11institutetext:  1Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841,
Republic of Korea
2Institute of Basic Science, 70, Yuseong-daero 1689-gil, Yuseong-gu, Daejeon,
Republic of Korea
3Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain
4Departamento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna,
Tenerife, Spain
5Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, Miyagi 980-8577, Japan
6The High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba,
Ibaraki 305-0801, Japan
7The Graduate University for Advanced Studies (SOKENDAI), Shonan Village,
Hayama, Kanagawa 240-0193, Japan
8Kyoto University, Yoshidahonmachi, Sakyo Ward, Kyoto, 606-8501 Japan
9Institute of Physical and Chemical Research, 2-1 Hirasawa, Wako, Saitama
351-0198, Japan
10Delft University of Technology, Mekelweg 5, 2628 CD Delft, Netherland
11The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8654, Japan
12Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama 338-8570, Japan
13National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan
14Japan Aerospace Exploration Agency (JAXA), 7-44-1 Jindaiji, Higashi-machi,
Chofu-shi, Tokyo 182-8522, Japan
11email<EMAIL_ADDRESS>
# GroundBIRD : A CMB polarization experiment with MKID arrays

K. Lee1 J. Choi2 R. T. Génova-Santos3,4 M. Hattori5 M. Hazumi6,7 S. Honda8 T.
Ikemitsu8 H. Ishida5,9 H. Ishitsuka7 Y. Jo1 K. Karatsu10 K. Kiuchi11 J.
Komine8 R. Koyano12 H. Kutsuma5,9 S. Mima9 M. Minowa11 J. Moon1 M. Nagai13 T.
Nagasaki9 M. Naruse12 S. Oguri9 C. Otani9 M. Peel3,4 R. Rebolo3,4 J. A.
Rubiño-Martín3,4 Y. Sekimoto14 J. Suzuki8 T. Taino12 O. Tajima8 N. Tomita11 T.
Uchida6,7 E. Won1 M. Yoshida6,7

###### Abstract

GroundBIRD is a ground-based experiment for the precise observation of the
polarization of the cosmic microwave background (CMB). To achieve high
sensitivity at large angular scale, we adopt three features in this
experiment: fast rotation scanning, microwave kinetic inductance detector
(MKID) and cold optics. The rotation scanning strategy has the advantage to
suppress $1/f$ noise. It also provides a large sky coverage of 40%, which
corresponds to the large angular scales of $l\sim 6$. This allows us to
constrain the tensor-to-scalar ratio by using low $l$ B-mode spectrum. The
focal plane consists of 7 MKID arrays for two target frequencies, 145 GHz and
220 GHz band. There are 161 pixels in total, of which 138 are for 144 GHz and
23 are for 220 GHz. This array is currently under development and the
prototype will soon be evaluated in telescope. The GroundBIRD telescope will
observe the CMB at the Teide observatory. The telescope was moved from Japan
to Tenerife and is now under test. We present the status and plan of the
GroundBIRD experiment.

###### Keywords:

cosmic microwave background, microwave kinetic inductance detector

††journal: Journal of Low Temperature Physics

##  1 Introduction

The cosmic microwave background (CMB) is a isotropic blackbody radiation at
the temperature of 2.7255 K WMAP . The CMB has small anisotropies at the
$10^{-5}~{}\mathrm{K}$ level in temperature and sub-$\mu\mathrm{K}$ level in
polarization. The CMB anisotropies can be characterized by the angular power
spectrum, and it provides constraints on numerous cosmological parameters
Dodelson .

The E-modes of CMB polarization have been well observed by various experiments
WMAP  and lensing B-modes have been measured by other experiments, PolarBEAR
and BICEP2 Polarbear ; Bicep2 . The B-modes larger than degree scale,
generated by primordial gravitational waves predicted by the inflation model,
has not been detected yet Guth1981 . The observation of B-modes from the
primordial gravitational waves will reveal information about the very early
universe hidden behind the background radiations.

GroundBIRD is aiming to measure the B-modes of CMB polarization Tajima2012 .
The observation site is Teide observatory. Earlier this year, we transported
the GroundBIRD telescope from Japan to Instituto de Astrofísica de Canarias
(IAC). In IAC, the telescope is assembled and is now under test. We are
preparing for installation of the telescope at the Teide observatory in
parallel. Once the assembly and tests are finished at IAC, we will move the
telescope to the Teide observatory and continue the commissioning of the
telescope with sky signals.

##  2 Overview of GroundBIRD

The GroundBIRD telescope is designed for the precise observation of CMB
polarization. The target sensitivity is to constrain the tensor-to-scalar
ratio $r$ with a precision of $\sigma_{r}\sim 0.01$ with three year
observation Oguri2016 . To achieve high enough sensitivity to observe B-modes
for large sky coverage, we employ three features: fast rotation scanning,
microwave kinetic inductance detector (MKID) and cold optics Tajima2012 .

Aside from the thermal noise, the main source of systematics in CMB
observations at large angular scale is $1/f$ noise due to the atmospheric
fluctuations. To suppress this, the sky should be measured repeatedly with a
scanning rate faster than the typical value of $f_{knee}=0.1$ Hz Tajima2012 .
The telescope scans the sky with continuous rotations at 20 rpm (revolutions
per minute) and an elevation of 60 degrees. The rotation speed results the
scanning rate of 0.33 Hz which means that the telescope scans a ring every 3
seconds. A sky area of $10^{-6}$ sr (steradian) is observed on average 1000
times per day assuming a pixelization which divides the whole sky into
12,582,912 pixels.

The MKID is a photon sensor using superconducting resonators Day . An MKID
array for the focal plane is under fabrication in RIKEN (Institute of Physical
and Chemical Research). Figure 1(a) shows a full scale design of our MKID. An
orthomode transducer (OMT) in each pixel receives photons in two perpendicular
directions. The signals from the OMTs are transmitted to the MKIDs through the
identical circuits with transmission line converters and filters, and the
MKIDs detect the intensities of the photon in two directions. The meander line
at the end of the circuit is MKID as shown in Fig. 1(a). The resonant
frequencies for our MKIDs are controlled by their length, and are in the range
of 4 to 8 GHz. The OMTs will be coupled with a corrugated horn antennae or a
hemispherical lenslets for higher optical efficiency. Due to the fast response
and short dead time of MKID, our system can have a fast sampling rate of 1
kSps.

(a) A pixel on a fabricated wafer.

(b) Focal plane design of GroundBIRD.

Figure 1: The Focal plane of GroundBIRD. A pixel on a fabricated wafer and the
design of entire focal plane with 7 modules are shown in (a) and (b),
respectively.

The focal plane of the GroundBIRD telescope has 7 modules as shown in Fig.
1(b). GroundBIRD will cover two frequency bands, 145 GHz and 220 GHz, for the
separation of dust foreground. The module in the center is for 220 GHz and
outer 6 modules are for 145 GHz. Each module has 23 pixels and the total
number of pixels are 161 of which 138 for 145 GHz and 23 for 220 GHz. The
sensitivity for the sky signal is determined by the integration time for
observation per pixel. The expected sky pixel noise level is 21 $\mu$K arcmin
by assuming three-year observation, 138 pixels and sky noise equivalent
temperature (NET) of 300 $\mu$K. By combining with the large sky coverage,
this value guarantees that the GroundBIRD can measure the tensor-to-scalar
ratio from B-mode polarizations patterns at large angular scales ($l\sim 6$).

(a) The GroundBIRD telescope.

(b) Inside the cryostat

Figure 2: 3D modeling of GroundBIRD. The entire telescope is shown in (a), and
(b) shows inside the cryostat.

The cryostat of the GroundBIRD telescope is a cylindrical vacuum chamber with
diameter of 1.5 m and 1.5 m in height as shown in Fig. 2(a). Within the
chamber, there are 40 K and 4 K stages. The temperatures of theses stages are
maintained by two-stage pulse tube cooler (PTC). The 4 K stage contains the
main part of the telescope including optics and focal plane. It is insulated
by 40 K and 4 K shields. The shields are covered with multilayer insulation
(MLI) sheets. In addition, these are wrapped in anti-magnetic sheets to reduce
the effect of external magnetic fields on the MKID array Kutsuma2018 . The
aperture windows of 40 K and 4 K shields are covered by low-pass metal mesh
filters Metalmesh  and radio transparent multilayer insulator (RT-MLI)
Choi2013 .

The crossed-Dragone reflector is the main optics of the GroundBIRD telescope
Tajima2012 . The incident radiations are reflected on the primary and
secondary mirrors. The two reflectors focus a plane wave to a point on the
focal plane. The resulting beam width for a pixel in the sky is about 0.5
degrees by this focusing. The crossed-Dragone reflector is cooled down on the
the 4 K stage. By cooling the reflector, thermal noise caused by the surface
emission can be suppressed Tajima2012 .

The telescope is rotated by our rotation table. It consists of the rotation
stage in the center, supporting legs and the upper plate. The rotation stage
rotates the upper structure. A custom-made rotary joint is used for the
electricity supply, and helium gas circulation during the rotation Tajima2012
.

The observation site is Teide Observatory in Tenerife, Canary islands, Spain.
The altitude and longitude are $28\degree 18^{\prime}$N and $16\degree
30^{\prime}$W and the altitude is 2390 m.

Even for only one day observation, we can make a sky map of a strip centered
at declination of $28\degree$ with width of $60\degree$. The resulting
coverage is 40% of the full sky and angular resolution is 0.5 degrees. To
improve the signal-to-noise ratio, we will stack the maps for year-scale. We
expect to obtain the angular power spectrum for the angular scales of
$6<l<300$, which enables to observe both recombination ($l\sim 100$) and
reionization ($l<10$) bumps in B-mode spectrum Tajima2012 .

##  3 Status of GroundBIRD

In the first half of this year, the GroundBIRD telescope has been transported
from Japan to IAC in Tenerife, Spain. We are assembling the telescope and
preparing its deployment on the Teide observatory. We shipped our telescope
after we confirmed the receiver performances with rotation on the telescope
mount. The temperature of focal plane and cold optics during rotation at 20
rpm. For 16 hours after the start of rotation, the cold optics temperature is
maintained at 3.5 K and the temperature of the focal plane is maintained
between 250 and 270 mK. Since the detector temperature varies much slower than
the rotation of the telescope, the drift in detector response correlated with
the temperature change is suppressed with the $1/f$ noise removal algorithm
during the data analysis. Additionally the resonant frequencies of MKIDs are
calibrated every hour to compensate the shift of the resonance peak due to the
temperature change.

###  3.1 Detectors

We fabricated a simplified version of MKID for the test. The mask design of
the test version MKID is shown in 3(a). The resonant frequencies are designed
between 4 to 8 GHz. Each pixel has a single polarization antenna, coupled with
a lenslet which is shown in Fig. 3(b). The hemispherical lenslet diameter is 6
mm, which determines the spacing between the detectors.

(a) Mask design of test version MKID for 145 GHz. There are 10 MKIDs, each
connected with a single polarized dipole antenna.

(b) Fabricated test version MKID with lenslets. The hemispheres are the
lenslets. Three pixels are coupled with the lenslets. It is mounted on the
module for the focal plane.

Figure 3: Test version MKID. The mask design and fabricated wafer and lenslet
are shown in (a) and (b) respectively.

The lab test of the test version MKID are shown in Fig. 4. The amplitude of
S21 is measured with a vector network analyzer (VNA). The 9 resonances are
observed between 4 and 4.7 GHz where one missing peak is just outside this
range. The resonance peaks are spaced approximately 100 MHz from each other
except for one peak. The phase noise power spectral density (PSD) of an MKID
is shown in Fig. 4(b). The knee frequency is 1 Hz which is below the scanning
rate of 3 Hz.

(a) S21 of the test version MKID. The 9 peaks are observed between 4.0 to 4.7
GHz. The peaks are spaced approximately 100 MHz from each other except for the
KID05.

(b) The phase noise PSD of an MKID. The knee frequency is 1 Hz and the roll-
off frequency is $10^{4}$ Hz. The flat noise psd level is

Figure 4: Lab tests of the test version MKID. The S21, measured with VNA, and
the phase noise PSD for an MKID are shown in (a) and (b) respectively.

The MKID arrays are set up in the telescope as shown in Fig. 5(a) and will be
used for the first light and in early stages of the commissioning.

(a) MKID arrays on the focal plane. Three MKID arrays are installed. The plate
on the center array is a metal mesh filter.

(b) Installed telescope in the dome.

Figure 5: The current setup of the focal plane and the installed telescope.

###  3.2 Telescope installation and commissioning

A Very Small Array (VSA) enclosure is used for the GroundBIRD experiment. A
dome built by baader planetarium company baader  is placed at the site and the
rotation table was installed inside the dome. The telescope was mounted on the
rotation table as shown in Fig. 5(b) and the installation is finished. The
commissioning for calibrations of the detector and pointing of the telescope
is being performed by observing the moon and the tau A.

##  4 Conclusions

GroundBIRD aims to observe the CMB B-mode polarization and measure the tensor-
to-scalar ratio $r$ with a precision of $\sigma_{r}\sim 0.01$. The GroundBIRD
telescope was moved from Japan to Tenerife and installed at the Teide
observatory. The commissioning of the telescope is on going and the MKID array
for the actual observation is under development aiming to start observation in
2020.

###### Acknowledgements.

This work was supported by JSPS KAKENHI Grant Numbers JP15H05743, JP16J09435,
JP18H05539, JP18J01831, JP15K13491, JP19H01916, JPR2804, and by the NRF Grant
Number NRF-2017R1A2B3001968. This was also supported by Kyoto University and
MEXT SPIRITS grant. We also thank Hisao Kawano, Noboru Furukawa, Hiroshi
Watanabe, Advanced Technology Center of National Astronomical Observatory of
Japan, and Advanced Manufacturing Support Team of RIKEN.

## References

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DOI:10.1103/PhysRevLett.113.021301

  * (4) BICEP2 Collaboration, Phys. Rev. Lett. 112, 241101 (2013)
DOI:10.1103/PhysRevLett.112.241101

  * (5) A. H. Guth, Phys. Rev. D 23, 347 (1981)
DOI:10.1103/PhysRevD.23.347

  * (6) O. Tajima et al. Proc. SPIE 8452, 84521M (2012)
DOI:10.1117/12.925816

  * (7) S. Oguri, J. Choi, T. Damayanthi et al., J. Low. Temp. Phys. 184, 786 (2016)
DOI:10.1007/s10909-015-1420-9

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DOI:10.1063/1.4827081

  * (12) https://healpix.sourceforge.io/
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# Bidiagonal Triads and the Tetrahedron Algebra

Darren Funk-Neubauer

###### Abstract

We introduce a linear algebraic object called a bidiagonal triad. A bidiagonal
triad is a modification of the previously studied and similarly defined
concept of bidiagonal triple. A bidiagonal triad and a bidiagonal triple both
consist of three diagonalizable linear transformations on a finite-dimensional
vector space, each of which acts in a bidiagonal fashion on the eigenspaces of
the other two. A triad differs from a triple in the way these bidiagonal
actions are defined. We modify a number of theorems about bidiagonal triples
to the case of bidiagonal triads. We also describe the close relationship
between bidiagonal triads and the representation theory of the tetrahedron Lie
algebra.

Keywords: bidiagonal triad, bidiagonal triple, bidiagonal pair, tridiagonal
pair, Tetrahedron Lie algebra, Lie algebra $\mathfrak{sl}_{2}$

Mathematics Subject Classification: 15A21, 15A30, 17B10

##  1 Introduction

In this paper we introduce a linear algebraic object called a bidiagonal
triad. Roughly speaking, a bidiagonal triad is an ordered list of three
diagonalizable linear transformations on a finite-dimensional vector space,
each of which acts in a bidiagonal fashion on the eigenspaces of the other two
(see Definition 2.8 for the precise definition). This concept arose as a
modification of the similarly defined concept of a bidiagonal triple (see
Definition 2.6). Furthermore, the idea of a bidiagonal triple arose as a
generalization of the concept of a bidiagonal pair (see Definition 2.4). The
theory of bidiagonal pairs and triples was developed in [14, 15]. The
relationship between bidiagonal pairs, triples, and triads is as follows. If
$A,\,A^{\prime},\,A^{\prime\prime}$ is a bidiagonal triple then
$A,\,A^{\prime}$ and $A^{\prime},\,A^{\prime\prime}$ and
$A^{\prime\prime},\,A$ are each bidiagonal pairs. However, the converse is not
true. That is, if $A,\,A^{\prime}$ and $A^{\prime},\,A^{\prime\prime}$ and
$A^{\prime\prime},\,A$ are each bidiagonal pairs then
$A,\,A^{\prime},\,A^{\prime\prime}$ does not have to be a bidiagonal triple.
This is the genesis of the concept of bidiagonal triad. Three bidiagonal pairs
can be interwoven together to produce a bidiagonal triple, but if the
interweaving process is altered then a bidiagonal triad will be produced. For
a triad the three transformations have a common eigenvector. However, for a
triple there does not exist a common eigenvector for all three
transformations, but any two of the three have a common eigenvector.

The main purpose of this paper is twofold. First we show how some results
about bidiagonal triples in [15] can be modified to hold for bidiagonal
triads. In particular, we show that the eigenvalues of the three linear
transformations in a bidiagonal triad satisfy a linear recurrence (see Theorem
4.1). We then define a type of bidiagonal triad called reduced based on the
solutions to this recurrence (see Definition 4.2). We show that reduced
bidiagonal triads are canonical in the sense that every bidiagonal triad is
equivalent to a reduced one (see Definition 2.13 and Theorem 4.3). The second
main purpose of this paper is to explain the connection between bidiagonal
triads and representations of the tetrahedron algebra. In Theorem 4.5 we show
that every finite-dimensional irreducible module for the tetrahedron algebra
provides numerous examples of reduced bidiagonal triads. Our last main result
is a partial converse of Theorem 4.5. In Theorem 4.6 we show how an
irreducible module for the tetrahedron algebra can be constructed starting
from a certain type of reduced bidiagonal triad called thin (see Definition
2.12). The stronger non-thin version of Theorem 4.6 is false as we demonstrate
with a counterexample at the end of the paper.

We now offer some background on the tetrahedron algebra. The tetrahedron
algebra, denoted $\boxtimes$, was introduced in [17]. $\boxtimes$ has
essentially six generators, each of which can be identified with the six edges
of a tetrahedron. In other words, each generator is indexed by a pair of
vertices in the tetrahedron. The symmetric group $S_{4}$ acts on $\boxtimes$
as a group of automorphisms [17, Section 2]. $\boxtimes$ is isomorphic to the
three-point $\mathfrak{sl}_{2}$ loop algebra [17, Theorem 11.5]. Any five of
the six edges of the tetrahedron generate a subalgebra of $\boxtimes$ which is
isomorphic to the $\mathfrak{sl}_{2}$ loop algebra [17, Corollary 12.6]. The
three surrounding edges in each of the four faces of the tetrahedron form a
basis for a subalgebra of $\boxtimes$ isomorphic to $\mathfrak{sl}_{2}$ [17,
Corollary 12.4]. Each of the three pairs of opposite edges of the tetrahedron
generate a subalgebra of $\boxtimes$ isomorphic to the Onsager algebra [17,
Corollary 12.5]. $\boxtimes$ decomposes into the direct sum of these three
Onsager subalgebras [17, Theorem 11.6]. For more information on the algebras
mentioned in the previous five sentences see the references in [17, Section
1]. The finite-dimensional irreducible $\boxtimes$-modules were classified in
[16, Theorems 1.7, 1.8]. This classification invokes the classification of the
finite-dimensional irreducible modules for the Onsager algebra (see [7, 8, 9]
as well as the summary in [16, Theorems 1.3, 1.4, 1.6]). For more information
on how the tetrahedron algebra arises in representation theory see [2, 6, 11,
25, 32]. See [18, 30, 31] for how the tetrahedron algebra has been used in
algebraic combinatorics and graph theory.

We now offer some additional information on bidiagonal pairs and triples
because of their close connection to bidiagonal triads. Bidiagonal pairs and
triples have been used to study representations of the well-known algebras
$\mathfrak{sl}_{2}$, $U_{q}(\mathfrak{sl}_{2})$,
$U_{q}(\widehat{\mathfrak{sl}}_{2})$, $\boxtimes$, and $\boxtimes_{q}$. The
$q$-tetrahedron algebra $\boxtimes_{q}$ is a quantum analog of the tetrahedron
algebra [21]. Bidiagonal pairs and triples appear implicitly in [5, 12, 13,
22, 27, 36]. However, they were not explicitly defined until [14, 15] which
contain systematic studies of bidiagonal pairs and triples as well as
classification theorems. Finite-dimensional irreducible $\boxtimes$-modules
provide numerous examples of bidiagonal pairs and triples. The actions of the
three generators corresponding to the surrounding edges of each face of the
tetrahedron act as a bidiagonal triple [15, Theorem 4.8], [16, Theorem 3.8],
[17, Corollary 12.4]. The actions of the two generators corresponding to any
two edges of the tetrahedron that meet in a common vertex act as a bidiagonal
pair [15, Theorem 5.8]. See also Note 10.4 of this paper. Similarly, finite-
dimensional irreducible $\boxtimes_{q}$-modules also provide numerous examples
of bidiagonal pairs and triples [15, 21]. See [3, 4, 29, 33, 37, 38, 39, 40,
41] for more references to bidiagonal pairs and triples.

We now remark on the initial motivation for investigating bidiagonal pairs,
triples, and triads as well as the tetrahedron algebra. These objects each had
their genesis in the study of tridiagonal pairs. Roughly speaking, a
tridiagonal pair is an ordered pair of diagonalizable linear transformations
on a finite-dimensional vector space, each of which acts in a tridiagonal
fashion on the eigenspaces of the other (see [20, Definition 1.1] for the
precise definition). Tridiagonal pairs originally arose in algebraic
combinatorics through the study of a combinatorial object called a P- and
Q-polynomial association scheme [20]. Tridiagonal pairs appear in a wide
variety of mathematical subjects including representation theory, orthogonal
polynomials, special functions, partially ordered sets, statistical mechanics,
and classical mechanics. See [14, Introduction] for the appropriate
references. A major classification result involving tridiagonal pairs appears
in [26]. Finite-dimensional irreducible $\boxtimes$-modules provide numerous
examples of tridiagonal pairs. The actions of each pair of generators
corresponding to opposite edges of the tetrahedron act as a tridiagonal pair
[16, Theorem 1.7, Corollary 2.7]. These tridiagonal pairs are said to have
Krawtchouk type [23]. See also Note 10.4 of this paper. Similarly, finite-
dimensional irreducible $\boxtimes_{q}$-modules also provide numerous examples
of tridiagonal pairs [21, Theorem 10.3], [24, Theorem 2.7]. These tridiagonal
pairs are said to have $q$-geometric type [24, Definition 2.6]. A certain
special case of a tridiagonal pair, called a Leonard pair, has also been the
subject of much research. The Leonard pairs are classified up to isomorphism
[34, Theorem 1.9]. See [35] for a survey of Leonard pairs.

This paper is organized as follows. In Section 2 we recall the definitions of
bidiagonal pair and triple and then define bidiagonal triad along with a
number of related definitions. In Section 3 we recall the definition of the
tetrahedron algebra $\boxtimes$ and review some facts about
$\boxtimes$-modules. We also describe the relationship between $\boxtimes$ and
the Lie algebra $\mathfrak{sl}_{2}$. Section 4 contains statements of the main
results of the paper. In Section 5 we develop the tools needed to prove our
main results including some additional properties of bidiagonal triads.
Sections 6 through 10 contain the proofs of the main results stated in Section
4. In Section 11 we provide a counterexample to show that a stronger version
of Theorem 4.6 is false.

##  2 Bidiagonal Triads

###### Notation 2.1

Throughout this paper we adopt the following notation. Let $d$ denote a
nonnegative integer. Let $\mathbb{K}$ denote a field. Let $V$ denote a vector
space over $\mathbb{K}$ with finite positive dimension. For linear
transformations $X:V\to V$ and $Y:V\to V$ we define $[X,Y]:=XY-YX$.

###### Note 2.2

With reference to Notation 2.1 observe that $[X,X]=0$ and $[X,Y]=-[Y,X]$.
Also, $[\,\,,\,\,]$ is bilinear. That is, for $c_{1},c_{2}\in\mathbb{K}$ and a
linear transformation $Z:V\to V$ we have
$[X,c_{1}Y+c_{2}Z]=c_{1}[X,Y]+c_{2}[X,Z]$ and
$[c_{1}X+c_{2}Y,Z]=c_{1}[X,Z]+c_{2}[Y,Z]$.

We present the following lemma in order to motivate the definitions of
bidiagonal pair, triple, and triad.

###### Lemma 2.3

[15, Lemma 2.2] With reference to Notation 2.1 the following (i),(ii) hold.

  1. (i)

Suppose that there exists an ordering $\\{Y_{i}\\}_{i=0}^{d}$ of the
eigenspaces of $Y$ with
$XY_{i}\subseteq Y_{i}+Y_{i+1}\,(0\leq i\leq d)$, where $Y_{d+1}=0$. Then for
$0\leq i\leq d$, the restriction $[X,Y]|_{Y_{i}}$ maps $Y_{i}$ into $Y_{i+1}$.

  2. (ii)

Suppose that there exists an ordering $\\{Y_{i}\\}_{i=0}^{d}$ of the
eigenspaces of $Y$ with
$XY_{i}\subseteq Y_{i-1}+Y_{i}\,(0\leq i\leq d)$, where $Y_{-1}=0$. Then for
$0\leq i\leq d$, the restriction $[X,Y]|_{Y_{i}}$ maps $Y_{i}$ into $Y_{i-1}$.

###### Definition 2.4

[14, Definition 2.2] A bidiagonal pair (BD pair) on $V$ is an ordered pair of
linear transformations $A:V\to V$ and $A^{\prime}:V\to V$ that satisfy the
following three conditions.

  1. (i)

Each of $A,\,A^{\prime}$ is diagonalizable.

  2. (ii)

There exists an ordering $\\{V_{i}\\}_{i=0}^{d}$ (resp.
$\\{V^{\prime}_{i}\\}_{i=0}^{D}$) of the eigenspaces of $A$ (resp.
$A^{\prime}$) with

$\displaystyle A^{\prime}V_{i}\subseteq V_{i}+V_{i+1}\qquad(0\leq i\leq d),$
(1) $\displaystyle AV^{\prime}_{i}\subseteq
V^{\prime}_{i}+V^{\prime}_{i+1}\qquad(0\leq i\leq D),$ (2)

where each of $V_{d+1},\,V^{\prime}_{D+1}$ is equal to $0$.

  3. (iii)

The restrictions

$\displaystyle[A^{\prime},A]^{d-2i}|_{V_{i}}:V_{i}\rightarrow
V_{d-i}\qquad(0\leq i\leq d/2),$ (3)
$\displaystyle[A,A^{\prime}]^{D-2i}|_{V^{\prime}_{i}}:V^{\prime}_{i}\rightarrow
V^{\prime}_{D-i}\qquad(0\leq i\leq D/2),$ (4)

are bijections.

###### Lemma 2.5

[14, Lemma 2.4] With reference to Definition 2.4, we have $d=D$.

###### Definition 2.6

[15, Definition 2.5] A bidiagonal triple (BD triple) on $V$ is an ordered
triple of linear transformations $A:V\to V$, $A^{\prime}:V\to V$, and
$A^{\prime\prime}:V\to V$ that satisfy the following three conditions.

  1. (i)

Each of $A,\,A^{\prime},\,A^{\prime\prime}$ is diagonalizable.

  2. (ii)

There exists an ordering $\\{V_{i}\\}_{i=0}^{d}$ (resp.
$\\{V^{\prime}_{i}\\}_{i=0}^{D}$) (resp.
$\\{V^{\prime\prime}_{i}\\}_{i=0}^{\delta}$) of the eigenspaces of $A$ (resp.
$A^{\prime}$) (resp. $A^{\prime\prime}$) with

$\displaystyle A^{\prime}V_{i}\subseteq V_{i}+V_{i+1},\qquad
A^{\prime\prime}V_{i}\subseteq V_{i-1}+V_{i}\qquad(0\leq i\leq d),$ (5)
$\displaystyle A^{\prime\prime}V^{\prime}_{i}\subseteq
V^{\prime}_{i}+V^{\prime}_{i+1},\qquad AV^{\prime}_{i}\subseteq
V^{\prime}_{i-1}+V^{\prime}_{i}\qquad(0\leq i\leq D),$ (6) $\displaystyle
AV^{\prime\prime}_{i}\subseteq
V^{\prime\prime}_{i}+V^{\prime\prime}_{i+1},\qquad
A^{\prime}V^{\prime\prime}_{i}\subseteq
V^{\prime\prime}_{i-1}+V^{\prime\prime}_{i}\qquad(0\leq i\leq\delta),$ (7)

where each of
$V_{d+1},\,V_{-1},\,V^{\prime}_{D+1},\,V^{\prime}_{-1},\,V^{\prime\prime}_{\delta+1},\,V^{\prime\prime}_{-1}$
is equal to $0$.

  3. (iii)

The restrictions

$\displaystyle[A^{\prime},A]^{d-2i}|_{V_{i}}:V_{i}\rightarrow
V_{d-i},\qquad[A^{\prime\prime},A]^{d-2i}|_{V_{d-i}}:V_{d-i}\rightarrow
V_{i}\qquad(0\leq i\leq d/2),$ (8)
$\displaystyle[A^{\prime\prime},A^{\prime}]^{D-2i}|_{V^{\prime}_{i}}:V^{\prime}_{i}\rightarrow
V^{\prime}_{D-i},\qquad[A,A^{\prime}]^{D-2i}|_{V^{\prime}_{D-i}}:V^{\prime}_{D-i}\rightarrow
V^{\prime}_{i}\,\,\,\,\,\,(0\leq i\leq D/2),$ (9)
$\displaystyle[A,A^{\prime\prime}]^{\delta-2i}|_{V^{\prime\prime}_{i}}:V^{\prime\prime}_{i}\rightarrow
V^{\prime\prime}_{\delta-i},\qquad[A^{\prime},A^{\prime\prime}]^{\delta-2i}|_{V^{\prime\prime}_{\delta-i}}:V^{\prime\prime}_{\delta-i}\rightarrow
V^{\prime\prime}_{i}\qquad(0\leq i\leq\delta/2),$ (10)

are bijections.

###### Lemma 2.7

[15, Lemma 2.6] With reference to Definition 2.6, we have $d=D=\delta$.

###### Definition 2.8

A bidiagonal triad (BD triad) on $V$ is an ordered triple of linear
transformations $A:V\to V$, $A^{\prime}:V\to V$, and $A^{\prime\prime}:V\to V$
that satisfy the following three conditions.

  1. (i)

Each of $A,\,A^{\prime},\,A^{\prime\prime}$ is diagonalizable.

  2. (ii)

There exists an ordering $\\{V_{i}\\}_{i=0}^{d}$ (resp.
$\\{V^{\prime}_{i}\\}_{i=0}^{D}$) (resp.
$\\{V^{\prime\prime}_{i}\\}_{i=0}^{\delta}$) of the eigenspaces of $A$ (resp.
$A^{\prime}$) (resp. $A^{\prime\prime}$) with

$\displaystyle A^{\prime}V_{i}\subseteq V_{i}+V_{i+1},\qquad
A^{\prime\prime}V_{i}\subseteq V_{i}+V_{i+1}\qquad(0\leq i\leq d),$ (11)
$\displaystyle A^{\prime\prime}V^{\prime}_{i}\subseteq
V^{\prime}_{i}+V^{\prime}_{i+1},\qquad AV^{\prime}_{i}\subseteq
V^{\prime}_{i}+V^{\prime}_{i+1}\qquad(0\leq i\leq D),$ (12) $\displaystyle
AV^{\prime\prime}_{i}\subseteq
V^{\prime\prime}_{i}+V^{\prime\prime}_{i+1},\qquad
A^{\prime}V^{\prime\prime}_{i}\subseteq
V^{\prime\prime}_{i}+V^{\prime\prime}_{i+1}\qquad(0\leq i\leq\delta),$ (13)

where each of $V_{d+1},\,V^{\prime}_{D+1},\,V^{\prime\prime}_{\delta+1}$ is
equal to $0$.

  3. (iii)

The restrictions

$\displaystyle[A^{\prime},A]^{d-2i}|_{V_{i}}:V_{i}\rightarrow
V_{d-i},\qquad[A^{\prime\prime},A]^{d-2i}|_{V_{i}}:V_{i}\rightarrow
V_{d-i}\qquad(0\leq i\leq d/2),$ (14)
$\displaystyle[A^{\prime\prime},A^{\prime}]^{D-2i}|_{V^{\prime}_{i}}:V^{\prime}_{i}\rightarrow
V^{\prime}_{D-i},\qquad[A,A^{\prime}]^{D-2i}|_{V^{\prime}_{i}}:V^{\prime}_{i}\rightarrow
V^{\prime}_{D-i}\,\,\,\,\,\,(0\leq i\leq D/2),$ (15)
$\displaystyle[A,A^{\prime\prime}]^{\delta-2i}|_{V^{\prime\prime}_{i}}:V^{\prime\prime}_{i}\rightarrow
V^{\prime\prime}_{\delta-i},\qquad[A^{\prime},A^{\prime\prime}]^{\delta-2i}|_{V^{\prime\prime}_{i}}:V^{\prime\prime}_{i}\rightarrow
V^{\prime\prime}_{\delta-i}\qquad(0\leq i\leq\delta/2),$ (16)

are bijections.

###### Lemma 2.9

With reference to Definition 2.8, we have $d=D=\delta$.

Proof: Let $A,\,A^{\prime},\,A^{\prime\prime}$ denote a BD triad. Then
$A,\,A^{\prime}$ is a BD pair. So by Lemma 2.5 we have $d=D$. Also,
$A^{\prime},\,A^{\prime\prime}$ is a BD pair. So by Lemma 2.5 we have
$D=\delta$. $\Box$

With reference to Definition 2.8 we call $V$ the vector space underlying
$A,\,A^{\prime},\,A^{\prime\prime}$. We say
$A,\,A^{\prime},\,A^{\prime\prime}$ is over $\mathbb{K}$. In view of Lemma
2.9, for the remainder of this paper we use $d$ to index the eigenspaces of
$A$, $A^{\prime}$, and $A^{\prime\prime}$. We call $d$ the diameter of
$A,\,A^{\prime},\,A^{\prime\prime}$. The diameter of a BD pair and BD triple
are defined similarly.

For the remainder of this paper we assume that the field $\mathbb{K}$ is
algebraically closed and characteristic zero.

Let $A,\,A^{\prime},\,A^{\prime\prime}$ denote a BD triad of diameter $d$. An
ordering of the eigenspaces of $A$ (resp. $A^{\prime}$) (resp.
$A^{\prime\prime}$) is called standard whenever this ordering satisfies (11)
(resp. (12)) (resp. (13)). Let $\\{V_{i}\\}_{i=0}^{d}$ (resp.
$\\{V^{\prime}_{i}\\}_{i=0}^{d}$) (resp.
$\\{V^{\prime\prime}_{i}\\}_{i=0}^{d}$) denote a standard ordering of the
eigenspaces of $A$ (resp. $A^{\prime}$) (resp. $A^{\prime\prime}$). Then no
other ordering of these eigenspaces is standard, and so the BD triad
$A,\,A^{\prime},\,A^{\prime\prime}$ uniquely determines these three standard
orderings. For $0\leq i\leq d$, let $\theta_{i}$ (resp. $\theta^{\prime}_{i}$)
(resp. $\theta^{\prime\prime}_{i}$) denote the eigenvalue of $A$ (resp.
$A^{\prime}$) (resp. $A^{\prime\prime}$) corresponding to $V_{i}$ (resp.
$V^{\prime}_{i}$) (resp. $V^{\prime\prime}_{i}$). We call
$\\{\theta_{i}\\}_{i=0}^{d}$ (resp. $\\{\theta^{\prime}_{i}\\}_{i=0}^{d}$)
(resp. $\\{\theta^{\prime\prime}_{i}\\}_{i=0}^{d}$) the first (resp. second)
(resp. third) eigenvalue sequence of $A,\,A^{\prime},\,A^{\prime\prime}$. For
the remainder of this section, we adopt the notation from this paragraph. All
the terms defined in this paragraph have analogs for BD pairs and BD triples,
see [14, Section 2] and [15, Section 2].

###### Lemma 2.10

For $0\leq i\leq d$, the spaces $V_{i}$, $V_{d-i}$, $V^{\prime}_{i}$,
$V^{\prime}_{d-i}$, $V^{\prime\prime}_{i}$, $V^{\prime\prime}_{d-i}$ all have
the same dimension.

Proof: Since $A,\,A^{\prime}$ is a BD pair then, by [14, Lemma 2.7], the
spaces $V_{i}$, $V_{d-i}$, $V^{\prime}_{i}$, $V^{\prime}_{d-i}$ all have the
same dimension. Since $A^{\prime},\,A^{\prime\prime}$ is a BD pair then, by
[14, Lemma 2.7], the spaces $V^{\prime}_{i}$, $V^{\prime}_{d-i}$,
$V^{\prime\prime}_{i}$, $V^{\prime\prime}_{d-i}$ all have the same dimension.
$\Box$

###### Definition 2.11

With reference to Lemma 2.10, for $0\leq i\leq d$, let $\rho_{i}$ denote the
common dimension of $V_{i}$, $V_{d-i}$, $V^{\prime}_{i}$, $V^{\prime}_{d-i}$,
$V^{\prime\prime}_{i}$, $V^{\prime\prime}_{d-i}$. We refer to the sequence
$\\{\rho_{i}\\}_{i=0}^{d}$ as the shape of
$A,\,A^{\prime},\,A^{\prime\prime}$.

The shape of a BD pair and BD triple are defined similarly. See [14, Lemma
2.8] and [15, Lemma 2.9].

The following definitions will be used in stating two of our main results (see
Theorem 4.6 and Theorem 4.3).

###### Definition 2.12

Let $\\{\rho_{i}\\}_{i=0}^{d}$ denote the shape of
$A,\,A^{\prime},\,A^{\prime\prime}$. We say that
$A,\,A^{\prime},\,A^{\prime\prime}$ is thin whenever $\rho_{i}=1$ for $0\leq
i\leq d$.

Similarly, a BD pair or BD triple is called thin whenever its shape is
$(1,1,1,\ldots,1)$.

###### Definition 2.13

Let $\\{\sigma_{i}\\}_{i=0}^{d}$ and $\\{\tau_{i}\\}_{i=0}^{d}$ each denote a
sequence of scalars taken from $\mathbb{K}$. Let $X:V\to V$ and $Y:V\to V$
denote linear transformations. Let $B,\,B^{\prime},\,B^{\prime\prime}$ denote
a BD triad on $V$.

  1. (i)

We say $\\{\sigma_{i}\\}_{i=0}^{d}$ is affine equivalent to
$\\{\tau_{i}\\}_{i=0}^{d}$, denoted
$\\{\sigma_{i}\\}_{i=0}^{d}\sim\\{\tau_{i}\\}_{i=0}^{d}$, whenever there exist
$r,s$ in $\mathbb{K}$ with $r\neq 0$ such that $\sigma_{i}=r\tau_{i}+s$
($0\leq i\leq d$).

  2. (ii)

We say $X$ is affine equivalent to $Y$, denoted $X\sim Y$, whenever there
exist $r,s$ in $\mathbb{K}$ with $r\neq 0$ such that $X=rY+sI$.

  3. (iii)

We say $A,\,A^{\prime},\,A^{\prime\prime}$ is affine equivalent to
$B,\,B^{\prime},\,B^{\prime\prime}$ whenever $A\sim B$, $A^{\prime}\sim
B^{\prime}$, and $A^{\prime\prime}\sim B^{\prime\prime}$.

##  3 The Tetrahedron Algebra

###### Definition 3.1

[17, Definition 1.1] Let $\boxtimes$ denote the Lie algebra over $\mathbb{K}$
that has generators

$\\{X_{ij}\,|\,i,j\in\mathbb{I},i\neq j\\}\qquad\mathbb{I}=\\{0,1,2,3\\}$

and the following relations:

  1. (i)

For distinct $i,j\in\mathbb{I}$,

$X_{ij}+X_{ji}=0.$

  2. (ii)

For mutually distinct $h,i,j\in\mathbb{I}$,

$[X_{hi},X_{ij}]=2X_{hi}+2X_{ij}.$

  3. (iii)

For mutually distinct $h,i,j,k\in\mathbb{I}$,

$[X_{hi},[X_{hi},[X_{hi},X_{jk}]]]=4[X_{hi},X_{jk}].$

We call $\boxtimes$ the tetrahedron algebra and pronounce the symbol
$\boxtimes$ as ”tet”.

The relations in Definition 3.1(iii) are the well-known Dolan-Grady relations.
See [1, 7, 8, 9, 10, 28].

The following definition will be used in stating two of our main results (see
Theorem 4.5 and Theorem 4.6).

###### Definition 3.2

We call $X_{ru},X_{su},X_{tu}$ a corner triad in $\boxtimes$ whenever
$X_{ru},X_{su},X_{tu}$ are mutually distinct generators of $\boxtimes$.

The name ”corner triad” comes from the fact that the generators
$X_{ru},X_{su},X_{tu}$ correspond to the three edges of the tetrahedron that
meet at the vertex labeled by $u$.

The following information concerning $\boxtimes$-modules will be useful in
proving one of our main results (Theorem 4.5).

###### Lemma 3.3

[16, Theorem 3.8] Let $V$ denote a finite-dimensional irreducible
$\boxtimes$-module. Then the following (i), (ii) hold.

  1. (i)

For distinct $r,s\in\mathbb{I}$ the generator $X_{rs}$ is diagonalizable on
$V$.

  2. (ii)

There exists an integer $d\geq 0$ such that for all distinct
$r,s\in\mathbb{I}$ the eigenvalues for $X_{rs}$ on $V$ are
$\\{d-2i\\}_{i=0}^{d}$.

###### Definition 3.4

[16, Definition 3.9] Let $V$ denote a finite-dimensional irreducible
$\boxtimes$-module. By the diameter of $V$ we mean the nonnegative integer $d$
from Lemma 3.3(ii).

###### Notation 3.5

Let $V$ denote a finite-dimensional irreducible $\boxtimes$-module of diameter
$d$. For distinct $r,s\in\mathbb{I}$ and $0\leq i\leq d$ we let $V_{rs}(d-2i)$
denote the eigenspace of $X_{rs}$ on $V$ corresponding to eigenvalue $d-2i$.
That is,

$\displaystyle V_{rs}(d-2i):=\\{v\in V\,|\,X_{rs}v=(d-2i)v\\}.$

Note that by Definition 3.1(i) we have

$\displaystyle X_{sr}=-X_{rs},$ (17)

and so

$\displaystyle V_{rs}(d-2i)=V_{sr}(2i-d).$ (18)

We now discuss the Lie algebra $\mathfrak{sl}_{2}$ and its relationship to
$\boxtimes$.

###### Definition 3.6

Let $\mathfrak{sl}_{2}$ denote the Lie algebra over $\mathbb{K}$ that has a
basis $h,\,e,\,f$ and Lie bracket

$\displaystyle[h,e]=2e,\qquad[h,f]=-2f,\qquad[e,f]=h.$

The following two lemmas give a description of all finite-dimensional
$\mathfrak{sl}_{2}$-modules.

###### Lemma 3.7

[19, Theorem 6.3] Each finite-dimensional $\mathfrak{sl}_{2}$-module $V$ is
completely reducible; this means that $V$ is a direct sum of irreducible
$\mathfrak{sl}_{2}$-modules.

The finite-dimensional irreducible $\mathfrak{sl}_{2}$-modules are described
as follows.

###### Lemma 3.8

[19, Theorem 7.2] There exists a family of finite-dimensional irreducible
$\mathfrak{sl}_{2}$-modules

$\displaystyle V(d),\qquad d=0,1,2,\ldots$

with the following properties: $V(d)$ has a basis $\\{v_{i}\\}_{i=0}^{d}$ such
that $h.v_{i}=(d-2i)v_{i}$ for $0\leq i\leq d$, $f.v_{i}=(i+1)v_{i+1}$ for
$0\leq i\leq d$, where $v_{d+1}=0$, and $e.v_{i}=(d-i+1)v_{i-1}$ for $0\leq
i\leq d$, where $v_{-1}=0$. Moreover, every finite-dimensional irreducible
$\mathfrak{sl}_{2}$-module is isomorphic to exactly one of the modules $V(d)$.

###### Lemma 3.9

[14, Lemma 3.7] Let $V$ denote an $\mathfrak{sl}_{2}$-module with finite
positive dimension. Define

$\displaystyle V_{\rm{even}}$ $\displaystyle:=$
$\displaystyle\operatorname{span}\\{\,v\in
V\,|\,h.v=i\,v,\,\,i\in\mathbb{Z},\,i\,even\,\\},$ $\displaystyle
V_{\rm{odd}}$ $\displaystyle:=$ $\displaystyle\operatorname{span}\\{\,v\in
V\,|\,h.v=i\,v,\,\,i\in\mathbb{Z},\,i\,odd\,\\}.$

Then $V_{\rm{even}}$ and $V_{\rm{odd}}$ are $\mathfrak{sl}_{2}$-modules, and
$V=V_{\rm{even}}+V_{\rm{odd}}$ (direct sum).

###### Definition 3.10

[14, Definition 3.8] Let $V$ denote an $\mathfrak{sl}_{2}$-module with finite
positive dimension. With reference to Lemma 3.9, we say $V$ is segregated
whenever $V=V_{\rm{even}}$ or $V=V_{\rm{odd}}$.

The following two lemmas explain the relationship between $\mathfrak{sl}_{2}$
and $\boxtimes$.

###### Lemma 3.11

[17, Lemma 3.2] The Lie algebra $\mathfrak{sl}_{2}$ is isomorphic to the Lie
algebra over $\mathbb{K}$ that has basis $X,\,Y,\,Z$ and Lie bracket

$\displaystyle[X,Y]=2X+2Y,\qquad[Y,Z]=2Y+2Z,\qquad[Z,X]=2Z+2X.$

An isomorphism with the presentation in Definition 3.6 is given by:

$\displaystyle X\rightarrow 2e-h,\qquad Y\rightarrow-2f-h,\qquad Z\rightarrow
h.$

The inverse of this isomorphism is given by:

$\displaystyle e\rightarrow(X+Z)/2,\qquad f\rightarrow-(Y+Z)/2,\qquad
h\rightarrow Z.$

###### Definition 3.12

[15, Definition 3.3] Let $X,\,Y,\,Z$ denote a basis for $\mathfrak{sl}_{2}$
satisfying the relations from Lemma 3.11. We refer to the ordered triple
$X,\,Y,\,Z$ as an equitable triple in $\mathfrak{sl}_{2}$.

###### Lemma 3.13

[17, Proposition 3.6] Let $h,i,j$ denote mutually distinct elements of
$\mathbb{I}$. Then there exists a unique Lie algebra homomorphism from
$\mathfrak{sl}_{2}$ to $\boxtimes$ that sends

$\displaystyle X\rightarrow X_{hi},\qquad Y\rightarrow X_{ij},\qquad
Z\rightarrow X_{jh}.$

##  4 The Main Theorems

The four theorems in this section make up the main conclusions of the paper.

###### Theorem 4.1

Let $A,\,A^{\prime},\,A^{\prime\prime}$ denote a BD triad on $V$ of diameter
$d$. Let $\\{\theta_{i}\\}_{i=0}^{d}$, $\\{\theta^{\prime}_{i}\\}_{i=0}^{d}$,
and $\\{\theta^{\prime\prime}_{i}\\}_{i=0}^{d}$ denote the first, second, and
third eigenvalue sequences of $A,\,A^{\prime},\,A^{\prime\prime}$
respectively. Suppose that $d\geq 2$. Then for $1\leq i\leq d-1$ we have

$\displaystyle\frac{\theta_{i+1}-\theta_{i}}{\theta_{i}-\theta_{i-1}}=\frac{\theta^{\prime}_{i+1}-\theta^{\prime}_{i}}{\theta^{\prime}_{i}-\theta^{\prime}_{i-1}}=\frac{\theta^{\prime\prime}_{i+1}-\theta^{\prime\prime}_{i}}{\theta^{\prime\prime}_{i}-\theta^{\prime\prime}_{i-1}}=1.$
(19)

We refer to Theorem 4.1 as the recurrence theorem. The following definition
will be used to state the next three theorems.

###### Definition 4.2

Let $A,\,A^{\prime},\,A^{\prime\prime}$ denote a BD triad of diameter $d$. We
say $A,\,A^{\prime},\,A^{\prime\prime}$ is reduced if the first, second, and
third eigenvalue sequences of $A,\,A^{\prime},\,A^{\prime\prime}$ are each
$\\{2i-d\\}_{i=0}^{d}$.

See [14, Definition 5.7] (resp. [15, Definition 4.5]) for the corresponding
definition of a reduced BD pair (resp. BD triple).

###### Theorem 4.3

Every BD triad is affine equivalent to a reduced BD triad.

We refer to Theorem 4.3 as the reducibility theorem.

###### Note 4.4

We make the following observation in order to motivate the next two theorems.
Let $X_{ru},X_{su},X_{tu}$ denote a corner triad in $\boxtimes$. By Definition
3.1(i),(ii) we have

$\displaystyle[X_{ru},X_{su}]$ $\displaystyle=$
$\displaystyle-2X_{ru}+2X_{su},$ $\displaystyle{[X_{su},X_{tu}]}$
$\displaystyle=$ $\displaystyle-2X_{su}+2X_{tu}$
$\displaystyle{[X_{tu},X_{ru}]}$ $\displaystyle=$
$\displaystyle-2X_{tu}+2X_{ru}.$

Let $A,\,A^{\prime},\,A^{\prime\prime}$ denote a reduced BD triad. It is shown
in Lemma 5.7 that

$\displaystyle[A,A^{\prime}]$ $\displaystyle=$ $\displaystyle-2A+2A^{\prime},$
$\displaystyle{[A^{\prime},A^{\prime\prime}]}$ $\displaystyle=$
$\displaystyle-2A^{\prime}+2A^{\prime\prime},$
$\displaystyle{[A^{\prime\prime},A]}$ $\displaystyle=$
$\displaystyle-2A^{\prime\prime}+2A.$

The following two theorems provide a correspondence between reduced BD triads
and irreducible $\boxtimes$-modules.

###### Theorem 4.5

Let $V$ denote a finite-dimensional irreducible $\boxtimes$-module. Then each
corner triad in $\boxtimes$ acts on $V$ as a reduced BD triad.

We refer to the result in Theorem 4.5 as corner triads act as BD triads. The
next theorem can be thought of as a partial converse of Theorem 4.5.

###### Theorem 4.6

Let $A,\,A^{\prime},\,A^{\prime\prime}$ denote a reduced thin BD triad on $V$.
Let $X_{ru},X_{su},X_{tu}$ denote a corner triad in $\boxtimes$. Then there
exists a $\boxtimes$-module structure on $V$ such that $(X_{ru}-A)V=0$,
$(X_{su}-A^{\prime})V=0$, and $(X_{tu}-A^{\prime\prime})V=0$. This
$\boxtimes$-module structure on $V$ is irreducible.

We refer to the result in Theorem 4.6 as BD triads act as corner triads.
Theorem 4.3 and Theorem 4.6 together show that, in the thin case, every BD
triad is affine equivalent to a BD triad of the type constructed in Theorem
4.5.

The $\boxtimes$-module constructed in Theorem 4.6 is an evaluation module for
$\boxtimes$ as defined in [25, Definition 5.7]. See also [25, Proposition
6.4]. Evaluation modules are the building blocks for all finite-dimensional
irreducible $\boxtimes$-modules as described in [25, Theorem 15.2].

In Theorem 4.6 the assumption that $A,\,A^{\prime},\,A^{\prime\prime}$ is thin
is needed to guarantee the conclusion of the theorem. In Section 11 we provide
a counterexample to demonstrate this.

##  5 Preliminaries

In this section we develop some tools needed to prove our main results.

###### Definition 5.1

Suppose that $d\geq 2$ and let $\\{\sigma_{i}\\}_{i=0}^{d}$ denote a sequence
of distinct scalars taken from $\mathbb{K}$. This sequence is called
$1$-recurrent whenever

$\displaystyle\frac{\sigma_{i+1}-\sigma_{i}}{\sigma_{i}-\sigma_{i-1}}=1\qquad(1\leq
i\leq d-1).$

###### Lemma 5.2

[15, Lemma 5.2] Suppose that $d\geq 2$ and let $\\{\sigma_{i}\\}_{i=0}^{d}$
and $\\{\tau_{i}\\}_{i=0}^{d}$ each denote a sequence of distinct scalars
taken from $\mathbb{K}$. Assume $\\{\sigma_{i}\\}_{i=0}^{d}$ and
$\\{\tau_{i}\\}_{i=0}^{d}$ are each $1$-recurrent. Then
$\\{\sigma_{i}\\}_{i=0}^{d}\sim\\{\tau_{i}\\}_{i=0}^{d}$.

###### Lemma 5.3

[15, Lemma 5.3] Suppose that $d<2$ and let $\\{\sigma_{i}\\}_{i=0}^{d}$ and
$\\{\tau_{i}\\}_{i=0}^{d}$ each denote a sequence of distinct scalars taken
from $\mathbb{K}$. Then
$\\{\sigma_{i}\\}_{i=0}^{d}\sim\\{\tau_{i}\\}_{i=0}^{d}$.

###### Definition 5.4

[15, Definition 5.4] A decomposition of $V$ is a sequence
$\\{U_{i}\\}_{i=0}^{d}$ consisting of nonzero subspaces of $V$ such that
$V=\sum_{i=0}^{d}U_{i}$ (direct sum). For any decomposition of $V$ we adopt
the convention that $U_{-1}:=0$ and $U_{d+1}:=0$.

###### Lemma 5.5

[15, Lemma 5.5] Let $\\{U_{i}\\}_{i=0}^{d}$ denote a decomposition of $V$. Let
$\\{\sigma_{i}\\}_{i=0}^{d}$ and $\\{\tau_{i}\\}_{i=0}^{d}$ each denote a
sequence of distinct scalars taken from $\mathbb{K}$. Let $X:V\to V$ (resp.
$Y:V\to V$) denote the linear transformation such that for $0\leq i\leq d$,
$U_{i}$ is an eigenspace for $X$ (resp. $Y$) with eigenvalue $\sigma_{i}$
(resp. $\tau_{i}$). Then
$\\{\sigma_{i}\\}_{i=0}^{d}\sim\\{\tau_{i}\\}_{i=0}^{d}$ if and only if $X\sim
Y$.

We now develop some more properties of BD triads.

###### Lemma 5.6

Let $A,\,A^{\prime},\,A^{\prime\prime}$ denote a BD triad on $V$ of diameter
$d$. Let $\\{\theta_{i}\\}_{i=0}^{d}$, $\\{\theta^{\prime}_{i}\\}_{i=0}^{d}$,
$\\{\theta^{\prime\prime}_{i}\\}_{i=0}^{d}$ denote the first, second, third
eigenvalue sequences of $A,\,A^{\prime},\,A^{\prime\prime}$ respectively. Let
$r,\,s,\,t,\,u,\,v,\,w$ denote scalars in $\mathbb{K}$ with $r,\,t,\,v$ each
nonzero. Then $rA+sI,\,tA^{\prime}+uI,\,vA^{\prime\prime}+wI$ is a BD triad on
$V$. Moreover, $\\{r\theta_{i}+s\\}_{i=0}^{d}$,
$\\{t\theta^{\prime}_{i}+u\\}_{i=0}^{d}$,
$\\{v\theta^{\prime\prime}_{i}+w\\}_{i=0}^{d}$ are the first, second, third
eigenvalue sequences of $rA+sI,\,tA^{\prime}+uI,\,vA^{\prime\prime}+wI$
respectively.

Proof: Imitate the proof of [14, Lemma 2.10]. $\Box$

###### Lemma 5.7

Let $A,\,A^{\prime},\,A^{\prime\prime}$ denote a reduced BD triad. Then the
following hold.

$\displaystyle[A,A^{\prime}]$ $\displaystyle=$ $\displaystyle-2A+2A^{\prime},$
(20) $\displaystyle{[A^{\prime},A^{\prime\prime}]}$ $\displaystyle=$
$\displaystyle-2A^{\prime}+2A^{\prime\prime},$ (21)
$\displaystyle{[A^{\prime\prime},A]}$ $\displaystyle=$
$\displaystyle-2A^{\prime\prime}+2A.$ (22)

Proof: We show (20). The proofs of (21) and (22) are similar. Let
$\\{\theta_{i}\\}_{i=0}^{d}$ (resp. $\\{\theta^{\prime}_{i}\\}_{i=0}^{d}$)
denote the first (resp. second) eigenvalue sequence of
$A,\,A^{\prime},\,A^{\prime\prime}$. By Definition 4.2 we have

$\displaystyle\theta_{i}=2i-d=\theta^{\prime}_{i}\qquad(0\leq i\leq d).$ (23)

First assume that $d=0$. Observe that $A=\theta_{0}I$ and
$A^{\prime}=\theta^{\prime}_{0}I$. By (23), $\theta_{0}=0$ and
$\theta^{\prime}_{0}=0$. Combining the previous two sentences we obtain (20).
Now assume that $d\geq 1$. Observe that $A,\,A^{\prime}$ is a BD pair. So by
[14, Theorem 5.3] there exist scalars $b,\alpha,\alpha^{*},\gamma$ such that

$\displaystyle AA^{\prime}-bA^{\prime}A-\alpha A-\alpha^{*}A^{\prime}-\gamma
I=0.$ (24)

When $d=1$ then $b=1$ by [14, Definition 5.5]. When $d\geq 2$ then $b=1$ by
(23) and [14, Lemma 9.1]. Combining this with [14, Theorem 8.1] and (23) we
have that $\alpha=-2,\alpha^{*}=2,\gamma=0$. Substituting these values into
(24) we obtain (20). $\Box$

##  6 The Proofs of the Recurrence and Reducibility
Theorems

In this section we prove Theorem 4.1 and Theorem 4.3. First we prove the
recurrence theorem.

Proof of Theorem 4.1: Let $A,\,A^{\prime},\,A^{\prime\prime}$ denote a BD
triad on $V$ of diameter $d$. Let $\\{\theta_{i}\\}_{i=0}^{d}$,
$\\{\theta^{\prime}_{i}\\}_{i=0}^{d}$, and
$\\{\theta^{\prime\prime}_{i}\\}_{i=0}^{d}$ denote the first, second, and
third eigenvalue sequences of $A,\,A^{\prime},\,A^{\prime\prime}$
respectively. Suppose that $d\geq 2$. For $1\leq i\leq d-1$ abbreviate
$\frac{\theta_{i+1}-\theta_{i}}{\theta_{i}-\theta_{i-1}}$,
$\frac{\theta^{\prime}_{i+1}-\theta^{\prime}_{i}}{\theta^{\prime}_{i}-\theta^{\prime}_{i-1}}$,
and
$\frac{\theta^{\prime\prime}_{i+1}-\theta^{\prime\prime}_{i}}{\theta^{\prime\prime}_{i}-\theta^{\prime\prime}_{i-1}}$
as $T_{i}$, $T^{\prime}_{i}$, and $T^{\prime\prime}_{i}$ respectively. By
construction $\\{\theta_{i}\\}_{i=0}^{d}$ is a list of distinct scalars. So
$T_{i}\neq 0$. Similarly, $T^{\prime}_{i}\neq 0$ and $T^{\prime\prime}_{i}\neq
0$. Observe that $A,\,A^{\prime}$ is a BD pair. So by [14, Theorem 5.1] we
have

$\displaystyle T_{i}=T^{\prime-1}_{i}\qquad(1\leq i\leq d-1).$ (25)

Observe that $A^{\prime},\,A^{\prime\prime}$ is a BD pair. So by [14, Theorem
5.1] we have

$\displaystyle T^{\prime}_{i}=T^{\prime\prime-1}_{i}\qquad(1\leq i\leq d-1).$
(26)

Observe that $A^{\prime\prime},\,A$ is a BD pair. So by [14, Theorem 5.1] we
have

$\displaystyle T^{\prime\prime}_{i}=T^{-1}_{i}\qquad(1\leq i\leq d-1).$ (27)

Combining (25), (26), (27) we have that $T_{i}=T^{-1}_{i}$. So $T_{i}$ must
equal $1$ or $-1$. But $\\{\theta_{i}\\}_{i=0}^{d}$ is a list of distinct
scalars so $T_{i}\neq-1$. Thus $T_{i}=1$ for $1\leq i\leq d-1$. Similarly
$T^{\prime}_{i}=1$ and $T^{\prime\prime}_{i}=1$ for $1\leq i\leq d-1$. $\Box$

We now prove the reducibility theorem. The following proof is essentially the
same as the proof of [15, Theorem 4.6]. For the sake of completeness and
accessibility we reproduce the argument here in full.

Proof of Theorem 4.3: Let $A,\,A^{\prime},\,A^{\prime\prime}$ denote a BD
triad on $V$ of diameter $d$. Let $\\{\theta_{i}\\}_{i=0}^{d}$,
$\\{\theta^{\prime}_{i}\\}_{i=0}^{d}$, and
$\\{\theta^{\prime\prime}_{i}\\}_{i=0}^{d}$ denote the first, second, and
third eigenvalue sequences of $A,\,A^{\prime},\,A^{\prime\prime}$
respectively. Let $\\{V_{i}\\}_{i=0}^{d}$, $\\{V^{\prime}_{i}\\}_{i=0}^{d}$,
and $\\{V^{\prime\prime}_{i}\\}_{i=0}^{d}$ denote the standard orderings of
the eigenspaces of $A,\,A^{\prime},\,A^{\prime\prime}$ respectively. First we
show that

$\displaystyle{\rm\,\,each\,\,of\,\,}\\{\theta_{i}\\}_{i=0}^{d},\\{\theta^{\prime}_{i}\\}_{i=0}^{d},\\{\theta^{\prime\prime}_{i}\\}_{i=0}^{d}{\rm\,\,is\,\,affine\,\,equivalent\,\,to\,\,}\\{2i-d\\}_{i=0}^{d}.$
(28)

If $d<2$ then (28) is immediate from Lemma 5.3. Now assume that $d\geq 2$. By
Theorem 4.1, each of the sequences $\\{\theta_{i}\\}_{i=0}^{d}$,
$\\{\theta^{\prime}_{i}\\}_{i=0}^{d}$,
$\\{\theta^{\prime\prime}_{i}\\}_{i=0}^{d}$ is $1$-recurrent. Observe that the
sequence $\\{2i-d\\}_{i=0}^{d}$ is also $1$-recurrent. So by Lemma 5.2 we
obtain (28). Let $B:V\to V$ (resp. $B^{\prime}:V\to V$) (resp.
$B^{\prime\prime}:V\to V$) be the linear transformation such that for $0\leq
i\leq d$, $V_{i}$ (resp. $V^{\prime}_{i}$) (resp. $V^{\prime\prime}_{i}$) is
an eigenspace for $B$ (resp. $B^{\prime}$) (resp. $B^{\prime\prime}$) with
eigenvalue $2i-d$. From (28) and Lemma 5.5 we have $A\sim B$, $A^{\prime}\sim
B^{\prime}$, and $A^{\prime\prime}\sim B^{\prime\prime}$. Combining this with
Lemma 5.6 we find that $B,\,B^{\prime},\,B^{\prime\prime}$ is a BD triad on
$V$ of diameter $d$. Also $A,\,A^{\prime},\,A^{\prime\prime}$ is affine
equivalent to $B,\,B^{\prime},\,B^{\prime\prime}$. Lemma 5.5 and Lemma 5.6
also show that each of the first, second, and third eigenvalue sequences of
$B,\,B^{\prime},\,B^{\prime\prime}$ is $\\{2i-d\\}_{i=0}^{d}$. Hence,
$B,\,B^{\prime},\,B^{\prime\prime}$ is reduced. So
$A,\,A^{\prime},\,A^{\prime\prime}$ is affine equivalent to a reduced BD
triad. $\Box$

##  7 The Proof that Corner Triads Act as BD Triads

In this section we prove Theorem 4.5.

###### Note 7.1

Throughout the remainder of the paper we will refer to the base of a BD pair
and BD triple. See [14, Definition 5.5] and [15, Definition 2.13] for the
definitions of base.

Proof of Theorem 4.5: Let $V$ denote a finite-dimensional irreducible
$\boxtimes$-module of diameter $d$. Let $X_{ru},X_{su},X_{tu}$ denote a corner
triad in $\boxtimes$. We show $X_{ru},X_{su},X_{tu}$ acts on $V$ as a reduced
BD triad. First, observe that by Lemma 3.3

$\displaystyle{\rm
each\,\,of\,\,}X_{ru},X_{su},X_{tu}{\rm\,\,is\,\,diagonalizable\,\,on\,\,}V.$
(29)

By Lemma 3.13, there exists an $\mathfrak{sl}_{2}$-module structure on $V$
such that $(X-X_{tu})V=0,(Y-X_{us})V=0,(Z-X_{st})V=0$ where $X,Y,Z$ is an
equitable triple in $\mathfrak{sl}_{2}$. We now check that this
$\mathfrak{sl}_{2}$-module structure on $V$ is segregated. By Lemma 3.3 the
action of $X_{st}$ on $V$ is diagonalizable with eigenvalues
$\\{d-2i\\}_{i=0}^{d}$. Using the $\mathfrak{sl}_{2}$ isomorphism from Lemma
3.11 we have $(h-Z)V=0$. Thus, $(h-X_{st})V=0$ and so the action of $h$ on $V$
has eigenvalues $\\{d-2i\\}_{i=0}^{d}$. Hence, if $d$ is even (resp. odd) then
$V=V_{\rm{even}}$ (resp. $V=V_{\rm{odd}}$). This shows the
$\mathfrak{sl}_{2}$-module structure on $V$ is segregated. Applying [14,
Theorem 5.10] we have that $X_{tu},X_{us}$ acts on $V$ as a reduced BD pair of
diameter $d$ and base $1$. By [14, Definition 5.7] the eigenvalue (resp. dual
eigenvalue) sequence of $X_{tu},X_{us}$ is $\\{2i-d\\}_{i=0}^{d}$ (resp.
$\\{d-2i\\}_{i=0}^{d}$). See [14, Section 2] for the definitions of eigenvalue
sequence and dual eigenvalue sequence of a BD pair. From this, Definition
2.4(ii), and Notation 3.5 we have

$\displaystyle X_{tu}V_{us}(d-2i)\subseteq
V_{us}(d-2i)+V_{us}(d-2(i+1))\qquad(0\leq i\leq d),$ $\displaystyle
X_{us}V_{tu}(2i-d)\subseteq V_{tu}(2i-d)+V_{tu}(2(i+1)-d)\qquad(0\leq i\leq
d).$

Combining the previous sentence with (17) and (18) we have

$\displaystyle X_{tu}V_{su}(2i-d)\subseteq
V_{su}(2i-d)+V_{su}(2(i+1)-d)\qquad(0\leq i\leq d),$ (30) $\displaystyle
X_{su}V_{tu}(2i-d)\subseteq V_{tu}(2i-d)+V_{tu}(2(i+1)-d)\qquad(0\leq i\leq
d).$ (31)

From Definition 2.4(iii) and Notation 3.5 we have

$\displaystyle{\rm
the\,\,restriction\,\,}[X_{tu},X_{us}]^{d-2i}:V_{us}(d-2i)\rightarrow
V_{us}(2i-d){\rm\,\,is\,\,a\,\,bijection\,\,for\,\,}0\leq i\leq d/2,$
$\displaystyle{\rm
the\,\,restriction\,\,}[X_{tu},X_{us}]^{d-2i}:V_{tu}(2i-d)\rightarrow
V_{tu}(d-2i){\rm\,\,is\,\,a\,\,bijection\,\,for\,\,}0\leq i\leq d/2.$

Combining the previous sentence with (17) and (18) we have

$\displaystyle{\rm
the\,\,restriction\,\,}[X_{tu},X_{su}]^{d-2i}:V_{su}(2i-d)\rightarrow
V_{su}(d-2i){\rm\,\,is\,\,a\,\,bijection\,\,for\,\,}0\leq i\leq d/2,$ (32)
$\displaystyle{\rm
the\,\,restriction\,\,}[X_{su},X_{tu}]^{d-2i}:V_{tu}(2i-d)\rightarrow
V_{tu}(d-2i){\rm\,\,is\,\,a\,\,bijection\,\,for\,\,}0\leq i\leq d/2.$ (33)

By Lemma 3.13, there exists an $\mathfrak{sl}_{2}$-module structure on $V$
such that $(X-X_{su})V=0,(Y-X_{ur})V=0,(Z-X_{rs})V=0$ where $X,Y,Z$ is an
equitable triple in $\mathfrak{sl}_{2}$. Repeating the argument from the
previous paragraph in this case gives

$\displaystyle X_{su}V_{ru}(2i-d)\subseteq
V_{ru}(2i-d)+V_{ru}(2(i+1)-d)\qquad(0\leq i\leq d),$ (34) $\displaystyle
X_{ru}V_{su}(2i-d)\subseteq V_{su}(2i-d)+V_{su}(2(i+1)-d)\qquad(0\leq i\leq
d),$ (35) $\displaystyle{\rm
the\,\,restriction\,\,}[X_{su},X_{ru}]^{d-2i}:V_{ru}(2i-d)\rightarrow
V_{ru}(d-2i){\rm\,\,is\,\,a\,\,bijection\,\,for\,\,}0\leq i\leq d/2,$ (36)
$\displaystyle{\rm
the\,\,restriction\,\,}[X_{ru},X_{su}]^{d-2i}:V_{su}(2i-d)\rightarrow
V_{su}(d-2i){\rm\,\,is\,\,a\,\,bijection\,\,for\,\,}0\leq i\leq d/2.$ (37)

By Lemma 3.13, there exists an $\mathfrak{sl}_{2}$-module structure on $V$
such that $(X-X_{tu})V=0,(Y-X_{ur})V=0,(Z-X_{rt})V=0$ where $X,Y,Z$ is an
equitable triple in $\mathfrak{sl}_{2}$. Repeating the argument from the first
paragraph again in this case gives

$\displaystyle X_{tu}V_{ru}(2i-d)\subseteq
V_{ru}(2i-d)+V_{ru}(2(i+1)-d)\qquad(0\leq i\leq d),$ (38) $\displaystyle
X_{ru}V_{tu}(2i-d)\subseteq V_{tu}(2i-d)+V_{tu}(2(i+1)-d)\qquad(0\leq i\leq
d),$ (39) $\displaystyle{\rm
the\,\,restriction\,\,}[X_{tu},X_{ru}]^{d-2i}:V_{ru}(2i-d)\rightarrow
V_{ru}(d-2i){\rm\,\,is\,\,a\,\,bijection\,\,for\,\,}0\leq i\leq d/2,$ (40)
$\displaystyle{\rm
the\,\,restriction\,\,}[X_{ru},X_{tu}]^{d-2i}:V_{tu}(2i-d)\rightarrow
V_{tu}(d-2i){\rm\,\,is\,\,a\,\,bijection\,\,for\,\,}0\leq i\leq d/2.$ (41)

Combining (29) – (41) we have that $X_{ru},X_{su},X_{tu}$ acts on $V$ as a BD
triad. It is immediate from (30), (31), (34), (35), (38), (39) that the first,
second, and third eigenvalue sequences of $X_{ru},X_{su},X_{tu}$ are each
$\\{2i-d\\}_{i=0}^{d}$. Therefore, the BD triad $X_{ru},X_{su},X_{tu}$ is
reduced. $\Box$

##  8 The Raising Maps $R$ and $r$

This section and the next two are devoted to the proof of Theorem 4.6.
Throughout the remainder of the paper we will refer to the following
assumption.

###### Assumption 8.1

Let $A,\,A^{\prime},\,A^{\prime\prime}$ denote a reduced BD triad on $V$ of
diameter $d$. Let $\\{V_{i}\\}_{i=0}^{d}$, $\\{V^{\prime}_{i}\\}_{i=0}^{d}$,
and $\\{V^{\prime\prime}_{i}\\}_{i=0}^{d}$ denote the standard orderings of
the eigenspaces of $A,\,A^{\prime},\,A^{\prime\prime}$ respectively.

###### Definition 8.2

With reference to Assumption 8.1, let $R:V\rightarrow V$ and $r:V\rightarrow
V$ denote the following linear transformations:

$\displaystyle R:=A-A^{\prime\prime},\qquad\qquad
r:=A^{\prime}-A^{\prime\prime}.$

###### Note 8.3

With reference to Assumption 8.1 note that by Definition 2.8(iii) $A\neq
A^{\prime\prime}$, and so $R\neq 0$. Similarly, $A^{\prime}\neq
A^{\prime\prime}$ and so $r\neq 0$. Lastly, $A\neq A^{\prime}$ and so $R\neq
r$.

We refer to the linear transformations from Definition 8.2 as raising maps.
The following lemma explains this terminology.

###### Lemma 8.4

With reference to Definition 8.2 the following (i), (ii) hold.

  1. (i)

$RV^{\prime\prime}_{i}\subseteq V^{\prime\prime}_{i+1}\qquad(0\leq i\leq d)$.

  2. (ii)

$rV^{\prime\prime}_{i}\subseteq V^{\prime\prime}_{i+1}\qquad(0\leq i\leq d)$.

Proof: (i) Adopt Assumption 8.1. Observe that $A,\,-A^{\prime\prime}$ is a
reduced BD pair of base $1$. Combining this with [14, Lemma 6.7(i)] we have,
for $0\leq i\leq d$ and $v\in V^{\prime\prime}_{i}$,

$\displaystyle[A,-A^{\prime\prime}]v$ $\displaystyle=$ $\displaystyle
2(A-(2i-d)I)v$ $\displaystyle=$ $\displaystyle 2(A-A^{\prime\prime})v.$

Combining this with Lemma 2.3(i) and Definition 8.2 we obtain (i).
(ii) Similar to (i). $\Box$

###### Lemma 8.5

With reference to Definition 8.2 we have $Rr=rR$.

Proof: Adopt Assumption 8.1. Observe that $A,\,-A^{\prime}$ is a reduced BD
pair of base $1$. Combining this with [14, Lemma 12.2] we have

$\displaystyle[A,-A^{\prime}]=2A-2A^{\prime}.$ (42)

From Definition 8.2 we have $A=R+A^{\prime\prime}$ and
$A^{\prime}=r+A^{\prime\prime}$. Substituting these into (42) and simplifying
gives

$\displaystyle[R,r]+[R,A^{\prime\prime}]+[A^{\prime\prime},r]+2R-2r=0.$ (43)

Recall for $0\leq i\leq d$ that $V^{\prime\prime}_{i}$ is the eigenspace for
$A^{\prime\prime}$ corresponding to the eigenvalue $2i-d$. Combining this with
Lemma 8.4 and (43) we have

$\displaystyle([R,r]+(2i-d)R-(2i+2-d)R+(2i+2-d)r-(2i-d)r+2R-2r)V^{\prime\prime}_{i}=0.$

Simplifying this gives $(Rr-rR)V^{\prime\prime}_{i}=0$ for $0\leq i\leq d$.
Since $A^{\prime\prime}$ is diagonalizable then
$\\{V^{\prime\prime}_{i}\\}_{i=0}^{d}$ is a decomposition of $V$. Therefore,
$Rr-rR=0$ and the result follows. $\Box$

###### Lemma 8.6

Adopt Assumption 8.1 and assume that the BD triad
$A,\,A^{\prime},\,A^{\prime\prime}$ is thin. With reference to Definition 8.2
the following (i),(ii) hold.

  1. (i)

The restriction $R|_{V^{\prime\prime}_{i}}:V^{\prime\prime}_{i}\rightarrow
V^{\prime\prime}_{i+1}$ is a bijection for $0\leq i\leq d-1$.

  2. (ii)

The restriction $r|_{V^{\prime\prime}_{i}}:V^{\prime\prime}_{i}\rightarrow
V^{\prime\prime}_{i+1}$ is a bijection for $0\leq i\leq d-1$.

Proof: (i) Observe that $A,\,-A^{\prime\prime}$ is a reduced BD pair of base
$1$. Combining this with [14, Lemma 6.7(i)] and Definition 8.2 we have

$\displaystyle[A,-A^{\prime\prime}]|_{V^{\prime\prime}_{i}}=2R|_{V^{\prime\prime}_{i}}\qquad(0\leq
i\leq d).$ (44)

Case 1: $0\leq i<d/2$. Combining [14, Lemma 2.5] with (44) shows that the
restriction $R|_{V^{\prime\prime}_{i}}:V^{\prime\prime}_{i}\rightarrow
V^{\prime\prime}_{i+1}$ is an injection. Since
$A,\,A^{\prime},\,A^{\prime\prime}$ is thin then ${\rm
dim}(V^{\prime\prime}_{i})=1={\rm dim}(V^{\prime\prime}_{i+1})$. Thus, the
restriction $R|_{V^{\prime\prime}_{i}}:V^{\prime\prime}_{i}\rightarrow
V^{\prime\prime}_{i+1}$ is also a surjection and hence a bijection.
Case 2: $d/2\leq i\leq d-1$. Combining [14, Lemma 2.5] with (44) shows that
the restriction $R|_{V^{\prime\prime}_{i}}:V^{\prime\prime}_{i}\rightarrow
V^{\prime\prime}_{i+1}$ is a surjection. Since
$A,\,A^{\prime},\,A^{\prime\prime}$ is thin then ${\rm
dim}(V^{\prime\prime}_{i})=1={\rm dim}(V^{\prime\prime}_{i+1})$. Thus, the
restriction $R|_{V^{\prime\prime}_{i}}:V^{\prime\prime}_{i}\rightarrow
V^{\prime\prime}_{i+1}$ is also an injection and hence a bijection.
(ii) Similar to (i). $\Box$

###### Lemma 8.7

Adopt Assumption 8.1 and assume that the BD triad
$A,\,A^{\prime},\,A^{\prime\prime}$ is thin. With reference to Definition 8.2
there exists a nonzero $c\in\mathbb{K}$ such that $r=cR$.

Proof: Since $A^{\prime\prime}$ is diagonalizable then
$\\{V^{\prime\prime}_{i}\\}_{i=0}^{d}$ is a decomposition of $V$. So to prove
the desired result it suffices to show

$\displaystyle{\rm
there\,\,exists\,\,a\,\,nonzero\,\,}c\in\mathbb{K}{\rm\,\,such\,\,that\,\,}(r-cR)V^{\prime\prime}_{i}=0{\rm\,\,for\,\,}0\leq
i\leq d.$ (45)

Recall that since $A,\,A^{\prime},\,A^{\prime\prime}$ is thin we have

$\displaystyle{\rm dim}(V^{\prime\prime}_{i})=1\qquad 0\leq i\leq d.$ (46)

We show (45) by induction on $i$. First we show (45) holds for $i=0$. Let $v$
denote a nonzero vector in $V^{\prime\prime}_{0}$. Note that $\\{v\\}$ is a
basis for $V^{\prime\prime}_{0}$ by (46). Observe that $Rv\in
V^{\prime\prime}_{1}$ and $Rv\neq 0$ by Lemma 8.6(i). Also $rv\in
V^{\prime\prime}_{1}$ and $rv\neq 0$ by Lemma 8.6(ii). Combining the previous
two sentences with (46) shows there exists a nonzero $c\in\mathbb{K}$ such
that $rv=cRv$. This shows (45) holds for $i=0$ since $\\{v\\}$ is a basis for
$V^{\prime\prime}_{0}$. Now let $i\geq 1$ and assume (45) holds for $i$. Let
$w$ denote a nonzero vector in $V^{\prime\prime}_{i}$. Note that $Rw\in
V^{\prime\prime}_{i+1}$ and $Rw\neq 0$ by Lemma 8.6(i). Combining this with
(46) shows $\\{Rw\\}$ is a basis for $V^{\prime\prime}_{i+1}$. By Lemma 8.5
and the induction hypothesis we have $(r-cR)Rw=R(r-cR)w=0$. This shows
$(\ref{raise4a})$ holds for $i+1$ since $\\{Rw\\}$ is a basis for
$V^{\prime\prime}_{i+1}$. $\Box$

##  9 The Linear Transformations $B,\,B^{\prime},\,B^{\prime\prime}$

###### Lemma 9.1

Adopt Assumption 8.1. Then there exists a linear transformation
$B:V\rightarrow V$ such that $A^{\prime},\,-A^{\prime\prime},\,B$ is a reduced
BD triple on $V$ of base $1$. Moreover, if the BD triad
$A,\,A^{\prime},\,A^{\prime\prime}$ is thin then the BD triple
$A^{\prime},\,-A^{\prime\prime},\,B$ is also thin.

Proof: Observe that $A^{\prime},\,-A^{\prime\prime}$ is a reduced BD pair on
$V$ of base $1$. So by [15, Theorem 4.1] there exists a linear transformation
$\overline{B}:V\rightarrow V$ such that
$A^{\prime},\,-A^{\prime\prime},\,\overline{B}$ is a BD triple on $V$ of base
$1$. Moreover, $\\{2i-d\\}_{i=0}^{d}$ is both the first and second eigenvalue
sequence of $A^{\prime},\,-A^{\prime\prime},\,\overline{B}$. Let
$\\{\overline{\beta_{i}}\\}_{i=0}^{d}$ denote the third eigenvalue sequence of
$A^{\prime},\,-A^{\prime\prime},\,\overline{B}$. See [15, Section 2] for the
definition of first, second, third eigenvalue sequence of a BD triple. By [15,
Lemma 2.12], Lemma 5.2, and Lemma 5.3 we have
$\\{2i-d\\}_{i=0}^{d}\sim\\{\overline{\beta_{i}}\\}_{i=0}^{d}$. So by
Definition 2.13(i) there exist $r,s$ in $\mathbb{K}$ with $r\neq 0$ such that
$2i-d=r\overline{\beta_{i}}+s$ for $0\leq i\leq d$. Define the linear
transformation $B:V\rightarrow V$ as $B:=r\overline{B}+sI$. Thus, by [15,
Lemma 5.7] we have that $A^{\prime},\,-A^{\prime\prime},\,B$ is a reduced BD
triple on $V$ of base $1$. Now suppose that the BD triad
$A,\,A^{\prime},\,A^{\prime\prime}$ is thin. So both the BD triad
$A,\,A^{\prime},\,A^{\prime\prime}$ and the BD pair
$A^{\prime},\,-A^{\prime\prime}$ have shape $(1,1,1,\ldots,1)$. From this and
[15, Theorem 4.1] the BD triple $A^{\prime},\,-A^{\prime\prime},\,B$ also has
shape $(1,1,1,\ldots,1)$. Thus, the BD triple
$A^{\prime},\,-A^{\prime\prime},\,B$ is thin. $\Box$

###### Definition 9.2

Let $c$ denote the nonzero scalar in $\mathbb{K}$ from Lemma 8.7. Define the
scalar $a$ in $\mathbb{K}$ as $a:=1-c^{-1}$.

###### Note 9.3

Let $c$ denote the scalar from Lemma 8.7, and let $a$ denote the scalar from
Definition 9.2. By Note 8.3 and Lemma 8.7 we have that $c\neq 1$. Thus, $a\neq
0$. Also, it is clear from Definition 9.2 that $a\neq 1$.

###### Definition 9.4

With reference to Lemma 9.1 and Definition 9.2, let $B^{\prime}:V\rightarrow
V$ and $B^{\prime\prime}:V\rightarrow V$ denote the following linear
transformations:

$\displaystyle B^{\prime}$ $\displaystyle:=$
$\displaystyle(a^{-1}-1)^{-1}A^{\prime\prime}+(a-1)^{-1}B,$ (47)
$\displaystyle B^{\prime\prime}$ $\displaystyle:=$
$\displaystyle(1-a^{-1})A^{\prime}-a^{-1}B.$ (48)

###### Lemma 9.5

Adopt Assumption 8.1 and assume that the BD triad
$A,\,A^{\prime},\,A^{\prime\prime}$ is thin. With reference to Lemma 9.1 and
Definition 9.4 the following hold.

$\displaystyle A$ $\displaystyle=$
$\displaystyle(1-a)A^{\prime}+aA^{\prime\prime},$ (49) $\displaystyle
A^{\prime}$ $\displaystyle=$
$\displaystyle(1-a^{-1})^{-1}A^{\prime\prime}+(1-a)^{-1}A,$ (50)
$\displaystyle A^{\prime\prime}$ $\displaystyle=$ $\displaystyle
a^{-1}A+(1-a^{-1})A^{\prime},$ (51) $\displaystyle B$ $\displaystyle=$
$\displaystyle(a-1)B^{\prime}+aA^{\prime\prime},$ (52) $\displaystyle
A^{\prime\prime}$ $\displaystyle=$ $\displaystyle
a^{-1}B+(a^{-1}-1)B^{\prime},$ (53) $\displaystyle B$ $\displaystyle=$
$\displaystyle-aB^{\prime\prime}+(a-1)A^{\prime},$ (54) $\displaystyle
A^{\prime}$ $\displaystyle=$
$\displaystyle(a-1)^{-1}B+(1-a^{-1})^{-1}B^{\prime\prime},$ (55)
$\displaystyle A$ $\displaystyle=$
$\displaystyle(1-a)B^{\prime}-aB^{\prime\prime},$ (56) $\displaystyle
B^{\prime}$ $\displaystyle=$
$\displaystyle(a^{-1}-1)^{-1}B^{\prime\prime}+(1-a)^{-1}A,$ (57)
$\displaystyle B^{\prime\prime}$ $\displaystyle=$
$\displaystyle-a^{-1}A+(a^{-1}-1)B^{\prime}.$ (58)

Proof: By Definition 9.2 we have $c=(1-a)^{-1}$. Combining this with
Definition 8.2 and Lemma 8.7 gives
$A^{\prime}-A^{\prime\prime}=(1-a)^{-1}(A-A^{\prime\prime})$. Solving this for
$A$ gives (49). Solving (49) for $A^{\prime}$ gives (50). Solving (49) for
$A^{\prime\prime}$ gives (51). Solving (47) for $B$ gives (52). Solving (47)
for $A^{\prime\prime}$ gives (53). Solving (48) for $B$ gives (54). Solving
(48) for $A^{\prime}$ gives (55). Substituting (53),(55) into (49) and
simplifying gives (56). Solving (56) for $B^{\prime}$ gives (57). Solving (56)
for $B^{\prime\prime}$ gives (58). $\Box$

###### Lemma 9.6

Adopt Assumption 8.1. With reference to Lemma 9.1 the following hold.

$\displaystyle[A^{\prime\prime},B]$ $\displaystyle=$ $\displaystyle
2A^{\prime\prime}-2B,$ (59) $\displaystyle[B,A^{\prime}]$ $\displaystyle=$
$\displaystyle 2B+2A^{\prime}.$ (60)

Proof: Immediate from Lemma 9.1 and [15, Corollary 8.2(i)]. $\Box$

###### Lemma 9.7

Adopt Assumption 8.1 and assume that the BD triad
$A,\,A^{\prime},\,A^{\prime\prime}$ is thin. With reference to Lemma 9.1 and
Definition 9.4 the following hold.

$\displaystyle[B^{\prime},A^{\prime\prime}]$ $\displaystyle=$ $\displaystyle
2B^{\prime}+2A^{\prime\prime},$ (61) $\displaystyle[B^{\prime},B]$
$\displaystyle=$ $\displaystyle 2B^{\prime}+2B,$ (62)
$\displaystyle[B,B^{\prime\prime}]$ $\displaystyle=$ $\displaystyle
2B+2B^{\prime\prime},$ (63) $\displaystyle[A^{\prime},B^{\prime\prime}]$
$\displaystyle=$ $\displaystyle 2A^{\prime}-2B^{\prime\prime},$ (64)
$\displaystyle[B^{\prime\prime},B^{\prime}]$ $\displaystyle=$ $\displaystyle
2B^{\prime\prime}+2B^{\prime},$ (65) $\displaystyle[B^{\prime\prime},A]$
$\displaystyle=$ $\displaystyle 2B^{\prime\prime}+2A,$ (66)
$\displaystyle[A,B^{\prime}]$ $\displaystyle=$ $\displaystyle 2A-2B^{\prime}.$
(67)

Proof: Substituting (52) into (59) and simplifying gives (61). Substituting
(53) into (59) and simplifying gives (62). Substituting (55) into (60) and
simplifying gives (63). Substituting (54) into (60) and simplifying gives
(64). Substituting (53),(55) into (21) and simplifying using (62),(63) gives
(65). Substituting (51),(54) into (59) and simplifying using (20),(64) gives
(66). Substituting (50),(52) into (60) and simplifying using (22),(61) gives
(67). $\Box$

###### Lemma 9.8

Adopt Assumption 8.1 and assume that the BD triad
$A,\,A^{\prime},\,A^{\prime\prime}$ is thin. With reference to Lemma 9.1 and
Definition 9.4 the following (i)–(vi) hold.

  1. (i)

$[A,[A,[A,B]]]=4[A,B].$

  2. (ii)

$[B,[B,[B,A]]]=4[B,A].$

  3. (iii)

$[A^{\prime},[A^{\prime},[A^{\prime},B^{\prime}]]]=4[A^{\prime},B^{\prime}].$

  4. (iv)

$[B^{\prime},[B^{\prime},[B^{\prime},A^{\prime}]]]=4[B^{\prime},A^{\prime}].$

  5. (v)

$[A^{\prime\prime},[A^{\prime\prime},[A^{\prime\prime},B^{\prime\prime}]]]=4[A^{\prime\prime},B^{\prime\prime}].$

  6. (vi)

$[B^{\prime\prime},[B^{\prime\prime},[B^{\prime\prime},A^{\prime\prime}]]]=4[B^{\prime\prime},A^{\prime\prime}].$

Proof: (i) We have

$\displaystyle[A,[A,[A,B]]]$ $\displaystyle=$
$\displaystyle[A,[A,[A,(a-1)B^{\prime}+aA^{\prime\prime}]]]\qquad{\rm
by\,\,(\ref{correl4})}$ $\displaystyle=$
$\displaystyle[A,[A,(a-1)(2A-2B^{\prime})+a(-2A+2A^{\prime\prime})]]\qquad{\rm
by\,\,(\ref{eq3}),(\ref{brack2g})}$ $\displaystyle=$
$\displaystyle[A,-2(a-1)[A,B^{\prime}]+2a[A,A^{\prime\prime}]]$
$\displaystyle=$ $\displaystyle
4(a-1)[A,B^{\prime}]+4a[A,A^{\prime\prime}]\qquad{\rm
by\,\,(\ref{eq3}),(\ref{brack2g})}$ $\displaystyle=$ $\displaystyle
4[A,B]\qquad{\rm by\,\,(\ref{correl4})}.$

(ii) We have

$\displaystyle[B,[B,[B,A]]]$ $\displaystyle=$
$\displaystyle[B,[B,[B,(1-a)A^{\prime}+aA^{\prime\prime}]]]\qquad{\rm
by\,\,(\ref{correl1})}$ $\displaystyle=$
$\displaystyle[B,[B,(1-a)(2B+2A^{\prime})+a(2B-2A^{\prime\prime})]]\qquad{\rm
by\,\,(\ref{brack1a}),(\ref{brack1b})}$ $\displaystyle=$
$\displaystyle[B,2(1-a)[B,A^{\prime}]-2a[B,A^{\prime\prime}]]$
$\displaystyle=$ $\displaystyle
4(1-a)[B,A^{\prime}]+4a[B,A^{\prime\prime}]\qquad{\rm
by\,\,(\ref{brack1a}),(\ref{brack1b})}$ $\displaystyle=$ $\displaystyle
4[B,A]\qquad{\rm by\,\,(\ref{correl1})}.$

(iii),(v) Similar to (i).
(iv),(vi) Similar to (ii). $\Box$

##  10 The Proof that BD triads Act as Corner Triads

In this section we prove Theorem 4.6.

###### Lemma 10.1

With reference to Notation 2.1 the following (i),(ii) are equivalent.

  1. (i)

$[X,Y]=2X+2Y.$

  2. (ii)

$[-Y,-X]=-2Y-2X.$

Proof: Immediate from Note 2.2. $\Box$

###### Lemma 10.2

With reference to Notation 2.1 the following (i)–(iv) are equivalent.

  1. (i)

$[X,[X,[X,Y]]]=4[X,Y].$

  2. (ii)

$[X,[X,[X,-Y]]]=4[X,-Y].$

  3. (iii)

$[-X,[-X,[-X,Y]]]=4[-X,Y].$

  4. (iv)

$[-X,[-X,[-X,-Y]]]=4[-X,-Y].$

Proof: Immediate from Note 2.2. $\Box$

###### Lemma 10.3

Let $X,\,Y,\,Z$ denote a thin BD triple on $V$. Let $\\{Y_{i}\\}_{i=0}^{d}$
denote the standard ordering of the eigenspaces of $Y$. Then the following
(i),(ii) hold.

  1. (i)

The restriction $[X,Y]^{j}|_{Y_{i}}:Y_{i}\rightarrow Y_{i-j}$ is a bijection
for $0\leq i\leq d$ and $0\leq j\leq i$.

  2. (ii)

The restriction $[Z,Y]^{j}|_{Y_{i}}:Y_{i}\rightarrow Y_{i+j}$ is a bijection
for $0\leq i\leq d$ and $0\leq j\leq d-i$.

Proof: (i) Recall by Lemma 2.3(ii) and (6) that the restriction
$[X,Y]^{j}|_{Y_{i}}$ maps $Y_{i}$ to $Y_{i-j}$ for $0\leq i\leq d$ and $0\leq
j\leq i$. We now show that this mapping is a bijection. Since $X,\,Y,\,Z$ is
thin we have

$\displaystyle{\rm dim}(Y_{i})=1{\rm\,\,for\,\,}0\leq i\leq d.$ (68)

The result is trivially true for $i=0$. Now let $i\geq 1$. Since the
composition of two bijections is also a bijection it suffices to show that

$\displaystyle{\rm The\,\,restriction\,\,}[X,Y]|_{Y_{i}}:Y_{i}\rightarrow
Y_{i-1}{\rm\,\,is\,\,a\,\,bijection\,\,for\,\,}1\leq i\leq d.$ (69)

Case 1: $1\leq i\leq d/2$. First we show that the restriction
$[X,Y]|_{Y_{i}}:Y_{i}\rightarrow Y_{i-1}$ is a surjection. Let $v\in Y_{i-1}$.
By (9) there exists $w\in Y_{d-i+1}$ such that $[X,Y]^{d-2i+2}w=v$. Observe
that $[X,Y]^{d-2i+1}w\in Y_{i}$ and $[X,Y][X,Y]^{d-2i+1}w=v$. This shows that
the restriction $[X,Y]|_{Y_{i}}:Y_{i}\rightarrow Y_{i-1}$ is a surjection.
From this and (68) we have (69).
Case 2: $d/2<i\leq d$. First we show that the restriction
$[X,Y]|_{Y_{i}}:Y_{i}\rightarrow Y_{i-1}$ is an injection. Let $v\in{\rm
ker}([X,Y]|_{Y_{i}})$. So, $[X,Y]v=0$ and thus $[X,Y]^{2i-d}v=0$. From this
and (9) we have $v=0$. This shows that the restriction
$[X,Y]|_{Y_{i}}:Y_{i}\rightarrow Y_{i-1}$ is an injection. From this and (68)
we have (69).
(ii) Similar to (i). $\Box$

Proof of Theorem 4.6: Adopt Assumption 8.1 and assume that the BD triad
$A,\,A^{\prime},\,A^{\prime\prime}$ is thin. Let $X_{ru},X_{su},X_{tu}$ denote
a corner triad in $\boxtimes$. Let $B$ denote the linear transformation from
Lemma 9.1. Let $B^{\prime},\,B^{\prime\prime}$ denote the linear
transformations from Definition 9.4. Comparing Definition 3.1 with Lemmas 5.7,
9.6, 9.7, 9.8, 10.1, and 10.2 we find there exists a $\boxtimes$-module
structure on $V$ such that

$\displaystyle(X_{ru}-A)V=0,\qquad(X_{ur}+A)V=0,\qquad(X_{su}-A^{\prime})V=0,\qquad(X_{us}+A^{\prime})V=0,$
$\displaystyle(X_{tu}-A^{\prime\prime})V=0,\qquad(X_{ut}+A^{\prime\prime})V=0,\qquad(X_{ts}-B)V=0,\qquad(X_{st}+B)V=0,$
$\displaystyle(X_{rt}-B^{\prime})V=0,\qquad(X_{tr}+B^{\prime})V=0,\qquad(X_{sr}-B^{\prime\prime})V=0,\qquad(X_{rs}+B^{\prime\prime})V=0.$

We now show that this $\boxtimes$-module structure on $V$ is irreducible. Let
$W$ denote a nonzero $\boxtimes$-submodule of $V$. We show that $W=V$. Let
$0\neq w\in W$. Since $A^{\prime\prime}$ is diagonalizable then
$\\{V^{\prime\prime}_{i}\\}_{i=0}^{d}$ is a decomposition of $V$. So for
$0\leq i\leq d$ there exists $v_{i}^{\prime\prime}\in V_{i}^{\prime\prime}$
such that $w=\sum_{i=0}^{d}v_{i}^{\prime\prime}$. Define $t:={\rm
min}\\{i\,|\,0\leq i\leq d,\,v_{i}^{\prime\prime}\neq 0\\}$, and observe that

$\displaystyle w=\sum_{i=t}^{d}v_{i}^{\prime\prime}.$ (70)

Recall by Lemma 9.1 that $A^{\prime},\,-A^{\prime\prime},\,B$ is a thin BD
triple and note that $\\{V_{d-i}^{\prime\prime}\\}_{i=0}^{d}$ is the standard
ordering of the eigenspaces of $-A^{\prime\prime}$. By Lemma 10.3(i) we have
that the restriction
$[A^{\prime},-A^{\prime\prime}]^{d-t}|_{V_{t}^{\prime\prime}}:V_{t}^{\prime\prime}\rightarrow
V_{d}^{\prime\prime}$ is a bijection. Abbreviate the vector
$[A^{\prime},-A^{\prime\prime}]^{d-t}v_{t}^{\prime\prime}$ as $v$. Recall
$0\neq v_{t}^{\prime\prime}\in V_{t}^{\prime\prime}$ by construction, and so

$\displaystyle 0\neq v\in V_{d}^{\prime\prime}.$ (71)

By Lemma 10.3(ii) we have that for $0\leq r\leq d$ the restriction
$[B,-A^{\prime\prime}]^{r}|_{V_{d}^{\prime\prime}}:V_{d}^{\prime\prime}\rightarrow
V_{d-r}^{\prime\prime}$ is a bijection. Combining this with (71) we have
$0\neq[B,-A^{\prime\prime}]^{r}v\in V_{d-r}^{\prime\prime}$ for $0\leq r\leq
d$. Since the BD triple $A^{\prime},\,-A^{\prime\prime},\,B$ is thin then
${\rm dim}(V_{d-r}^{\prime\prime})=1$ for $0\leq r\leq d$. Combining the
previous two sentences with the fact that $A^{\prime\prime}$ is diagonalizable
shows

$\displaystyle\\{[B,-A^{\prime\prime}]^{r}v\\}_{r=0}^{d}{\rm\,\,is\,\,a\,\,basis\,\,for\,\,}V.$
(72)

Since $W$ is a $\boxtimes$-submodule of $V$ then

$\displaystyle A^{\prime}W\subseteq W,\qquad-A^{\prime\prime}W\subseteq
W,\qquad BW\subseteq W.$ (73)

By (70) and Lemma 2.3(i) we have $[A^{\prime},-A^{\prime\prime}]^{d-t}w=v$.
From this and (73) we have that $v\in W$. Combining this with (72) and (73)
shows that $W=V$. Therefore, the $\boxtimes$-module structure on $V$ is
irreducible. $\Box$

###### Note 10.4

Adopt Assumption 8.1 and assume that the BD triad
$A,\,A^{\prime},\,A^{\prime\prime}$ is thin. Let $B$ denote the linear
transformation from Lemma 9.1. Let $B^{\prime},\,B^{\prime\prime}$ denote the
linear transformations from Definition 9.4. With reference to the irreducible
$\boxtimes$-module structure on $V$ constructed in the proof of Theorem 4.6 we
make the following observations. By Theorem 4.5 each of the following

$\displaystyle
A,\,A^{\prime},\,A^{\prime\prime}\qquad\qquad-A^{\prime\prime},\,B^{\prime},\,-B\qquad\qquad
B,\,-A^{\prime},\,-B^{\prime\prime}\qquad\qquad
B^{\prime\prime},\,-B^{\prime},\,-A$

acts on $V$ as a reduced BD triad. By [15, Theorem 4.8], [16, Theorem 3.8],
and [17, Corollary 12.4] each of the following

$\displaystyle A^{\prime},\,-A^{\prime\prime},\,B\qquad\qquad
A,\,-A^{\prime},\,B^{\prime\prime}\qquad\qquad
A,\,-A^{\prime\prime},\,-B^{\prime}\qquad\qquad
B^{\prime},\,B,\,B^{\prime\prime}$

acts on $V$ as a reduced BD triple of base $1$. From this and [15, Lemma 5.8]
each of the following

$\displaystyle
A^{\prime},\,-A^{\prime\prime}\qquad\qquad-A^{\prime\prime},\,B\qquad\qquad
B,\,A^{\prime}\qquad\qquad
A,\,-A^{\prime}\qquad\qquad-A^{\prime},\,B^{\prime\prime}\qquad\qquad
B^{\prime\prime},\,A$ $\displaystyle
A,\,-A^{\prime\prime}\qquad\qquad-A^{\prime\prime},\,-B^{\prime}\qquad\qquad-B^{\prime},\,A\qquad\qquad
B^{\prime},\,B\qquad\qquad B,\,B^{\prime\prime}\qquad\qquad
B^{\prime\prime},\,B^{\prime}$

acts on $V$ as a reduced BD pair of base $1$. By [16, Theorems 1.7, 2.4] each
of the following

$\displaystyle A,\,B\qquad\qquad A^{\prime},\,B^{\prime}\qquad\qquad
A^{\prime\prime},\,B^{\prime\prime}$

acts on $V$ as a tridiagonal pair of Krawtchouk type [23].

##  11 A Counterexample

In this section we provide a counterexample to show that the thin assumption
in Theorem 4.6 is needed. That is, we demonstrate that the conclusion of
Theorem 4.6 can fail to hold for BD triads which are not thin. Consider the
vector space $\mathbb{K}^{6}$ and let $A:\mathbb{K}^{6}\to\mathbb{K}^{6}$,
$A^{\prime}:\mathbb{K}^{6}\to\mathbb{K}^{6}$, and
$A^{\prime\prime}:\mathbb{K}^{6}\to\mathbb{K}^{6}$ denote the linear
transformations given by the following matrices:

$\displaystyle A=\begin{bmatrix}-3&0&0&0&0&0\\\ 1&-1&0&0&0&0\\\
1&0&-1&0&0&0\\\ 0&2&0&1&0&0\\\ 0&1&2&0&1&0\\\
0&0&0&3&0&3\end{bmatrix}A^{\prime}=\begin{bmatrix}-3&0&0&0&0&0\\\
-2&-1&0&0&0&0\\\ 0&0&-1&0&0&0\\\ 0&-4&0&1&0&0\\\ 0&0&-2&0&1&0\\\
0&0&0&-6&0&3\end{bmatrix}A^{\prime\prime}=\begin{bmatrix}-3&0&0&0&0&0\\\
0&-1&0&0&0&0\\\ 0&0&-1&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\
0&0&0&0&0&3\end{bmatrix}.$

It is straightforward to verify that $A,\,A^{\prime},\,A^{\prime\prime}$ is a
reduced BD triad on $\mathbb{K}^{6}$ whose first, second, and third eigenvalue
sequences are each $\\{2i-3\\}_{i=0}^{3}$. Notice that the shape of
$A,\,A^{\prime},\,A^{\prime\prime}$ is $(1,2,2,1)$, and so this BD triad is
not thin. Consider the corner triad $X_{03},X_{13},X_{23}$ in $\boxtimes$.
Suppose, towards a contradiction, that the conclusion of Theorem 4.6 holds.
That is, suppose there exists a $\boxtimes$-module structure on
$\mathbb{K}^{6}$ such that $(X_{03}-A)\mathbb{K}^{6}=0$,
$(X_{13}-A^{\prime})\mathbb{K}^{6}=0$, and
$(X_{23}-A^{\prime\prime})\mathbb{K}^{6}=0$. Then the matrix representing
$X_{02}$ is given by

$\displaystyle X_{02}=\begin{bmatrix}3&12&0&0&0&0\\\ 0&1&0&8&0&0\\\
0&0&1&5&2&0\\\ 0&0&0&-1&0&4\\\ 0&0&0&0&-1&6\\\ 0&0&0&0&0&-3\end{bmatrix}.$

However, direct calculation shows that

$[X_{13},[X_{13},[X_{13},X_{02}]]]\neq 4[X_{13},X_{02}].$

which contradicts Definition 3.1(iii). Thus, there does not exist a
$\boxtimes$-module structure on $\mathbb{K}^{6}$ such that
$(X_{03}-A)\mathbb{K}^{6}=0$, $(X_{13}-A^{\prime})\mathbb{K}^{6}=0$, and
$(X_{23}-A^{\prime\prime})\mathbb{K}^{6}=0$, and so the conclusion of Theorem
4.6 does not hold.

##  12 Acknowledgment

The author would like to thank Professor Paul Terwilliger for his advice and
many helpful suggestions. His comments greatly improved this paper. In
particular, his comment indicating the relevance of [25] to this paper was
especially helpful.

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Darren Funk-Neubauer Department of Mathematics and Physics Colorado State
University - Pueblo 2200 Bonforte Boulevard Pueblo, CO 81001 USA email:
<EMAIL_ADDRESS>phone: (719) 549 - 2693 fax: (719) 549 -
2962