crystal graph of $W$
that the weights of basis vectors in $W$ take the following forms, for
$i,j\in\\{1,2,3,-1,-2,-3\\}$ with $i,-i,j,-j$ all unique:

$\displaystyle\\{0,2L_{i},L_{i}+L_{j}\\}$

Note that two basis vectors in $W$ ($e_{1}\otimes e_{4}$ and $e_{2}\otimes
e_{5}$) have weight $0$, and each of the other eighteen basis vectors in $W$
has a unique weight corresponding to one of the forms above. By considering
possible sums of the weights above, the weights of vectors in $W\otimes W$
take one of the following forms, for $i,j,k\in\\{1,2,3,-1,-2,-3\\}$ with
$i,-i,j,-j,k,-k$ all unique:

$\displaystyle\\{0,L_{i}+L_{j},2L_{i},2L_{i}+L_{j}+L_{k},2L_{i}+2L_{j},3L_{i}+L_{j},4L_{i}\\}$

Now we count the number of weight spaces corresponding to each of the forms
above.

  * •

There is evidently one weight space with weight $0$.

  * •

There are $6\cdot 4=24$ possible pairs $(i,j)$. Since the pairs $(i,j)$ and
$(j,i)$ yield the same weight, there are twelve weight spaces with a weight of
the form $L_{i}+L_{j}$.

  * •

Since there are $6$ possible values for $i$, there are six weight spaces with
a weight of the form $2L_{i}$.

  * •

There are $6\cdot 4\cdot 2=48$ possible triplets $(i,j,k)$. Since the triplets
$(i,j,k)$ and $(i,k,j)$ yield the same weight, there are twenty-four weight
spaces with a weight of the form $2L_{i}+L_{j}+L_{k}$.

  * •

There are $6\cdot 4=24$ possible pairs $(i,j)$. Since the pairs $(i,j)$ and
$(j,i)$ yield the same weight, there are twelve weight spaces with a weight of
the form $2L_{i}+2L_{j}$.

  * •

There are $6\cdot 4=24$ possible pairs $(i,j)$. Since each pair yields a
distinct weight, there are twenty-four weight spaces with a weight of the form
$3L_{i}+L_{j}$.

  * •

Since there are $6$ possible values for $i$, there are six weight spaces with
a weight of the form $4L_{i}$.

Now we count the dimension of each type of weight space. We can do so by
counting the number of basis vectors $w_{1}\otimes w_{2}\in W\otimes W$ such
that the sum of the weights corresponding to $w_{1}$ and $w_{2}$ in $W$ equals
the desired weight. Denote the weights in $W$ corresponding to $w_{1}$ and
$w_{2}$ by $\lambda_{1}$ and $\lambda_{2}$, respectively.

  * •

If $\lambda_{1}+\lambda_{2}=0$, then either

    * $*$

$\lambda_{1}=0$. Then $\lambda_{2}=0$ as well. There are thus two options for
each of $w_{1}$ and $w_{2}$, and thus four total possibilities in this case.

    * $*$

$\lambda_{1}\neq 0$. Then there are eighteen possible values for $w_{1}$ with
nonzero weights, and exactly one possible value for $w_{2}$ such that
$\lambda_{2}=-\lambda_{1}$.

Thus, the dimension of the weight space of $W\otimes W$ with weight $0$ is
twenty-two.

  * •

If $\lambda_{1}+\lambda_{2}$ takes the form $L_{i}+L_{j}$, then either

    * $*$

$\lambda_{1}=L_{i}+L_{j}$ and $\lambda_{2}=0$; or $\lambda_{1}=0$ and
$\lambda_{2}=L_{i}+L_{j}$. Since the weight space of $W$ with weight $0$ has
dimension $2$, there are four total possibilities in this case.

    * $*$

$\lambda_{1}=L_{i}-L_{j}$ and $\lambda_{2}=2L_{j}$; or $\lambda_{1}=2L_{j}$
and $\lambda_{2}=L_{i}-L_{j}$; or $\lambda_{1}=-L_{i}+L_{j}$ and
$\lambda_{2}=2L_{i}$; or $\lambda_{1}=2L_{i}$ and $\lambda_{2}=-L_{i}+L_{j}$.
There are four total possibilities in this case.

    * $*$

$\lambda_{1}=L_{i}+L_{k}$ and $\lambda_{2}=-L_{j}-L_{k}$; or
$\lambda_{1}=-L_{j}-L_{k}$ and $\lambda_{2}=L_{i}+L_{k}$. Since there are two
possibilities for $k$ once $i$ and $j$ are fixed, there are four total
possibilities in this case.

Thus, each weight space of $W\otimes W$ with a weight of the form
$L_{i}+L_{j}$ has dimension twelve.

  * •

If $\lambda_{1}+\lambda_{2}$ takes the form $2L_{i}$, then either

    * $*$

$\lambda_{1}=2L_{i}$ and $\lambda_{2}=0$; or $\lambda_{1}=0$ and
$\lambda_{2}=2L_{i}$. Since the weight space of $W$ with weight $0$ has
dimension $2$, there are four total possibilities in this case.

    * $*$

$\lambda_{1}=L_{i}+L_{j}$ and $\lambda_{2}=L_{i}-L_{j}$. Since there are four
possibilities for $j$ once $i$ is fixed, there are four total possibilities in
this case.

Thus, each weight space of $W\otimes W$ with a weight of the form $2L_{i}$ has
dimension eight.

  * •

If $\lambda_{1}+\lambda_{2}$ takes the form $2L_{i}+L_{j}+L_{k}$, then either

    * $*$

$\lambda_{1}=2L_{i}$ and $\lambda_{2}=L_{j}+L_{k}$

    * $*$

$\lambda_{1}=L_{j}+L_{k}$ and $\lambda_{2}=2L_{i}$

    * $*$

$\lambda_{1}=L_{i}+L_{j}$ and $\lambda_{2}=L_{i}+L_{k}$

    * $*$

$\lambda_{1}=L_{i}+L_{k}$ and $\lambda_{2}=L_{i}+L_{j}$

Thus, each weight space of $W\otimes W$ with a weight of the form
$2L_{i}+L_{j}+L_{k}$ has dimension four.

  * •

If $\lambda_{1}+\lambda_{2}$ takes the form $2L_{i}+2L_{j}$, then either

    * $*$

$\lambda_{1}=2L_{i}$ and $\lambda_{2}=2L_{j}$

    * $*$

$\lambda_{1}=2L_{j}$ and $\lambda_{2}=2L_{i}$

    * $*$

$\lambda_{1}=L_{i}+L_{j}$ and $\lambda_{2}=L_{i}+L_{j}$

Thus, each weight space of $W\otimes W$ with a weight of the form
$2L_{i}+2L_{j}$ has dimension three.

  * •

If $\lambda_{1}+\lambda_{2}$ takes the form $3L_{i}+L_{j}$, then either

    * $*$

$\lambda_{1}=2L_{i}$ and $\lambda_{2}=L_{i}+L_{j}$

    * $*$

$\lambda_{1}=L_{i}+L_{j}$ and $\lambda_{2}=2L_{i}$

Thus, each weight space of $W\otimes W$ with a weight of the form
$3L_{i}+L_{j}$ has dimension two.

  * •

If $\lambda_{1}+\lambda_{2}$ takes the form $4L_{i}$, then
$\lambda_{1}=2L_{i}$ and $\lambda_{2}=2L_{i}$, so each weight space of
$W\otimes W$ with a weight of the form $4L_{i}$ has dimension one.

∎

###### Proof of Proposition 6.

This is a direct consequence of Lemma 9 and Lemma 10. ∎

###  6.3 Block form of $\pi_{W\otimes W}(C)$

####  6.3.1 Block sizes of $\pi_{W\otimes W}(C)$

Since $W\otimes W$ can be decomposed into a direct sum of weight spaces, any
matrix in $W\otimes W$ can be expressed as a direct sum of matrices in the
weight spaces of $W\otimes W$. This yields Proposition 7, restated as the
following corollary of Lemma 10:

######  Corollary 1.

$\pi_{W\otimes W}(C)$ admits a block diagonal decomposition into a direct sum
of one $22\times 22$ block, twelve $12\times 12$ blocks, six $8\times 8$
blocks, twenty-four $4\times 4$ blocks, twelve $3\times 3$ blocks, twenty-four
$2\times 2$ blocks, and six $1\times 1$ blocks. The basis in respect to which
results in $\pi_{W\otimes W}(C)$ blocked this way is included in Section 7.7.

######  Remark 4.

Note that since $\mathfrak{so}_{6}$ and $\mathfrak{sl}_{4}$ are isomorphic as
Lie algebras, computing the Kostka numbers for $\mathfrak{sl}_{4}$ would also
yield the block sizes found above. For example, the $22\times 22$ block can be
calculated by $22=3+6+7+2+3+1$ where each summand appears as $K_{\lambda\mu}$
for the following choices of $\lambda$ and $\mu.$

$\displaystyle K_{\lambda\mu}=7,$
$\displaystyle\quad\lambda=(4,3,1,0),\mu=(2,2,2,2)$ $\displaystyle
K_{\lambda\mu}=6,$ $\displaystyle\quad\lambda=(4,2,2,0),\mu=(2,2,2,2)$
$\displaystyle K_{\lambda\mu}=3,$
$\displaystyle\quad\lambda=(4,4,0,0),\mu=(2,2,2,2)$ $\displaystyle
K_{\lambda\mu}=2,$ $\displaystyle\quad\lambda=(2,2,0,0),\mu=(1,1,1,1)$
$\displaystyle K_{\lambda\mu}=3,$
$\displaystyle\quad\lambda=(2,1,1,0),\mu=(1,1,1,1)$ $\displaystyle
K_{\lambda\mu}=1,$ $\displaystyle\quad\lambda=(0,0,0,0),\mu=(0,0,0,0)$

####  6.3.2 Entries of $\pi_{W\otimes W}(C)$

Since the coproduct $\Delta$ is a homomorphism,
$\pi_{\mathbb{R}^{6}\otimes\mathbb{R}^{6}\otimes\mathbb{R}^{6}\otimes\mathbb{R}^{6}}(C)$
can be computed from Definition 7 by computing $\Delta^{3}(E_{i})$,
$\Delta^{3}(F_{i})$, and $\Delta^{3}(q^{H_{i}})$ for $1\leq i\leq 3$ and
substituting them for $E_{i}$, $F_{i}$, and $q^{H_{i}}$, respectively, in the
equation in Example 1. By Definition 6, for all $1\leq i\leq 3$,

$\displaystyle\Delta^{3}(E_{i})=E_{i}\otimes 1\otimes 1\otimes
1+q^{H_{i}}\otimes E_{i}\otimes 1\otimes 1+q^{H_{1}}\otimes q^{H_{1}}\otimes
E_{i}\otimes 1+q^{H_{1}}\otimes q^{H_{1}}\otimes q^{H_{1}}\otimes E_{1}$
$\displaystyle\Delta^{3}(F_{i})=1\otimes 1\otimes 1\otimes F_{i}+1\otimes
1\otimes F_{i}\otimes q^{-H_{i}}+1\otimes F_{i}\otimes q^{-H_{i}}\otimes
q^{-H_{i}}+F_{i}\otimes q^{-H_{i}}\otimes q^{-H_{i}}\otimes q^{-H_{i}}$
$\displaystyle\Delta^{3}(q^{H_{i}})=q^{H_{i}}\otimes q^{H_{i}}\otimes
q^{H_{i}}\otimes q^{H_{i}}$

Denote the matrix resulting from the above procedure by
$\pi_{\mathbb{R}^{1296}}(C)$.

Denote the basis of $W$ established in the proof of Lemma 8 by
$B_{W}\coloneqq(w_{1},w_{2},\ldots,w_{20})$. Recall the $36\times 36$ change
of basis matrix $Y$ defined in Example 1, and note that a change of basis by
$Y$ maps $w_{i}$ to the standard basis vector $e_{i}$ of
$\mathbb{R}^{6}\otimes\mathbb{R}^{6}$ for $1\leq i\leq 20$. Thus, a change of
basis by $Y\otimes Y$ in
$\mathbb{R}^{6}\otimes\mathbb{R}^{6}\otimes\mathbb{R}^{6}\otimes\mathbb{R}^{6}$
maps $B_{W}\otimes B_{W}$ to $\\{e_{i}\otimes e_{j}:1\leq i,j\leq 20\\}$.
Since for any $1\leq i,j\leq 36$, $e_{i}\otimes e_{j}$ is the $(36(i-1)+j)$th
basis vector of
$\mathbb{R}^{6}\otimes\mathbb{R}^{6}\otimes\mathbb{R}^{6}\otimes\mathbb{R}^{6}$,
$\pi_{W\otimes W}(C)=\pi_{\mathbb{R}^{1296}}(C)|_{W\otimes W}$ can be found by
taking the $(36(i-1)+j)$th row and column of $(Y^{-1}\otimes
Y^{-1})\pi_{\mathbb{R}^{1296}}(C)(Y\otimes Y)$ for all $1\leq i,j\leq 20$.
Since the $e_{i}\otimes e_{j}$th row or column of $\pi_{W\otimes W}(C)$
corresponds directly to the $(36(i-1)+j)$th row or column of $(Y^{-1}\otimes
Y^{-1})\pi_{\mathbb{R}^{1296}}(C)(Y\otimes Y)$, we only need to determine a
few of the values of $(Y^{-1}\otimes
Y^{-1})\pi_{\mathbb{R}^{1296}}(C)(Y\otimes Y)$ to compute some of the blocks
of $\pi_{W\otimes W}(C)$. Explicit values of $\pi_{W\otimes
W}(E_{i}),\pi_{W\otimes W}(F_{i}),\pi_{W\otimes W}(q^{H_{i}})$ for $1\leq
i\leq 3$ are given in Section 7.10

####  6.3.3 Eigenvalues of $\pi_{W\otimes W}(C)$

Given $W\otimes W$’s decomposition into highest weight
$\mathcal{U}_{q}(\mathfrak{so}_{6})$-modules, we are also able to deduce the
eigenvalues of $\pi_{W\otimes W}(C)$.

######  Corollary 2.

$\pi_{W\otimes W}(C)$ has eigenvalues of

$\displaystyle q^{12}+q^{2}+2+q^{-2}+q^{-12},$ $\displaystyle
q^{10}+q^{4}+2+q^{-4}+q^{-10},$ $\displaystyle q^{8}+q^{6}+2+q^{-6}+q^{-8},$
$\displaystyle q^{8}+q^{2}+2+q^{-2}+q^{-8}$ $\displaystyle
q^{6}+q^{4}+2+q^{-4}+q^{-6},$ $\displaystyle q^{4}+q^{2}+2+q^{-2}+q^{-4}$

with respective multiplicities of $105$, $175$, $84$, $20$, $15$, $1$.

###### Proof.

By Lemma 9,

$W\otimes W\cong V(4L_{1})\oplus V(3L_{1}+L_{2})\oplus V(2L_{1}+2L_{2})\oplus
V(2L_{1})\oplus V(L_{1}+L_{2})\oplus V(0)$

with respective dimensions of $105$, $175$, $84$, $20$, $15$, $1$. Since each
of these representations is irreducible,

$\pi_{W\otimes W}(C)|_{V(\lambda)}=c\mathrm{Id}$

for some $c\in\mathbb{R}[q,q^{-1}]$. Also, by construction of each
$V(\lambda)$, $v_{\lambda}$ is a basis vector of $V(\lambda)$. Therefore, in
changing the basis of $W\otimes W$ to compute $\pi_{W\otimes
W}(C)|_{V(\lambda)}$, we will have

$\pi_{W\otimes W}(C)|_{V(\lambda)}(1,1)=\langle v_{\lambda}|\pi_{W\otimes
W}(C)|_{V(\lambda)}|v_{\lambda}\rangle$

since we may express each highest weight vector $v_{\lambda}$ as being a
$400$-dimensional vector with a $1$ in the $i^{th}$ position and $0$s
elsewhere. Thus, to find the $c$ associated with each $v_{\lambda}$, we must
only find the $i^{th}$ diagonal entry of $\pi_{W\otimes W}(C)$. We will then
have that $c$ is an eigenvalue of multiplicity $\mathrm{dim}(V(\lambda))$ of
this matrix with changed basis. Since a change of basis does not impact the
eigenvalues of the matrix and $\pi_{W\otimes W}(C)$ is a block matrix by Lemma
10, $c$ is an eigenvalue of $\pi_{W\otimes W}(C)$ with multiplicity
$\mathrm{dim}(V(\lambda))$. Next, as $v_{\lambda}$ is a highest weight vector
of weight $\lambda$, by definition,

$E_{1}v_{\lambda}=E_{2}v_{\lambda}=E_{3}v_{\lambda}=0.$

Thus,

$\pi_{W\otimes W}(C)v_{\lambda}=\pi_{W\otimes
W}(q^{-4-2H_{1}-H_{2}-H_{3}}+q^{-2-H_{2}-H_{3}}+q^{H_{2}-H_{3}}+q^{H_{3}-H_{2}}+q^{2+H_{2}+H_{3}}+q^{4+2H_{1}+H_{2}+H_{3}})v_{\lambda}.$

Since each $q^{H_{i}}$ is a diagonal matrix, we may use Section 6.3 to find
the corresponding diagonal entry to each highest weight vector in a feasible
way computationally. For example, $v_{4L_{1}}=e_{1}^{\otimes 2}$ where $e_{1}$
is $20$-dimensional, and thus $\pi_{W\otimes W}(C)$ has an eigenvalue of

$\displaystyle c$ $\displaystyle=\langle e_{1}^{\otimes 2}|\pi_{W\otimes
W}(C)|e_{1}^{\otimes 2}\rangle$ $\displaystyle=\langle e_{1}^{\otimes
2}|\pi_{W\otimes
W}(q^{-4-2H_{1}-H_{2}-H_{3}}+q^{-2-H_{2}-H_{3}}+q^{H_{2}-H_{3}}+q^{H_{3}-H_{2}}+q^{2+H_{2}+H_{3}}+q^{4+2H_{1}+H_{2}+H_{3}})|e_{1}^{\otimes
2}\rangle$ $\displaystyle=q^{-4}\cdot q^{-4}\cdot q^{-4}\cdot 1\cdot
1+q^{-2}\cdot 1\cdot 1+1\cdot 1+1\cdot 1+q^{2}\cdot 1\cdot 1+q^{4}\cdot
q^{4}\cdot q^{4}\cdot 1\cdot 1$ $\displaystyle=q^{12}+q^{2}+2+q^{-2}+q^{-12}$

with multiplicity $\mathrm{dim}(V(4L_{1}))=105$. The remaining eigenvalues can
be found similarly.

∎

######  Remark 5.

Recall that $\mathfrak{so}_{6}$ is a rank 3 Lie algebra which is isomorphic to
the rank 3 Lie algebra $\mathfrak{sl}_{4}$. In this case, previous research in
[KZ23] allows us to express $C$ using generators from both $\mathfrak{so}_{6}$
and $\mathfrak{sl}_{4}$ in terms of the above eigenvalues. We choose not to
explore this, but someone who wishes to relate [KLLPZ20]’s central element $C$
to the Harish Chandra isomorphism may reference [KZ23].

###  6.4 Type $A$ Ground State Transformation

Following the path of previous research, we used Section 7.9 as a starting
point to form our ground state transformation. We perform a ground state
transformation so that the resulting matrix has rows which sum to $0$,
allowing us to use this transformed matrix as a generator for a Markov
process. We will then compare this generator to that of the probability
section.

######  Definition 10.

A ground state transformation is a map from a Hamiltonian matrix, $H$, to a
matrix whose rows sum to $0$. We define such a map to be of the following
form, where $G$ is a diagonal matrix and $a$ is in the underlying field.

$G^{-1}HG-a\mathrm{Id}$

We call $G$ a ground state transformation matrix of $H$.

Section 7.9 provides a ground state transformation that worked with Type
$A_{n}$ Lie algebras. However, as can be seen using the respective crystal
graphs, Type $D_{n}$ Lie algebras have more complicated behavior: the crystal
graph for a fundamental representation of $\mathcal{U}_{q}(\mathfrak{sl}_{4})$
is shown in Figure 6.

Figure 6: The crystal graph of the fundamental representation of
$\mathcal{U}_{q}(\mathfrak{sl}_{4})$

As a result, the crystal graph for $V\otimes V$ of
$\mathcal{U}_{q}(\mathfrak{sl}_{4})$ is the graph shown in Figure 7. Note that
the cycle from the highest weight vector to the lowest weight vector is
straightforward.

Figure 7: Crystal graph of $\mathcal{U}_{q}(\mathfrak{sl}_{4})$ in $V\otimes
V$

Instead, in $\mathcal{U}_{q}(\mathfrak{so}_{6})$, the crystal graph for
$V\otimes V$ shown in Figure 5 was much less simple due to the more complex
crystal graph for $V$ shown in Figure 3. In this case, the shortest path from
the highest weight vector to the lowest weight vector requires eight
applications of $F_{i}$s as shown in the third example within the proof of
Lemma 8.

Thus, intuitively, when we apply the tensor product rule again to $W$, the
path from the highest weight vector to the lowest weight vector should be even
more intricate. Therefore, the ground state transformation in Section 7.9
should include more complicated structures than solely permutations of
$F_{1}^{k_{1}}F_{2}^{k_{2}}F_{3}^{k_{3}}$ in order to generate a complete
matrix $G$. This result is formalized below. See 8

###### Proof.

By the crystal graph for $V$ shown in Figure 3, which is based on the
fundamental representation defined in Definition 7, we see that we will not be
able to compute $e_{4}$ without applying either $F_{1}F_{2}F_{3}F_{1}$ or
$F_{1}F_{3}F_{2}F_{1}$.

Now, note that $W$ is a subspace of $\mathrm{Sym}^{2}_{q}(\mathbb{R}^{6})$; in
particular, as shown in Lemma 8, $W$ has basis elements of the form:

$\displaystyle\\{p_{i}(q)$ $\displaystyle e_{i}\otimes
e_{i},p_{i}(q)e_{i}\otimes e_{j}+p_{j}(q)e_{j}\otimes e_{i},\mid i\neq
j,(i,j)\neq(1,4),(2,5),(3,6),p_{j}\text{ Laurent polynomial}\\}\cup$
$\displaystyle\\{-e_{1}\otimes e_{4}+q^{-1}e_{2}\otimes
e_{5}-q^{-2}e_{4}\otimes e_{1}+q^{-1}e_{5}\otimes e_{2},e_{2}\otimes
e_{5}-q^{-1}e_{3}\otimes e_{6}+q^{-2}e_{5}\otimes e_{2}-(q^{2}+1)e_{6}\otimes
e_{3}\\}$

Thus, applying a sequence of $F_{i}$s to obtain, for example,

$\langle e_{1}\otimes e_{2}|\pi_{W}(\Delta(F_{1}))|e_{1}\otimes
e_{1}\rangle=1$

will also yield

$\langle e_{2}\otimes e_{1}|\pi_{W}(\Delta(F_{1}))|e_{1}\otimes
e_{1}\rangle=q^{-1}$

because $e_{1}\otimes e_{2}+qe_{2}\otimes e_{1}$ is a basis vector of $W$.

However, the composition of $F_{i}$s required to obtain $e_{4}\otimes e_{4}$,
$\Pi_{i}(\Delta F_{i})$, will only produce a nonzero

$\langle e_{4}\otimes e_{4}|\pi_{W}(\Pi_{i}(\Delta(F_{i})))|e_{1}\otimes
e_{1}\rangle$

and result in $0$ when taking, for any other $(i,j)\neq(4,4)$,

$\langle e_{i}\otimes e_{j}|\pi_{W}(\Pi_{i}(\Delta(F_{i})))|e_{1}\otimes
e_{1}\rangle$

as $e_{4}\otimes e_{4}$ does not appear in any other basis vector of $W$.
Therefore, since $e_{4}\otimes e_{4}$ lies in the bottom right corner of
Figure 5, it is clear that in order to obtain a nonzero value corresponding to
$e_{4}\otimes e_{4}$, we must follow the arrows and apply a composition of the
form

$\langle e_{4}\otimes
e_{4}|\pi_{W}(\Delta(F_{1})\cdot\Pi_{i}(\Delta(F_{i}))\cdot\Delta(F_{1}))|e_{1}\otimes
e_{1}\rangle$

with, in particular,

$\langle e_{4}\otimes
e_{4}|\pi_{W}(\Delta(F_{1})\Delta(F_{1})\Delta(F_{3})\Delta(F_{3})\Delta(F_{2})\Delta(F_{2})\Delta(F_{1})\Delta(F_{1}))|e_{1}\otimes
e_{1}\rangle=q^{4}+4q^{2}+6+4q^{-2}+q^{-4}.$

Finally, since $W$ is a subset of a symmetric tensor, $W\otimes W$ will be as
well. Therefore, we must again find a composition $\Pi_{i}(\Delta(F_{i}))$
that produces exactly $e_{4}^{\otimes 4}$, and it will not be enough to
produce any other vector. By the tensor product rule, this will yet again
result in a nonzero value to an expression only in the following structure

$\langle e_{4}^{\otimes
4}|\pi_{W}(\Delta(F_{1})\cdot\Pi_{i}(\Delta(F_{i}))\cdot\Delta(F_{1}))|e_{1}^{\otimes
4}\rangle$

∎

###  6.5 Finding a Markov Process Generator

####  6.5.1 Partial Type $A$ Ground State Transformation

######  Definition 11.

We define a partial Type $A$ ground state transformation of $\pi_{W\otimes
W}(C)$ to be a map

$\displaystyle\pi_{W\otimes W}(C)\mapsto\left.\left(G^{-1}\pi_{W\otimes
W}(C)G-a\mathrm{Id}\right)\right|_{K}$

where $K$ is the vector subspace of $W\otimes W$ which is the span of all
basis vectors $u_{i}\otimes u_{j}$ for which there exists a triple
$(k_{1},k_{2},k_{3})$ of nonnegative integers satisfying

$\displaystyle\left\langle u_{i}\otimes
u_{j}\middle|\prod_{i=1}^{3}\Delta(F_{\ell_{i}})^{k_{\ell_{i}}}\middle|e_{1}\otimes
e_{1}\right\rangle\neq 0$

where $(\ell_{1},\ell_{2},\ell_{3})$ is some permutation of $(1,2,3)$, and
where $G$ is a $400\times 400$ diagonal matrix such that the diagonal entry of
$G$ corresponding to $u_{i}\otimes u_{j}$ is the unique nonzero value above.

Note that by Lemma 8, the partial Type $A$ ground state transformation is not
unique.

####  6.5.2 Selecting a Ground State Transformation

We now extend the partial Type $A$ ground state transformation to a ground
state transformation of $\pi_{W\otimes W}(C)$. Define $a\coloneqq
q^{12}+q^{2}+q^{-2}+q^{-12}$.

######  Lemma 11.

Let $G$ be a ground state transformation matrix for $\pi_{W\otimes W}(C)$,
i.e., a diagonal matrix such that each row of $a^{-1}G^{-1}\pi_{W\otimes
W}(C)G-\mathrm{Id}$ sums to $0$, and let $g_{i}$ denote the $i$th diagonal
entry of $G$ for $1\leq i\leq 400$. Then the vector

$\displaystyle\vec{g}\coloneqq\begin{pmatrix}g_{1}\\\ g_{2}\\\ \vdots\\\
g_{400}\end{pmatrix}$

satisfies $\pi_{W\otimes W}(C)\vec{g}=a\vec{g}$. This condition is both
sufficient and necessary, i.e., if $\vec{g}$ is an eigenvector of
$\pi_{W\otimes W}(G)$ with eigenvalue $a$, the diagonal matrix $G$ with
entries corresponding to $\vec{g}$ as above is such that the rows of
$a^{-1}G^{-1}\pi_{W\otimes W}G-\mathrm{Id}$ sum to $0$.

###### Proof.

Define the matrix $L$ by

$\displaystyle L\coloneqq a^{-1}G^{-1}\pi_{W\otimes W}(C)G-\mathrm{Id}$

By definition, each row of $L$ sums to $0$. Then we have

$\displaystyle G^{-1}(\pi_{W\otimes W}(C))G-a\mathrm{Id}$ $\displaystyle=aL$
$\displaystyle G^{-1}(\pi_{W\otimes W}(C))G$ $\displaystyle=aL+a\mathrm{Id}$
$\displaystyle\pi_{W\otimes W}(C)G$ $\displaystyle=aGL+aG$

Since $G$ is diagonal and each row of $L$ sums to $0$, each row of $aGL$ sums
to $0$. Thus, the sum of the elements in each row of $\pi_{W\otimes W}(C)G$
equals the sum of the elements in the same row of $aG$. Defining $g_{i}$ as
above, we can write the sum of the $i$th row of $\pi_{W\otimes W}(C)G$ as
$\sum_{j=1}^{400}\pi_{W\otimes W}(C)_{i,j}g_{j}$, and the sum of the $i$th row
of $aG$ as $ag_{i}$. Thus, for all $1\leq i\leq 400$,

$\displaystyle\sum_{j=1}^{400}\pi_{W\otimes W}(C)_{i,j}g_{j}$
$\displaystyle=ag_{i}$

This is equivalent to saying that $\pi_{W\otimes W}(C)\vec{g}=a\vec{g}$.

Reversing the steps above shows that the condition $\pi_{W\otimes
W}(C)\vec{g}=a\vec{g}$ is sufficient as well as necessary. ∎

Thus, any partial Type $A$ ground state transformation matrix $G$ of
$\pi_{W\otimes W}(C)$ which satisfies the condition in Lemma 11 is a ground
state transformation matrix of $\pi_{W\otimes W}(C)$. Finding diagonal entries
of $G$ as described in Definition 11 and solving the condition in Lemma 11 as
a system of linear equations yields a solution space $\mathcal{S}$ with $37$
unknowns.

######  Lemma 12.

Given any $G\in\mathcal{S}$,

$\displaystyle a^{-1}G^{-1}HG-\mathrm{Id}$

is a generator for a Markov process.

###### Proof.

Immediate by the definition of $\mathcal{S}$. ∎

Note that the proof of Theorem 4 is an immediate consequence of $\mathcal{S}$
being nonempty. We give one such generator in Section 7.8 by setting all
unknowns to $0$ for ease of computation. One detail to note is that this
generator has dimension $196=14^{2}$, which suggests that, when interpreted as
an interacting particle system, the process allows for $14$ different states —
all but four particles of the same class — at each site.

###  6.6 Difference with Probability Generator

To relate our algebraic findings to those of probability, we end by comparing
the two generators. First, note that the $2\times 2$ and $3\times 3$ blocks of
the algebraically-produced Markov generator in Section 7.8 are a constant
multiple of the $2\times 2$ and $3\times 3$ blocks of the probabilistically-
produced Markov generator in Section 7.3. However, we notice that any
generator matrix given by a ground state transformation as described in
Definition 10 will not produce a generator matrix for a Markov process
matching the probabilistically-produced generator matrix. We now prove
Proposition 9.

###### Proof.

We will show that no block of any ground state transformation of
$\pi_{W\otimes W}(C)$ can be a constant multiple of the $4\times 4$ block of
$L_{Q}$ in Section 7.3 corresponding to the communicating class

$\Bigl{(}\langle 3,0\rangle,\langle 1,2\rangle,\langle 2,1\rangle,\langle
0,3\rangle\Bigl{)}.$

Thus, $L_{Q}$ must be different from any Markov generator produced from
$\pi_{W\otimes W}(C)$.

Note that the above communicating class contains one class 1 particle and one
class 2 particle per Section 7.2. As proven in Proposition 2.5 of [KLLPZ20],
this communicating class must correspond to the weight

$2\lambda_{i}+\lambda_{j}+\lambda_{k}$

found in Section 6.2.2. As discussed in Lemma 10, there are $24$ weight spaces
of dimension $4$ associated with $2\lambda_{i}+\lambda_{j}+\lambda_{k}$, and
thus $2\lambda_{i}+\lambda_{j}+\lambda_{k}$ corresponds to the twenty-four
$4\times 4$ blocks in $\pi_{W\otimes W}$. It is therefore sufficient to show
that none of the $4\times 4$ blocks of $\pi_{W\otimes W}(C)$, which are listed
in Section 7.11, can be mapped to the $4\times 4$ block in $L_{Q}$
corresponding to the above communicating class.

By referencing Section 7.3, we notice that each $4\times 4$ block does not
contain any zero entries. However, every $4\times 4$ block listed in Section
7.11 has an off-diagonal entry which takes the value of $0$.

Since conjugation by a diagonal matrix will not change a $0$-entry in a matrix
and the $0$-entries are off-diagonal, Definition 10 confirms that no ground
state transformation of $\pi_{W\otimes W}(C)$ will result in $L_{Q}$. ∎

The computationally-heavy verification which involves disregarding the
relationship between weights and particles and instead checking $4\times 4$
blocks after trying all possible ground state transformations on
$\pi_{W\otimes W}(C)$ will yield the same result, as this proof is equivalent
to the one detailed above.

##  7 Appendix

###  7.1 Code for Probabilistic Sections

The SymPy code used to verify the generator matrix $L_{Q}$, limits of
component generator matrices, stationary distributions, spectral gaps, and
matrices of Markov duality can be found at the authors’ GitHub repository.
Because the largest communicating classes only have size 9, the matrices are
small enough that the calculations can be done by hand. Those details are
omitted from this paper, and the SymPy code only serves as an independent
verification.

###  7.2 List of Communicating Classes by Size

The following communicating classes are for the two-$\gamma$ lattice site Type
D ASEP.

The ordered 9-state communicating class follows:

$\Bigl{(}\langle 3,3\rangle,\langle 0,33\rangle,\langle 33,0\rangle,\langle
1,32\rangle,\langle 32,1\rangle,\langle 2,31\rangle,\langle
31,2\rangle,\langle 11,22\rangle,\langle 22,11\rangle\Bigl{)}$

The ordered 6-state communicating classes follow:

$\Bigl{(}\langle 1,3\rangle,\langle 3,1\rangle,\langle 0,31\rangle,\langle
31,0\rangle,\langle 2,11\rangle,\langle 11,2\rangle\Bigl{)}$ $\Bigl{(}\langle
2,3\rangle,\langle 3,2\rangle,\langle 0,32\rangle,\langle 32,0\rangle,\langle
1,22\rangle,\langle 22,1\rangle\Bigl{)}$ $\Bigl{(}\langle 3,31\rangle,\langle
31,3\rangle,\langle 1,33\rangle,\langle 33,1\rangle,\langle
11,32\rangle,\langle 32,11\rangle\Bigl{)}$ $\Bigl{(}\langle
3,32\rangle,\langle 32,3\rangle,\langle 2,33\rangle,\langle
33,2\rangle,\langle 22,31\rangle,\langle 31,22\rangle\Bigl{)}$

The ordered 4-state communicating classes follow:

$\Bigl{(}\langle 3,0\rangle,\langle 1,2\rangle,\langle 2,1\rangle,\langle
0,3\rangle\Bigl{)}$ $\Bigl{(}\langle 3,11\rangle,\langle 1,31\rangle,\langle
11,3\rangle,\langle 31,1\rangle\Bigl{)}$ $\Bigl{(}\langle 3,22\rangle,\langle
2,32\rangle,\langle 22,3\rangle,\langle 32,2\rangle\Bigl{)}$ $\Bigl{(}\langle
33,3\rangle,\langle 31,32\rangle,\langle 32,31\rangle,\langle
3,33\rangle\Bigl{)}$

The ordered 3-state communicating classes follow:

$\Bigl{(}\langle 1,1\rangle,\langle 0,11\rangle,\langle 11,0\rangle\Bigl{)}$
$\Bigl{(}\langle 2,2\rangle,\langle 0,22\rangle,\langle 22,0\rangle\Bigl{)}$
$\Bigl{(}\langle 31,31\rangle,\langle 11,33\rangle,\langle
33,11\rangle\Bigl{)}$ $\Bigl{(}\langle 32,32\rangle,\langle
22,33\rangle,\langle 33,22\rangle\Bigl{)}$

The ordered 2-state communicating classes follow:

$\Bigl{(}\langle 0,1\rangle,\langle 1,0\rangle\Bigl{)}$ $\Bigl{(}\langle
0,2\rangle,\langle 2,0\rangle\Bigl{)}$ $\Bigl{(}\langle 1,11\rangle,\langle
11,1\rangle\Bigl{)}$ $\Bigl{(}\langle 2,22\rangle,\langle 22,2\rangle\Bigl{)}$
$\Bigl{(}\langle 11,31\rangle,\langle 31,11\rangle\Bigl{)}$ $\Bigl{(}\langle