# Digitization and subduction of $SU(N)$ gauge theories

Benoît Assi<EMAIL_ADDRESS>Fermi National Accelerator Laboratory, Batavia,
Illinois, 60510, USA  Henry Lamm<EMAIL_ADDRESS>Fermi National Accelerator
Laboratory, Batavia, Illinois, 60510, USA

###### Abstract

The simulation of lattice gauge theories on quantum computers necessitates
digitizing gauge fields. One approach involves substituting the continuous
gauge group with a discrete subgroup, but the implications of this
approximation still need to be clarified. To gain insights, we investigate the
subduction of $SU(2)$ and $SU(3)$ to discrete crystal-like subgroups. Using
classical lattice calculations, we show that subduction offers valuable
information based on subduced direct sums, helping us identify additional
terms to incorporate into the lattice action that can mitigate the effects of
digitization. Furthermore, we compute the static potentials of all irreducible
representations of $\Sigma(360\times 3)$ at a fixed lattice spacing. Our
results reveal a percent-level agreement with the Casimir scaling of $SU(3)$
for irreducible representations that subduce to a single $\Sigma(360\times 3)$
irreducible representation. This provides a diagnostic measure of
approximation quality, as some irreducible representations closely match the
expected results while others exhibit significant deviations.

††preprint: FERMILAB-PUB-24-0177-T

##  I Introduction

Digital quantum computers offer promising opportunities for exploring lattice
gauge theories (LGTs) in regimes that classical computers struggle with due to
sign problems, such as real-time dynamics or the properties of matter at
finite density [1, 2, 3, 4, 5, 6, 7, 8]. However, these devices also bring
significant challenges, primarily the limited availability of qubits and the
shallow circuit depth before noise overtakes the calculations. This constraint
is reminiscent of the early days of Euclidean LGT on classical computers,
where storing floating-point representations of $SU(3)$ was computationally
prohibitive. This led to research on approximations [9, 10, 11, 12] and gauge-
fixing [13].

Many digitizations have been proposed to reduce this high cost [14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54,
55, 56, 57, 58, 59, 60, 61, 62, 63, 64]. Each makes choices that break
symmetries [65, 24] complicating the extrapolation back to original theory
[66, 67, 68, 69, 70, 71, 43, 44, 46, 72, 73, 74, 53]. Moreover, the
effectiveness of the digitization can be spacetime dimension-dependent [38,
75, 76, 77, 78]. Further, some digitizations can be simulated on classical
computers without sign problems, allowing for nonperturbative Euclidean
simulations or stochastic state preparation [79, 80, 81, 82]. Currently, there
is a significant gap in our understanding regarding the resource demands,
potential errors, and the feasibility of achieving continuum limits with these
techniques.

In this paper, we will consider the discrete subgroup approximation, which
uses comparatively few qubits (or bits) and shallow circuit depth by avoiding
fixed-point arithmetic. This digitization was first studied in Euclidean
simulations for reducing classical resources [9, 83, 84, 85, 86] including
with dynamical fermions [87, 88]. Today, this approximation’s potential for
use in quantum computing is under investigation [22, 24, 25, 26, 49, 27, 89,
28, 90, 77, 76, 91, 92, 93, 58, 94, 95, 96, 97].

This work will focus on how the subduction of the continuous $SU(2)$ and
$SU(3)$ groups to their discrete subgroups plays a role in the digitization
error. We will use the notation of $\mathcal{G}$ to represent a continuous
group and $\mathbb{G}$ to denote any of its discrete subgroups. Subduction
provides a mapping from irreducible representations (IR) of one group to one
or a direct sum of IRs of a subgroup. This builds on previous work, which
investigated how reducing continuous spacetime to a cubic lattice corresponds
to the subduction of $SU(2)$ to the binary octahedral group $\mathbb{BO}$ –
breaking in the rotational symmetry of LGT [98, 99]. From this understanding,
it has been possible to construct an improved operator with reduced signal-to-
noise and excited state contamination [100, 101, 102, 103]. We extend these
ideas to gauge digitization by providing subduction tables for crystal-like
subgroups of both $SU(2)$ and $SU(3)$ and use this to gain insight by
analyzing two lattice observables – the lattice energy density and Casimir
scaling of the nonperturbative static potentials. This quantitative
understanding of the breaking of the continuous gauge symmetry represents a
starting point to systematically interpret the discrete subgroups as
continuous groups broken by a Higgs mechanism [104, 105, 106, 107, 108, 109,
110].

This paper is organized as follows. Sec. II and III summarizing key properties
of, respectively, the $SU(2)$ and $SU(3)$ and their crystal-like subgroups. In
Sec. IV, we determine the subductions and inductions between the continuous
and discrete groups. Following that, in Sec. V, numerical results are
presented for the lattice energy density and Casimir scaling of the static
potentials; we then conclude in Sec. VI.

##  II $SU(2)$

The infinite set of IRs of $SU(2)$ are indexed by a half-integer $j$, and
their dimensionality is given by $d=2j+1$. In a given IR, the character of
group elements further depends on a rotation angle $\theta$:

$\chi^{(j)}(\theta)=\frac{\sin\left(\left[j+\frac{1}{2}\right]\theta\right)}{\sin\left(\frac{\theta}{2}\right)}.$
(1)

Table 1: Character Table of $\mathbb{BT}$222A typo in the 7th line of [77] has been corrected here. Size | 1 | 1 | 6 | 4 | 4 | 4 | 4
---|---|---|---|---|---|---|---
Order | 1 | 2 | 4 | 6 | 6 | 3 | 3
$A_{0}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1
$E_{1}$ | 1 | 1 | 1 | $\omega$ | $\omega^{2}$ | $\omega^{2}$ | $\omega$
$E_{2}$ | 1 | 1 | 1 | $\omega^{2}$ | $\omega$ | $\omega$ | $\omega^{2}$
$H$ | 2 | $-2$ | 0 | 1 | 1 | $-1$ | $-1$
$G_{1}$ | 2 | $-2$ | 0 | $\omega$ | $\omega^{2}$ | $-\omega^{2}$ | $-\omega$
$G_{2}$ | 2 | $-2$ | 0 | $\omega^{2}$ | $\omega$ | $-\omega$ | $-\omega^{2}$
$T$ | 3 | 3 | $-1$ | 0 | 0 | 0 | 0

$SU(2)$ has three crystal-like subgroups: the 24-element Binary Tetrahedral
group ($\mathbb{BT}$), the 48-element Binary Octahedral group ($\mathbb{BO}$),
and the 120-element Binary Icosahedral group ($\mathbb{BI}$). The character
tables for each group are given by Tables 2, 2, and 3 respectively and will be
used below to derive the subduction of $SU(2)$.

Table 2: Character table of $\mathbb{BO}$ Size | 1 | 1 | 12 | 6 | 8 | 8 | 6 | 6
---|---|---|---|---|---|---|---|---
Ord. | 1 | 2 | 4 | 4 | 6 | 3 | 8 | 8
$A_{1}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
$A_{2}$ | 1 | 1 | $-1$ | 1 | 1 | 1 | $-1$ | $-1$
$E$ | 2 | $2$ | 0 | 2 | $-1$ | $-1$ | 0 | 0
$G_{1}$ | 2 | $-2$ | 0 | 0 | 1 | $-1$ | $-\sqrt{2}$ | $\sqrt{2}$
$G_{2}$ | 2 | $-2$ | 0 | 0 | 1 | $-1$ | $\sqrt{2}$ | $-\sqrt{2}$
$T_{1}$ | 3 | 3 | $-1$ | $-1$ | 0 | 0 | 1 | 1
$T_{2}$ | 3 | 3 | 1 | $-1$ | 0 | 0 | $-1$ | $-1$
$H$ | 4 | $-4$ | 0 | 0 | $-1$ | 1 | 0 | 0
Table 3: Character table of $\mathbb{BI}$ Size | 1 | 1 | 20 | 30 | 12 | 12 | 20 | 12 | 12
---|---|---|---|---|---|---|---|---|---
Ord. | 1 | 2 | 3 | 4 | 5 | 5 | 6 | 10 | 10
$A_{0}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
$E_{1}$ | 2 | -2 | -1 | 0 | $\frac{-1+\sqrt{5}}{2}$ | $\frac{-1-\sqrt{5}}{2}$ | 1 | $\frac{1+\sqrt{5}}{2}$ | $\frac{1-\sqrt{5}}{2}$
$E_{2}$ | 2 | -2 | -1 | 0 | $\frac{-1-\sqrt{5}}{2}$ | $\frac{-1+\sqrt{5}}{2}$ | 1 | $\frac{1-\sqrt{5}}{2}$ | $\frac{1+\sqrt{5}}{2}$
$T_{1}$ | 3 | 3 | 0 | -1 | $\frac{1-\sqrt{5}}{2}$ | $\frac{1+\sqrt{5}}{2}$ | 0 | $\frac{1+\sqrt{5}}{2}$ | $\frac{1-\sqrt{5}}{2}$
$T_{2}$ | 3 | 3 | 0 | -1 | $\frac{1+\sqrt{5}}{2}$ | $\frac{1-\sqrt{5}}{2}$ | 0 | $\frac{1-\sqrt{5}}{2}$ | $\frac{1+\sqrt{5}}{2}$
$G_{1}$ | 4 | 4 | 1 | 0 | -1 | -1 | 1 | -1 | -1
$G_{2}$ | 4 | -4 | 1 | 0 | -1 | -1 | -1 | 1 | 1
$H$ | 5 | 5 | -1 | 1 | 0 | 0 | -1 | 0 | 0
$J$ | 6 | -6 | 0 | 0 | 1 | 1 | 0 | -1 | -1

##  III $SU(3)$

The infinite set of IRs of $SU(3)$, $\chi^{(p,q)}$, are indexed by non-
negative integers $p,q$, which correspond to the highest weights of the
representation, reflecting the symmetry properties of particle states under
$SU(3)$ transformations. The dimensionality of $\chi^{(p,q)}$ is
$d=\frac{1}{2}(p+1)(q+1)(p+q+2)$. In a given IR, the character of group
elements further depends on rotation angles $\theta,\phi$:

$\chi^{(p,q)}=e^{i(p+2q)\theta}\sum_{k=q}^{p+q}\sum_{l=0}^{q}e^{-3i(k+l)\frac{\theta}{2}}\frac{\sin\left(\left[k-l+1\right]\frac{\phi}{2}\right)}{\sin\left(\frac{\phi}{2}\right)},$
(2)

We classify the discrete subgroups of $SU(3)$ we are interested in and their
associated IRs. The finite, non-Abelian crystal-like subgroups of $SU(3)$ with
a $\mathbb{Z}_{3}$ center are $\Sigma(108)$, $\Sigma(216)$, $\Sigma(648)$, and
$\Sigma(1080)$, where the number in parentheses indicates the number of
elements. We provide the character tables for these discrete subgroups in
Tables 4, 5, 6 and 7, respectively.

Table 4: Character table of $\operatorname{\Sigma(108)}$ with $\omega=e^{2\pi i/3}$.  Size | 1 | 1 | 1 | 12 | 12 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Ord. | 1 | 3 | 3 | 3 | 3 | 2 | 6 | 6 | 4 | 12 | 12 | 4 | 12 | 12
$\mathbf{1}^{(0)}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
$\mathbf{1}^{(1)}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | $i$ | $i$ | $i$ | $\text{-}i$ | $\text{-}i$ | $\text{-}i$
$\mathbf{1}^{(2)}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$
$\mathbf{1}^{(3)}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}i$ | $\text{-}i$ | $\text{-}i$ | $i$ | $i$ | $i$
$\mathbf{3}^{(0)}$ | $3$ | $3\omega$ | $3\omega^{2}$ | $0$ | $0$ | $\text{-}1$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | $1$ | $\omega$ | $\omega^{2}$ | $1$ | $\omega$ | $\omega^{2}$
$\mathbf{3}^{(1)}$ | $3$ | $3\omega$ | $3\omega^{2}$ | $0$ | $0$ | $1$ | $\omega$ | $\omega^{2}$ | $i$ | $i\omega$ | $i\omega^{2}$ | $\text{-}i$ | $\text{-}i\omega$ | $\text{-}i\omega^{2}$
$\mathbf{3}^{(2)}$ | $3$ | $3\omega$ | $3\omega^{2}$ | $0$ | $0$ | $\text{-}1$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | $\text{-}1$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | $\text{-}1$ | $\text{-}\omega$ | $\text{-}\omega^{2}$
$\mathbf{3}^{(3)}$ | $3$ | $3\omega$ | $3\omega^{2}$ | $0$ | $0$ | $1$ | $\omega$ | $\omega^{2}$ | $\text{-}i$ | $\text{-}i\omega$ | $\text{-}i\omega^{2}$ | $i$ | $i\omega$ | $i\omega^{2}$
$\mathbf{3}^{(0)*}$ | $3$ | $3\omega^{2}$ | $3\omega$ | $0$ | $0$ | $\text{-}1$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | $1$ | $\omega^{2}$ | $\omega$ | $1$ | $\omega^{2}$ | $\omega$
$\mathbf{3}^{(1)^{*}}$ | $3$ | $3\omega^{2}$ | $3\omega$ | $0$ | $0$ | $1$ | $\omega^{2}$ | $\omega$ | $\text{-}i$ | $\text{-}i\omega^{2}$ | $\text{-}i\omega$ | $i$ | $i\omega^{2}$ | $i\omega$
$\mathbf{3}^{(2)*}$ | $3$ | $3\omega^{2}$ | $3\omega$ | $0$ | $0$ | $\text{-}1$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | $\text{-}1$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | $\text{-}1$ | $\text{-}\omega^{2}$ | $\text{-}\omega$
$\mathbf{3}^{(3)*}$ | $3$ | $3\omega^{2}$ | $3\omega$ | $0$ | $0$ | $1$ | $\omega^{2}$ | $\omega$ | $i$ | $i\omega^{2}$ | $i\omega$ | $\text{-}i$ | $\text{-}i\omega^{2}$ | $\text{-}i\omega$
$\mathbf{4}$ | $4$ | $4$ | $4$ | $1$ | $\text{-}2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\mathbf{4}^{\prime}$ | $4$ | $4$ | $4$ | $\text{-}2$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 5: Character table of $\operatorname{\Sigma(216)}$ with $\omega=e^{2\pi i/3}$.  Size | 1 | 1 | 1 | 24 | 9 | 9 | 9 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Ord | 1 | 3 | 3 | 3 | 2 | 6 | 6 | 4 | 12 | 12 | 4 | 12 | 12 | 12 | 12 | 4
$\mathbf{1}^{(0)}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
$\mathbf{1}^{(1)}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | $1$ | $1$ | $1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$
$\mathbf{1}^{(2)}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$
$\mathbf{1}^{(3)}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | $1$ | $1$ | $1$
$\mathbf{2}$ | $2$ | $2$ | $2$ | $2$ | $\text{-}2$ | $\text{-}2$ | $\text{-}2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\mathbf{3}^{(0)}$ | $3$ | $3\omega$ | $3\omega^{2}$ | $0$ | $\text{-}1$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | $1$ | $\omega$ | $\omega^{2}$ | $1$ | $\omega$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $1$
$\mathbf{3}^{(1)}$ | $3$ | $3\omega$ | $3\omega^{2}$ | $0$ | $\text{-}1$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | $\text{-}1$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | $1$ | $\omega$ | $\omega^{2}$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | $\text{-}1$
$\mathbf{3}^{(2)}$ | $3$ | $3\omega$ | $3\omega^{2}$ | $0$ | $\text{-}1$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | $1$ | $\omega$ | $\omega^{2}$ | $\text{-}1$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | $\text{-}1$
$\mathbf{3}^{(3)}$ | $3$ | $3\omega$ | $3\omega^{2}$ | $0$ | $\text{-}1$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | $\text{-}1$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | $\text{-}1$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | $\omega$ | $\omega^{2}$ | $1$
$\mathbf{3}^{(0)*}$ | $3$ | $3\omega^{2}$ | $3\omega$ | $0$ | $\text{-}1$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | $1$ | $\omega^{2}$ | $\omega$ | $1$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $\omega$ | $1$
$\mathbf{3}^{(1)*}$ | $3$ | $3\omega^{2}$ | $3\omega$ | $0$ | $\text{-}1$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | $\text{-}1$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | $1$ | $\omega^{2}$ | $\omega$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | $\text{-}1$
$\mathbf{3}^{(2)*}$ | $3$ | $3\omega^{2}$ | $3\omega$ | $0$ | $\text{-}1$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | $1$ | $\omega^{2}$ | $\omega$ | $\text{-}1$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | $\text{-}1$
$\mathbf{3}^{(3)*}$ | $3$ | $3\omega^{2}$ | $3\omega$ | $0$ | $\text{-}1$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | $\text{-}1$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | $\text{-}1$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | $\omega^{2}$ | $\omega$ | $1$
$\mathbf{6}$ | $6$ | $6\omega$ | $6\omega^{2}$ | $0$ | $2$ | $2\omega$ | $2\omega^{2}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\mathbf{6^{\ast}}$ | $6$ | $6\omega^{2}$ | $6\omega$ | $0$ | $2$ | $2\omega^{2}$ | $2\omega$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$\mathbf{8}$ | $8$ | $8$ | $8$ | $\text{-}1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
Table 6: Character table of $\operatorname{\Sigma(648)}$ with $\omega=e^{2\pi i/3}$, $\rho\equiv e^{2\pi i/9}$, $\sigma\equiv\rho(1+2\omega)$ Size | 1 | 1 | 1 | 24 | 9 | 9 | 9 | 54 | 54 | 54 | 72 | 72 | 12 | 12 | 12 | 12 | 12 | 12 | 36 | 36 | 36 | 36 | 36 | 36
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Ord. | 1 | 3 | 3 | 3 | 2 | 6 | 6 | 4 | 12 | 12 | 3 | 3 | 9 | 9 | 9 | 9 | 9 | 9 | 18 | 18 | 18 | 18 | 18 | 18
$\mathbf{1}^{(0)}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
$\mathbf{1}^{(1)}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | $\omega$ | $\omega^{2}$ | $\omega$ | $\omega$ | $\omega$ | $\omega^{2}$ | $\omega^{2}$ | $\omega^{2}$ | $\omega$ | $\omega$ | $\omega$ | $\omega^{2}$ | $\omega^{2}$ | $\omega^{2}$
$\mathbf{1}^{(2)}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $\omega^{2}$ | $\omega^{2}$ | $\omega$ | $\omega$ | $\omega$ | $\omega^{2}$ | $\omega^{2}$ | $\omega^{2}$ | $\omega$ | $\omega$ | $\omega$
$\mathbf{2}^{(0)}$ | 2 | 2 | 2 | 2 | $\text{-}2$ | $\text{-}2$ | $\text{-}2$ | 0 | 0 | 0 | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | 1 | 1 | 1 | 1 | 1 | 1
$\mathbf{2}^{(1)}$ | 2 | 2 | 2 | 2 | $\text{-}2$ | $\text{-}2$ | $\text{-}2$ | 0 | 0 | 0 | $\text{-}\omega$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | $\text{-}\omega$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | $\text{-}\omega^{2}$ | $\text{-}\omega^{2}$ | $\omega$ | $\omega$ | $\omega$ | $\omega^{2}$ | $\omega^{2}$ | $\omega^{2}$
$\mathbf{2}^{(2)}$ | 2 | 2 | 2 | 2 | $\text{-}2$ | $\text{-}2$ | $\text{-}2$ | 0 | 0 | 0 | $\text{-}\omega^{2}$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | $\text{-}\omega^{2}$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | $\text{-}\omega$ | $\text{-}\omega$ | $\omega^{2}$ | $\omega^{2}$ | $\omega^{2}$ | $\omega$ | $\omega$ | $\omega$
$\mathbf{3}^{(a)}$ | 3 | 3 | 3 | 3 | 3 | 3 | 3 | $\text{-}1$ | $\text{-}1$ | $\text{-}1$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\mathbf{3}^{(0)}$ | 3 | $3\omega$ | $3\omega^{2}$ | 0 | $\text{-}1$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | 1 | $\omega$ | $\omega^{2}$ | 0 | 0 | $\omega^{2}\sigma^{\ast}$ | $\sigma^{\ast}$ | $\omega\sigma^{\ast}$ | $\sigma$ | $\omega\sigma$ | $\omega^{2}\sigma$ | $\text{-}\rho^{2}$ | $\text{-}\rho^{4*}$ | $\text{-}\rho^{\ast}$ | $\text{-}\rho^{2*}$ | $\text{-}\rho$ | $\text{-}\rho^{4}$
$\mathbf{3}^{(1)}$ | 3 | $3\omega$ | $3\omega^{2}$ | 0 | $\text{-}1$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | 1 | $\omega$ | $\omega^{2}$ | 0 | 0 | $\sigma^{\ast}$ | $\omega\sigma^{\ast}$ | $\omega^{2}\sigma^{\ast}$ | $\omega^{2}\sigma$ | $\sigma$ | $\omega\sigma$ | $\text{-}\rho^{4*}$ | $\text{-}\rho^{\ast}$ | $\text{-}\rho^{2}$ | $\text{-}\rho^{4}$ | $\text{-}\rho^{2*}$ | $\text{-}\rho$
$\mathbf{3}^{(2)}$ | 3 | $3\omega$ | $3\omega^{2}$ | 0 | $\text{-}1$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | 1 | $\omega$ | $\omega^{2}$ | 0 | 0 | $\omega\sigma^{\ast}$ | $\omega^{2}\sigma^{\ast}$ | $\sigma^{\ast}$ | $\omega\sigma$ | $\omega^{2}\sigma$ | $\sigma$ | $\text{-}\rho^{\ast}$ | $\text{-}\rho^{2}$ | $\text{-}\rho^{4*}$ | $\text{-}\rho$ | $\text{-}\rho^{4}$ | $\text{-}\rho^{2*}$
$\mathbf{3}^{(0)*}$ | 3 | $3\omega^{2}$ | $3\omega$ | 0 | $\text{-}1$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | 1 | $\omega^{2}$ | $\omega$ | 0 | 0 | $\omega\sigma$ | $\sigma$ | $\omega^{2}\sigma$ | $\sigma^{\ast}$ | $\omega^{2}\sigma^{\ast}$ | $\omega\sigma^{\ast}$ | $\text{-}\rho^{2*}$ | $\text{-}\rho^{4}$ | $\text{-}\rho$ | $\text{-}\rho^{2}$ | $\text{-}\rho^{\ast}$ | $\text{-}\rho^{4*}$
$\mathbf{3}^{(1)*}$ | 3 | $3\omega^{2}$ | $3\omega$ | 0 | $\text{-}1$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | 1 | $\omega^{2}$ | $\omega$ | 0 | 0 | $\sigma$ | $\omega^{2}\sigma$ | $\omega\sigma$ | $\omega\sigma^{\ast}$ | $\sigma^{\ast}$ | $\omega^{2}\sigma^{\ast}$ | $\text{-}\rho^{4}$ | $\text{-}\rho$ | $\text{-}\rho^{2*}$ | $\text{-}\rho^{4*}$ | $\text{-}\rho^{2}$ | $\text{-}\rho^{\ast}$
$\mathbf{3}^{(2)*}$ | 3 | $3\omega^{2}$ | $3\omega$ | 0 | $\text{-}1$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | 1 | $\omega^{2}$ | $\omega$ | 0 | 0 | $\omega^{2}\sigma$ | $\omega\sigma$ | $\sigma$ | $\omega^{2}\sigma^{\ast}$ | $\omega\sigma^{\ast}$ | $\sigma^{\ast}$ | $\text{-}\rho$ | $\text{-}\rho^{2*}$ | $\text{-}\rho^{4}$ | $\text{-}\rho^{\ast}$ | $\text{-}\rho^{4*}$ | $\text{-}\rho^{2}$
$\mathbf{6}^{(0)}$ | 6 | $6\omega$ | $6\omega^{2}$ | 0 | 2 | $2\omega$ | $2\omega^{2}$ | 0 | 0 | 0 | 0 | 0 | $\text{-}\omega\sigma^{\ast}$ | $\text{-}\omega^{2}\sigma^{\ast}$ | $\text{-}\sigma^{\ast}$ | $\text{-}\omega\sigma$ | $\text{-}\omega^{2}\sigma$ | $\text{-}\sigma$ | $\text{-}\rho^{\ast}$ | $\text{-}\rho^{2}$ | $\text{-}\rho^{4*}$ | $\text{-}\rho$ | $\text{-}\rho^{4}$ | $\text{-}\rho^{2*}$
$\mathbf{6}^{(1)}$ | 6 | $6\omega$ | $6\omega^{2}$ | 0 | 2 | $2\omega$ | $2\omega^{2}$ | 0 | 0 | 0 | 0 | 0 | $\text{-}\omega^{2}\sigma^{\ast}$ | $\text{-}\sigma^{\ast}$ | $\text{-}\omega\sigma^{\ast}$ | $\text{-}\sigma$ | $\text{-}\omega\sigma$ | $\text{-}\omega^{2}\sigma$ | $\text{-}\rho^{2}$ | $\text{-}\rho^{4*}$ | $\text{-}\rho^{\ast}$ | $\text{-}\rho^{2*}$ | $\text{-}\rho$ | $\text{-}\rho^{4}$
$\mathbf{6}^{(2)}$ | 6 | $6\omega$ | $6\omega^{2}$ | 0 | 2 | $2\omega$ | $2\omega^{2}$ | 0 | 0 | 0 | 0 | 0 | $\text{-}\sigma^{\ast}$ | $\text{-}\omega\sigma^{\ast}$ | $\text{-}\omega^{2}\sigma^{\ast}$ | $\text{-}\omega^{2}\sigma$ | $\text{-}\sigma$ | $\text{-}\omega\sigma$ | $\text{-}\rho^{4*}$ | $\text{-}\rho^{\ast}$ | $\text{-}\rho^{2}$ | $\text{-}\rho^{4}$ | $\text{-}\rho^{2*}$ | $\text{-}\rho$
$\mathbf{6}^{(0)*}$ | 6 | $6\omega^{2}$ | $6\omega$ | 0 | 2 | $2\omega^{2}$ | $2\omega$ | 0 | 0 | 0 | 0 | 0 | $\text{-}\omega^{2}\sigma$ | $\text{-}\omega\sigma$ | $\text{-}\sigma$ | $\text{-}\omega^{2}\sigma^{\ast}$ | $\text{-}\omega\sigma^{\ast}$ | $\text{-}\sigma^{\ast}$ | $\text{-}\rho$ | $\text{-}\rho^{2*}$ | $\text{-}\rho^{4}$ | $\text{-}\rho^{\ast}$ | $\text{-}\rho^{4*}$ | $\text{-}\rho^{2}$
$\mathbf{6}^{(1)*}$ | 6 | $6\omega^{2}$ | $6\omega$ | 0 | 2 | $2\omega^{2}$ | $2\omega$ | 0 | 0 | 0 | 0 | 0 | $\text{-}\omega\sigma$ | $\text{-}\sigma$ | $\text{-}\omega^{2}\sigma$ | $\text{-}\sigma^{\ast}$ | $\text{-}\omega^{2}\sigma^{\ast}$ | $\text{-}\omega\sigma^{\ast}$ | $\text{-}\rho^{2*}$ | $\text{-}\rho^{4}$ | $\text{-}\rho$ | $\text{-}\rho^{2}$ | $\text{-}\rho^{\ast}$ | $\text{-}\rho^{4*}$
$\mathbf{6}^{(2)*}$ | 6 | $6\omega^{2}$ | $6\omega$ | 0 | 2 | $2\omega^{2}$ | $2\omega$ | 0 | 0 | 0 | 0 | 0 | $\text{-}\sigma$ | $\text{-}\omega^{2}\sigma$ | $\text{-}\omega\sigma$ | $\text{-}\omega\sigma^{\ast}$ | $\text{-}\sigma^{\ast}$ | $\text{-}\omega^{2}\sigma^{\ast}$ | $\text{-}\rho^{4}$ | $\text{-}\rho$ | $\text{-}\rho^{2*}$ | $\text{-}\rho^{4*}$ | $\text{-}\rho^{2}$ | $\text{-}\rho^{\ast}$
$\mathbf{8}^{(0)}$ | 8 | 8 | 8 | $\text{-}1$ | 0 | 0 | 0 | 0 | 0 | 0 | $\text{-}1$ | $\text{-}1$ | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0
$\mathbf{8}^{(1)}$ | 8 | 8 | 8 | $\text{-}1$ | 0 | 0 | 0 | 0 | 0 | 0 | $\text{-}\omega$ | $\text{-}\omega^{2}$ | $2\omega$ | $2\omega$ | $2\omega$ | $2\omega^{2}$ | $2\omega^{2}$ | $2\omega^{2}$ | 0 | 0 | 0 | 0 | 0 | 0
$\mathbf{8}^{(2)}$ | 8 | 8 | 8 | $\text{-}1$ | 0 | 0 | 0 | 0 | 0 | 0 | $\text{-}\omega^{2}$ | $\text{-}\omega$ | $2\omega^{2}$ | $2\omega^{2}$ | $2\omega^{2}$ | $2\omega$ | $2\omega$ | $2\omega$ | 0 | 0 | 0 | 0 | 0 | 0
$\mathbf{9}$ | 9 | $9\omega$ | $9\omega^{2}$ | 0 | $\text{-}3$ | $\text{-}3\omega$ | $\text{-}3\omega^{2}$ | $\text{-}1$ | $\text{-}\omega$ | $\text{-}\omega^{2}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$\mathbf{9}^{\ast}$ | 9 | $9\omega^{2}$ | $9\omega$ | 0 | $\text{-}3$ | $\text{-}3\omega^{2}$ | $\text{-}3\omega$ | $\text{-}1$ | $\text{-}\omega^{2}$ | $\text{-}\omega$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Table 7: Character table for $\operatorname{\Sigma(1080)}$ with $\omega=e^{2\pi i/3}$, $\mu_{1}=(1-\sqrt{5})/2$, $\mu_{2}=(1+\sqrt{5})/2$. Size | 1 | 72 | 90 | 45 | 45 | 72 | 72 | 120 | 120 | 90 | 90 | 72 | 72 | 72 | 45 | 1 | 1
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Ord | 1 | 5 | 4 | 6 | 6 | 15 | 15 | 3 | 3 | 12 | 12 | 5 | 15 | 15 | 2 | 3 | 3
$\mathbf{1}^{(0)}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
$\mathbf{3}^{(0)}$ | $3$ | $\mu_{2}$ | $1$ | -$\omega^{*}$ | -$\omega$ | $\mu_{1}\omega^{*}$ | $\mu_{1}\omega$ | $0$ | $0$ | $\omega$ | $\omega^{*}$ | $\mu_{1}$ | $\mu_{2}\omega$ | $\mu_{2}\omega^{*}$ | -$1$ | $3\omega$ | $3\omega^{*}$
$\mathbf{3}^{(0)*}$ | $3$ | $\mu_{2}$ | $1$ | -$\omega$ | -$\omega^{*}$ | $\mu_{1}\omega$ | $\mu_{1}\omega^{*}$ | $0$ | $0$ | $\omega^{*}$ | $\omega$ | $\mu_{1}$ | $\mu_{2}\omega^{*}$ | $\mu_{2}\omega$ | -$1$ | $3\omega^{*}$ | $3\omega$
$\mathbf{3}^{(1)}$ | $3$ | $\mu_{1}$ | $1$ | -$\omega^{*}$ | -$\omega$ | $\mu_{2}\omega^{*}$ | $\mu_{2}\omega$ | $0$ | $0$ | $\omega$ | $\omega^{*}$ | $\mu_{2}$ | $\mu_{1}\omega$ | $\mu_{1}\omega^{*}$ | -$1$ | $3\omega$ | $3\omega^{*}$
$\mathbf{3}^{(1)*}$ | $3$ | $\mu_{1}$ | $1$ | -$\omega$ | -$\omega^{*}$ | $\mu_{2}\omega$ | $\mu_{2}\omega^{*}$ | $0$ | $0$ | $\omega^{*}$ | $\omega$ | $\mu_{2}$ | $\mu_{1}\omega^{*}$ | $\mu_{1}\omega$ | -$1$ | $3\omega^{*}$ | $3\omega$
$\mathbf{5}^{(0)}$ | $5$ | $0$ | -$1$ | $1$ | $1$ | $0$ | $0$ | -$1$ | $2$ | -$1$ | -$1$ | $0$ | $0$ | $0$ | $1$ | $5$ | $5$
$\mathbf{5}^{(0)^{\prime}}$ | $5$ | $0$ | -$1$ | $1$ | $1$ | $0$ | $0$ | $2$ | -$1$ | -$1$ | -$1$ | $0$ | $0$ | $0$ | $1$ | $5$ | $5$
$\mathbf{6}^{(0)}$ | $6$ | $1$ | $0$ | $2\omega$ | $2\omega^{*}$ | $\omega$ | $\omega^{*}$ | $0$ | $0$ | $0$ | $0$ | $1$ | $\omega^{*}$ | $\omega$ | $2$ | $6\omega^{*}$ | $6\omega$
$\mathbf{6}^{(0)*}$ | $6$ | $1$ | $0$ | $2\omega^{*}$ | $2\omega$ | $\omega^{*}$ | $\omega$ | $0$ | $0$ | $0$ | $0$ | $1$ | $\omega$ | $\omega^{*}$ | $2$ | $6\omega$ | $6\omega^{*}$
$\mathbf{8}^{(0)}$ | $8$ | $\mu_{2}$ | $0$ | $0$ | $0$ | $\mu_{1}$ | $\mu_{1}$ | -$1$ | -$1$ | $0$ | $0$ | $\mu_{1}$ | $\mu_{2}$ | $\mu_{2}$ | $0$ | $8$ | $8$
$\mathbf{8}^{(0)^{\prime}}$ | $8$ | $\mu_{1}$ | $0$ | $0$ | $0$ | $\mu_{2}$ | $\mu_{2}$ | -$1$ | -$1$ | $0$ | $0$ | $\mu_{2}$ | $\mu_{1}$ | $\mu_{1}$ | $0$ | $8$ | $8$
$\mathbf{9}^{(0)}$ | $9$ | -$1$ | $1$ | $1$ | $1$ | -$1$ | -$1$ | $0$ | $0$ | $1$ | $1$ | -$1$ | -$1$ | -$1$ | $1$ | $9$ | $9$
$\mathbf{9}^{(1)}$ | $9$ | -$1$ | $1$ | $\omega$ | $\omega^{*}$ | -$\omega$ | -$\omega^{*}$ | $0$ | $0$ | $\omega^{*}$ | $\omega$ | -$1$ | -$\omega^{*}$ | -$\omega$ | $1$ | $9\omega^{*}$ | $9\omega$
$\mathbf{9}^{(1)*}$ | $9$ | -$1$ | $1$ | $\omega^{*}$ | $\omega$ | -$\omega^{*}$ | -$\omega$ | $0$ | $0$ | $\omega$ | $\omega^{*}$ | -$1$ | -$\omega$ | -$\omega^{*}$ | $1$ | $9\omega$ | $9\omega^{*}$
$\mathbf{10}^{(0)}$ | $10$ | $0$ | $0$ | -$2$ | -$2$ | $0$ | $0$ | $1$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | -$2$ | $10$ | $10$
$\mathbf{15}^{(0)}$ | $15$ | $0$ | -$1$ | -$\omega^{*}$ | -$\omega$ | $0$ | $0$ | $0$ | $0$ | -$\omega$ | -$\omega^{*}$ | $0$ | $0$ | $0$ | -$1$ | $15\omega$ | $15\omega^{*}$
$\mathbf{15}^{(0)*}$ | $15$ | $0$ | -$1$ | -$\omega$ | -$\omega^{*}$ | $0$ | $0$ | $0$ | $0$ | -$\omega^{*}$ | -$\omega$ | $0$ | $0$ | $0$ | -$1$ | $15\omega^{*}$ | $15\omega$

##  IV Basics of Subduction

Subduction in group theory refers to the decomposition of an IR of a group
into the IRs of one of its subgroups. In the case of $\mathcal{G}$ subducing
to $\mathbb{G}$, the infinite set of IRs of $\mathcal{G}$ must be subduced to
the finite set of IRs of $\mathbb{G}$. Consequently, only a limited number of
IRs can be subduced one-to-one, with higher dimensional IRs subducing to
direct sums of increasingly large numbers of IRs. The multiplicity of the
$r$-th IR of $\mathbb{G}$ in the subduction of the $j$-th $\mathcal{G}$ IR can
be determined by computing

$m_{j}^{(r)}=\frac{1}{g}\sum_{k}n_{k}\chi_{k}^{(r)}\chi_{k}^{(j)}$ (3)

where $g$ is the size of $\mathbb{G}$, the sum is over all $k$ conjugacy
classes, $n_{k}$ is the size in $k$ in $\mathbb{G}$, and $\chi_{k}^{(r)}$ and
$\chi_{k}^{(j)}$ are the characters of the $r$-th and $j$-th IRs,
respectively, evaluated for elements of the $k$-th conjugacy class. Using
$d_{r},d_{j}$ to indicate the dimension of the IRs, we have

$d_{r}=\sum_{j}m_{j}^{(r)}d_{j}$ (4)

The requirement that LFT observables must be gauge invariant can be restated
as saying they must be functions only of characters of IRs. We therefore
anticipate that observables in IRs that subduce to direct sums correspond to
larger discrepancies between $\mathcal{G}$ and $\mathbb{G}$. In what follows,
we will use the subduction for the IRs of discrete subgroups of $SU(2)$ and
$SU(3)$ to interrogate this idea.

###  IV.1 $SU(2)$ Subduction tables

By inputting the character tables of $\mathbb{BT},\mathbb{BO},\mathbb{BI}$ and
Eq. 1 into Eq. 3, we derive the multiplicity of each IR of $\mathbb{G}$ in the
subductions of $SU(2)$. Due to its relation to the reduction of the continuous
spacetime onto the lattice, the subduction of $\mathbb{BO}$ was previously
investigated in [98, 99], and they further consider how this affects states of
fixed angular momentum in LGT. It is included alongside $\mathbb{BT}$ and
$\mathbb{BI}$ in Table 8.

Each row of Table 8 corresponds to a specific $SU(2)$ IR, characterized by the
quantum number $j$, with the direct sum of the IRs from the finite subgroups
in its subduction. The dimension of the subduced IR must equal the sum of the
direct sum. Therefore, with only a finite set of IRs, as $j$ increase, the
number of IRs in the subduction and their multiplicities must grow, with
larger discrete groups growing more slowly.

Table 8: Subduction of $SU(2)$ to its crystal-like subgroups: $\mathbb{BT}$,$\mathbb{BO}$,$\mathbb{BI}$ j | $\mathbb{BT}$ | $\mathbb{BO}$ | $\mathbb{BI}$
---|---|---|---
0 | $A_{0}$ | $A_{1}$ | $A_{0}$
$\frac{1}{2}$ | $H$ | $G_{1}$ | $E_{1}$
1 | $T$ | $T_{1}$ | $T_{1}$
$\frac{3}{2}$ | $G_{1}\oplus G_{2}$ | $H$ | $G_{2}$
2 | $E_{1}\oplus E_{2}\oplus T$ | $E\oplus T_{2}$ | $H$
$\frac{5}{2}$ | $H\oplus G_{1}\oplus G_{2}$ | $H\oplus G_{2}$ | $J$
3 | $A_{0}\oplus 2T$ | $A_{2}\oplus T_{1}\oplus T_{2}$ | $G_{1}\oplus T_{2}$
$\frac{7}{2}$ | $2H\oplus G_{1}\oplus G_{2}$ | $H\oplus G_{1}\oplus G_{2}$ | $J\oplus E_{2}$
4 | $A_{0}\oplus E_{1}\oplus E_{2}\oplus 2T$ | $E\oplus A_{1}\oplus T_{1}\oplus T_{2}$ | $H\oplus G_{1}$
$\frac{9}{2}$ | $H\oplus 2G_{1}\oplus 2G_{1}$ | $2H\oplus G_{1}$ | $J\oplus G_{2}$
5 | $E_{1}\oplus E_{2}\oplus 3T$ | $E\oplus 2T_{1}\oplus T_{2}$ | $H\oplus T_{1}\oplus T_{2}$
$\frac{11}{2}$ | $2H\oplus 2G_{1}\oplus 2G_{2}$ | $2H\oplus G_{1}\oplus G_{2}$ | $J\oplus E_{1}\oplus G_{2}$
6 | $2A_{0}\oplus E_{1}\oplus E_{2}\oplus 3T$ | $E\oplus A_{1}\oplus A_{2}\oplus T_{1}\oplus 2T_{2}$ | $H\oplus A_{0}\oplus G_{1}\oplus T_{1}$
$\frac{13}{2}$ | $3H\oplus 2G_{1}\oplus 2G_{2}$ | $2H\oplus G_{1}\oplus 2G_{2}$ | $J\oplus E_{1}\oplus E_{2}\oplus G_{2}$
7 | $A_{0}\oplus E_{1}\oplus E_{2}\oplus 4T$ | $E\oplus A_{2}\oplus 2T_{1}\oplus 2T_{2}$ | $H\oplus G_{1}\oplus T_{1}\oplus T_{2}$
$\frac{15}{2}$ | $2H\oplus 3G_{1}\oplus 3G_{2}$ | $3H\oplus G_{1}\oplus G_{2}$ | $2J\oplus E_{2}\oplus G_{2}$
8 | $A_{0}\oplus 2E_{1}\oplus 2E_{2}\oplus 4T$ | $2E\oplus A_{1}\oplus 2T_{1}\oplus 2T_{2}$ | $2H\oplus H_{1}\oplus T_{2}$
$\frac{17}{2}$ | $3H\oplus 3G_{1}\oplus 3G_{2}$ | $3H\oplus 2G_{1}\oplus G_{2}$ | $2J\oplus E_{2}\oplus G_{2}$
9 | $2A_{0}\oplus E_{1}\oplus E_{2}\oplus 5T$ | $E\oplus A_{1}\oplus A_{2}\oplus 3T_{1}\oplus 2T_{2}$ | $H\oplus 2G_{1}\oplus T_{1}\oplus T_{2}$
$\frac{19}{2}$ | $4H\oplus 3G_{1}\oplus 3G_{2}$ | $3H\oplus 2G_{1}\oplus 2G_{2}$ | $2J\oplus E_{1}\oplus E_{2}\oplus G_{2}$
10 | $2A_{0}\oplus 2E_{1}\oplus 2E_{2}\oplus 5T$ | $2E\oplus A_{1}\oplus A_{2}\oplus 2T_{1}\oplus 3T_{2}$ | $2H\oplus A_{0}\oplus G_{1}\oplus T_{1}\oplus T_{2}$

The table shows that all three crystal-like subgroups have one-to-one
subductions of the lowest IRs of $SU(2)$, increasing from 3 for $\mathbb{BT}$
to 4 for $\mathbb{BO}$ to 6 for $\mathbb{BI}$. While the gauge-invariant
spectrum does not map precisely to the increasing dimension of IRs, it is
perhaps instructive to consider the similar example of rigid rotators [98, 99]
where the energy levels scale like $j^{2}$. Taking this heuristic, the
digitization approximation might be estimated to break down for the first $j$
for which the subduction is not one-to-one. This would suggest that the
relative breakdown of $\mathbb{BT}$ to $\mathbb{BO}$ and $\mathbb{BI}$ would
be 1:1.8:4. From Euclidean simulations in 2+1d and 3+1d, it has been observed
that the ratio of freezeout couplings for the three groups are 1:1.6:2.8 and
1:1.4:2.6 respectively [77, 58]. Thus, the subduction to multiple IRs does not
seem to be more than qualitatively related to the freezeout.

Similar to the question asked by [98, 99]: what is the spin content of a
lattice eigenstate that transforms as a single IR, we are interested in what
IRs of the continuous group are mixed in the discrete group. From this, one
could explore the hierarchy of states to motivate which higher IRs should be
added to the interactions to reduce the effects of digitization. As such, we
list here the induced IRs up to $j=10$ for each crystal-like group. To provide
the reader with a sense of the potential importance of different IRs of
$\mathbb{G}$, we have listed their dimensionality $d$ as well. An IR for which
$h$ multiple copies are required in the induction is denoted by $|_{h}$.

For the 7 IRs of $\mathbb{BT}$, we observe the number of IRs of $SU(2)$ which
mix into $\mathbb{BT}$ IRs are

$\displaystyle(d=1)\;$ $\displaystyle
A_{0}\rightarrow(j=0,3,4,6|_{2},7,8,9|_{2},10,\ldots)$ $\displaystyle(d=1)\;$
$\displaystyle E_{1},E_{2}\rightarrow(j=2,4,5,6,7,8|_{2},9,10|_{2},\ldots)$
$\displaystyle(d=2)\;$ $\displaystyle
H\rightarrow\textstyle(j=\frac{1}{2},\frac{5}{2},\frac{7}{2}|_{2},\frac{9}{2},\frac{11}{2}|_{2},\frac{13}{2}|_{3},$
$\displaystyle\phantom{xxxxxxxxxxxx}\textstyle\frac{15}{2}|_{2},\frac{17}{2}|_{3},\frac{19}{2}|_{4},\ldots)$
$\displaystyle(d=2)\;$ $\displaystyle
G_{1},G_{2}\rightarrow\textstyle(j=\frac{3}{2},\frac{5}{2},\frac{7}{2},\frac{9}{2}|_{2},\frac{11}{2}|_{2},\frac{13}{2}|_{2},$
$\displaystyle\phantom{xxxxxxxxxxxx}\textstyle\frac{15}{2}|_{3},\frac{17}{2}|_{3},\frac{19}{2}|_{3},\ldots)$
$\displaystyle(d=3)\;$ $\displaystyle
T\rightarrow\textstyle(j=1,2,3|_{2},4|_{2},5|_{3},6|_{3},7|_{4},8|_{4},$
$\displaystyle\phantom{xxxxxxxxxxxx}\textstyle 9|_{5},10|_{5},\ldots)$

Because $SU(2)$ has one 1d IR, we see that $E_{1},E_{2}$ arise only within
direct sums of IRs. Further, they always occur as a pair. A similar behavior
is observed in the 2d IRs of $\mathbb{BT}$. One of them is subduced from the
$j=1/2$ IR of $SU(2)$, while the other two only occur in pairs, first together
in the $j=3/2$ IR.

Given that $\mathbb{BT}$ is a subgroup of $\mathbb{BO}$, it is useful to
compare their IRs. In contrast to $\mathbb{BT}$, $\mathbb{BO}$ has only two 1d
IRs, an additional 3d one, and a new 4d IR that is the induction of
$G_{1},G_{2}$ of $\mathbb{BT}$. The set of $\mathbb{BO}$ IRs consist of a
different mixing of $SU(2)$ IRs, with generically greater separation in
$d_{j}$:

$\displaystyle(d=1)\;$ $\displaystyle A_{1}\rightarrow(j=0,4,6,8,9,10,\ldots)$
$\displaystyle(d=1)\;$ $\displaystyle A_{2}\rightarrow(j=3,6,7,9,10,\ldots)$
$\displaystyle(d=2)\;$ $\displaystyle
E\rightarrow(j=2,4,5,6,7,8|_{2},9,10|_{2},\ldots)$ $\displaystyle(d=2)\;$
$\displaystyle
G_{1}\rightarrow\textstyle(j=\frac{1}{2},\frac{7}{2},\frac{9}{2},\frac{11}{2},\frac{13}{2},\frac{15}{2},\frac{17}{2}|_{2},\frac{19}{2}|_{2},\ldots)$
$\displaystyle(d=2)\;$ $\displaystyle
G_{2}\rightarrow\textstyle(j=\frac{5}{2},\frac{7}{2},\frac{11}{2},\frac{13}{2}|_{2},\frac{15}{2},\frac{17}{2}|_{2},\frac{19}{2}|_{2},\ldots)$
$\displaystyle(d=3)\;$ $\displaystyle
T_{1}\rightarrow\textstyle(j=1,3,4,5|_{2},6,7|_{2},8|_{2},9|_{3},10|_{2},\ldots)$
$\displaystyle(d=3)\;$ $\displaystyle
T_{2}\rightarrow\textstyle(j=2,3,4,5,6|_{2},7|_{2},8|_{2},9|_{2},10|_{3},\ldots)$
$\displaystyle(d=4)\;$ $\displaystyle
H\rightarrow\textstyle(j=\frac{3}{2},\frac{5}{2},\frac{7}{2},\frac{9}{2}|_{2},\frac{11}{2}|_{2},\frac{13}{2}|_{2},$
$\displaystyle\phantom{xxxxxxxxxxxxxxx}\textstyle\frac{15}{2}|_{3},\frac{17}{2}|_{3},\frac{19}{2}|_{3},\ldots)$

By inspection, we see that, unlike $\mathbb{BT}$, no pairs of IRs exist in the
induction. Concerning the fundamental faithful IR used in the Wilson action,
$G_{1}$, we note that while it does induce to $j=1/2$ like $H$ for
$\mathbb{BT}$, it mixes with the higher dimension $j=7/2$ rather than $5/2$
for $H$. This should be related to a larger freezeout coupling and thus a
better approximation of $SU(2)$.

Finally, we can investigate $\mathbb{BI}$, observing that for each $d_{j}$,
one IR corresponds one-to-one with $SU(2)$, and the others occur in complex
mixings. The inductions of $\mathbb{BI}$ are:

$\displaystyle(d=1)\;$ $\displaystyle
A_{0}\rightarrow\textstyle(j=0,6,10\ldots)$ $\displaystyle(d=2)\;$
$\displaystyle
E_{1}\rightarrow\textstyle(j=\frac{1}{2},\frac{11}{2},\frac{13}{2},\frac{19}{2},\ldots)$
$\displaystyle(d=2)\;$ $\displaystyle
E_{2}\rightarrow\textstyle(j=\frac{7}{2},\frac{13}{2},\frac{17}{2},\frac{19}{2}\ldots)$
$\displaystyle(d=3)\;$ $\displaystyle
T_{1}\rightarrow\textstyle(j=1,5,6,7,9,10\ldots)$ $\displaystyle(d=3)\;$
$\displaystyle T_{2}\rightarrow\textstyle(j=3,5,7,8,9,10\ldots)$
$\displaystyle(d=4)\;$ $\displaystyle
G_{1}\rightarrow\textstyle(j=3,4,6,7,8,9|_{2},10,\ldots)$
$\displaystyle(d=4)\;$ $\displaystyle
G_{2}\rightarrow\textstyle(j=\frac{3}{2},\frac{9}{2},\frac{11}{2},\frac{13}{2},\frac{15}{2},\frac{17}{2},\frac{19}{2},\ldots)$
$\displaystyle(d=5)\;$ $\displaystyle
H\rightarrow\textstyle(j=2,4,5,6,7,8|_{2},9,10|_{2}\ldots)$
$\displaystyle(d=6)\;$ $\displaystyle
J\rightarrow\textstyle(j=\frac{5}{2},\frac{7}{2},\frac{9}{2},\frac{11}{2},\frac{13}{2},\frac{15}{2}|_{2},\frac{17}{2}|_{2},\frac{19}{2}|_{2},\ldots)$

Looking again at the 2d IR used in the Wilson action, $E_{1}$, we see the next
state after $j=1/2$ to mix into it has increased further to $j=11/2$. Previous
results using group space decimation suggested in [49] would suggest and IR
enter as $\beta^{d_{j}-1}$ in the strong-coupling expansion and thus the
subduction gives a relation to breaking at strong coupling. A dedicated study
of excited state glueball masses should be undertaken to study how the
dynamics of strong-coupling eigenstates of a given IR of a subgroup are
affected by their mixing into multiple IRs of $SU(2)$, especially when
$m_{j}^{r}>1$.

###  IV.2 $SU(3)$ Subduction tables

Table 9: Subduction of $SU(3)$ to crystal-like subgroups: $\Sigma(108)$ ,
$\Sigma(216)$ , $\Sigma(648)$, $\Sigma(1080)$

$(p,q)$ | $\Sigma(108)$ | $\Sigma(216)$ | $\Sigma(648)$ | $\Sigma(1080)$
---|---|---|---|---
(0,0) | $\mathbf{1}^{(0)}$ | $\mathbf{1}^{(0)}$ | $\mathbf{1}^{(0)}$ | $\mathbf{1}^{(0)}$
(1,0) | $\mathbf{3}^{(0)*}$ | $\mathbf{3}^{(0)*}$ | $\mathbf{3}^{(0)*}$ | $\mathbf{3}^{(0)*}$
(0,1) | $\mathbf{3}^{(0)}$ | $\mathbf{3}^{(0)}$ | $\mathbf{3}^{(0)}$ | $\mathbf{3}^{(0)}$
(2,0) | $\mathbf{3}^{(1)}\oplus\mathbf{3}^{(3)}$ | $\mathbf{6}^{(0)}$ | $\mathbf{6}^{(0)}$ | $\mathbf{6}^{(0)*}$
(1,1) | $\mathbf{4}^{(0)}\oplus\mathbf{4}^{(0)^{\prime}}$ | $\mathbf{8}^{(0)}$ | $\mathbf{8}^{(0)}$ | $\mathbf{8}^{(0)}$
(0,2) | $\mathbf{3}^{(1)*}\oplus\mathbf{3}^{(3)*}$ | $\mathbf{6}^{(0)*}$ | $\mathbf{6}^{(0)*}$ | $\mathbf{6}^{(0)}$
(3,0) | $\mathbf{1}^{(1)}\oplus\mathbf{1}^{(3)}\oplus\mathbf{4}^{(0)}\oplus\mathbf{4}^{(0)^{\prime}}$ | $\mathbf{2}^{(0)}\oplus\mathbf{8}^{(0)}$ | $\mathbf{2}^{(2)}\oplus\mathbf{8}^{(1)}$ | $\mathbf{10}^{(0)}$
(2,1) | $\mathbf{3}^{(0)*}\oplus\mathbf{3}^{(1)*}\oplus\mathbf{3}^{(3)*}\oplus 2\,\mathbf{3}^{(2)*}$ | $\mathbf{3}^{(1)*}\oplus\mathbf{3}^{(2)*}\oplus\mathbf{3}^{(3)*}\oplus\mathbf{6}^{(0)*}$ | $\mathbf{6}^{(2)*}\oplus\mathbf{9}^{(0)*}$ | $\mathbf{15}^{(0)*}$
(1,2) | $\mathbf{3}^{(0)}\oplus\mathbf{3}^{(1)}\oplus\mathbf{3}^{(3)}\oplus 2\,\mathbf{3}^{(2)}$ | $\mathbf{3}^{(1)}\oplus\mathbf{3}^{(2)}\oplus\mathbf{3}^{(3)}\oplus\mathbf{6}^{(0)}$ | $\mathbf{6}^{(2)}\oplus\mathbf{9}^{(0)}$ | $\mathbf{15}^{(0)}$
(0,3) | $\mathbf{1}^{(1)}\oplus\mathbf{1}^{(3)}\oplus\mathbf{4}^{(0)}\oplus\mathbf{4}^{(0)^{\prime}}$ | $\mathbf{2}^{(0)}\oplus\mathbf{8}^{(0)}$ | $\mathbf{2}^{(1)}\oplus\mathbf{8}^{(2)}$ | $\mathbf{10}^{(0)}$
(4,0) | $\mathbf{3}^{(0)*}\oplus 2\,\mathbf{3}^{(1)*}\oplus 2\,\mathbf{3}^{(3)*}$ | $\mathbf{3}^{(0)*}\oplus 2\,\mathbf{6}^{(0)*}$ | $\mathbf{3}^{(2)*}\oplus\mathbf{6}^{(1)*}\oplus\mathbf{6}^{(2)*}$ | $\mathbf{6}^{(0)}\oplus\mathbf{9}^{(1)}$
(3,1) | $2\,\mathbf{3}^{(0)}\oplus 2\,\mathbf{3}^{(1)}\oplus 2\,\mathbf{3}^{(2)}\oplus 2\,\mathbf{3}^{(3)}$ | $\mathbf{3}^{(0)}\oplus\mathbf{3}^{(1)}\oplus\mathbf{3}^{(2)}\oplus\mathbf{3}^{(3)}\oplus 2\,\mathbf{6}^{(0)}$ | $\mathbf{3}^{(1)}\oplus\mathbf{6}^{(0)}\oplus\mathbf{6}^{(1)}\oplus\mathbf{9}^{(0)}$ | $\mathbf{15}^{(0)}\oplus\mathbf{9}^{(1)*}$
(2,2) | $\mathbf{1}^{(0)}\oplus 2\,\mathbf{1}^{(2)}\oplus 3\,\mathbf{4}^{(0)}\oplus 3\,\mathbf{4}^{(0)^{\prime}}$ | $\mathbf{1}^{(1)}\oplus\mathbf{1}^{(2)}\oplus\mathbf{1}^{(3)}\oplus 3\,\mathbf{8}^{(0)}$ | $\mathbf{3}^{(a)}\oplus\mathbf{8}^{(0)}\oplus\mathbf{8}^{(1)}\oplus\mathbf{8}^{(2)}$ | $\mathbf{5}^{(0)^{\prime}}\oplus\mathbf{5}^{(0)}\oplus\mathbf{8}^{(0)^{\prime}}\oplus\mathbf{9}^{(0)}$
(1,3) | $2\,\mathbf{3}^{(0)*}\oplus 2\,\mathbf{3}^{(1)*}\oplus 2\,\mathbf{3}^{(2)*}\oplus 2\,\mathbf{3}^{(3)*}$ | $\mathbf{3}^{(0)*}\oplus\mathbf{3}^{(1)*}\oplus\mathbf{3}^{(2)*}\oplus\mathbf{3}^{(3)*}\oplus 2\,\mathbf{6}^{(0)*}$ | $\mathbf{3}^{(1)*}\oplus\mathbf{6}^{(0)*}\oplus\mathbf{6}^{(1)*}\oplus\mathbf{9}^{(0)*}$ | $\mathbf{15}^{(0)*}\oplus\mathbf{9}^{(1)}$
(0,4) | $\mathbf{3}^{(0)}\oplus 2\,\mathbf{3}^{(1)}\oplus 2\,\mathbf{3}^{(3)}$ | $\mathbf{3}^{(0)}\oplus 2\,\mathbf{6}^{(0)}$ | $\mathbf{3}^{(2)}\oplus\mathbf{6}^{(1)}\oplus\mathbf{6}^{(2)}$ | $\mathbf{6}^{(0)*}\oplus\mathbf{9}^{(1)*}$
(5,0) | $\mathbf{3}^{(1)}\oplus\mathbf{3}^{(3)}\oplus 2\,\mathbf{3}^{(2)}\oplus 3\,\mathbf{3}^{(0)}$ | $\mathbf{3}^{(1)}\oplus\mathbf{3}^{(2)}\oplus\mathbf{3}^{(3)}\oplus\mathbf{6}^{(0)}\oplus 2\,\mathbf{3}^{(0)}$ | $\mathbf{3}^{(1)}\oplus\mathbf{3}^{(2)}\oplus\mathbf{6}^{(1)}\oplus\mathbf{9}^{(0)}$ | $\mathbf{15}^{(0)}\oplus\mathbf{3}^{(0)}\oplus\mathbf{3}^{(1)}$
(4,1) | $\mathbf{1}^{(0)}\oplus\mathbf{1}^{(1)}\oplus\mathbf{1}^{(3)}\oplus 4\,\mathbf{4}^{(0)}\oplus 4\,\mathbf{4}^{(0)^{\prime}}$ | $\mathbf{1}^{(0)}\oplus\mathbf{2}^{(0)}\oplus 4\,\mathbf{8}^{(0)}$ | $\mathbf{1}^{(1)}\oplus\mathbf{2}^{(0)}\oplus\mathbf{8}^{(0)}\oplus\mathbf{8}^{(2)}\oplus 2\,\mathbf{8}^{(1)}$ | $\mathbf{10}^{(0)}\oplus\mathbf{8}^{(0)^{\prime}}\oplus\mathbf{8}^{(0)}\oplus\mathbf{9}^{(0)}$
(3,2) | $3\,\mathbf{3}^{(1)*}\oplus 3\,\mathbf{3}^{(3)*}\oplus 4\,\mathbf{3}^{(0)*}\oplus 4\,\mathbf{3}^{(2)*}$ | $2\,\mathbf{3}^{(0)*}\oplus 2\,\mathbf{3}^{(1)*}\oplus 2\,\mathbf{3}^{(2)*}\oplus 2\,\mathbf{3}^{(3)*}\oplus 3\,\mathbf{6}^{(0)*}$ | $\mathbf{3}^{(0)*}\oplus\mathbf{3}^{(2)*}\oplus\mathbf{6}^{(0)*}\oplus\mathbf{6}^{(1)*}\oplus\mathbf{6}^{(2)*}\oplus 2\,\mathbf{9}^{(0)*}$ | $\mathbf{3}^{(1)*}\oplus\mathbf{9}^{(1)}\oplus 2\,\mathbf{15}^{(0)*}$
(2,3) | $3\,\mathbf{3}^{(1)}\oplus 3\,\mathbf{3}^{(3)}\oplus 4\,\mathbf{3}^{(0)}\oplus 4\,\mathbf{3}^{(2)}$ | $2\,\mathbf{3}^{(0)}\oplus 2\,\mathbf{3}^{(1)}\oplus 2\,\mathbf{3}^{(2)}\oplus 2\,\mathbf{3}^{(3)}\oplus 3\,\mathbf{6}^{(0)}$ | $\mathbf{3}^{(0)}\oplus\mathbf{3}^{(2)}\oplus\mathbf{6}^{(0)}\oplus\mathbf{6}^{(1)}\oplus\mathbf{6}^{(2)}\oplus 2\,\mathbf{9}^{(0)}$ | $\mathbf{3}^{(1)}\oplus\mathbf{9}^{(1)*}\oplus 2\,\mathbf{15}^{(0)}$
(1,4) | $\mathbf{1}^{(0)}\oplus\mathbf{1}^{(1)}\oplus\mathbf{1}^{(3)}\oplus 4\,\mathbf{4}^{(0)}\oplus 4\,\mathbf{4}^{(0)^{\prime}}$ | $\mathbf{1}^{(0)}\oplus\mathbf{2}^{(0)}\oplus 4\,\mathbf{8}^{(0)}$ | $\mathbf{1}^{(2)}\oplus\mathbf{2}^{(0)}\oplus\mathbf{8}^{(0)}\oplus\mathbf{8}^{(1)}\oplus 2\,\mathbf{8}^{(2)}$ | $\mathbf{10}^{(0)}\oplus\mathbf{8}^{(0)^{\prime}}\oplus\mathbf{8}^{(0)}\oplus\mathbf{9}^{(0)}$
(0,5) | $\mathbf{3}^{(1)*}\oplus\mathbf{3}^{(3)*}\oplus 2\,\mathbf{3}^{(2)*}\oplus 3\,\mathbf{3}^{(0)*}$ | $\mathbf{3}^{(1)*}\oplus\mathbf{3}^{(2)*}\oplus\mathbf{3}^{(3)*}\oplus\mathbf{6}^{(0)*}\oplus 2\,\mathbf{3}^{(0)*}$ | $\mathbf{3}^{(1)*}\oplus\mathbf{3}^{(2)*}\oplus\mathbf{6}^{(1)*}\oplus\mathbf{9}^{(0)*}$ | $\mathbf{15}^{(0)*}\oplus\mathbf{3}^{(0)*}\oplus\mathbf{3}^{(1)*}$
(6,0) | $2\,\mathbf{1}^{(0)}\oplus 2\,\mathbf{1}^{(2)}\oplus 3\,\mathbf{4}^{(0)}\oplus 3\,\mathbf{4}^{(0)^{\prime}}$ | $\mathbf{1}^{(0)}\oplus\mathbf{1}^{(1)}\oplus\mathbf{1}^{(2)}\oplus\mathbf{1}^{(3)}\oplus 3\,\mathbf{8}^{(0)}$ | $\mathbf{1}^{(2)}\oplus\mathbf{3}^{(a)}\oplus\mathbf{8}^{(1)}\oplus 2\,\mathbf{8}^{(2)}$ | $\mathbf{1}^{(0)}\oplus\mathbf{5}^{(0)^{\prime}}\oplus\mathbf{5}^{(0)}\oplus\mathbf{8}^{(0)}\oplus\mathbf{9}^{(0)}$
(5,1) | $4\,\mathbf{3}^{(0)*}\oplus 4\,\mathbf{3}^{(1)*}\oplus 4\,\mathbf{3}^{(2)*}\oplus 4\,\mathbf{3}^{(3)*}$ | $2\,\mathbf{3}^{(0)*}\oplus 2\,\mathbf{3}^{(1)*}\oplus 2\,\mathbf{3}^{(2)*}\oplus 2\,\mathbf{3}^{(3)*}\oplus 4\,\mathbf{6}^{(0)*}$ | $\mathbf{3}^{(1)*}\oplus\mathbf{3}^{(2)*}\oplus\mathbf{6}^{(0)*}\oplus\mathbf{6}^{(2)*}\oplus 2\,\mathbf{6}^{(1)*}\oplus 2\,\mathbf{9}^{(0)*}$ | $\mathbf{3}^{(0)*}\oplus\mathbf{6}^{(0)}\oplus\mathbf{9}^{(1)}\oplus 2\,\mathbf{15}^{(0)*}$
(4,2) | $4\,\mathbf{3}^{(0)}\oplus 4\,\mathbf{3}^{(2)}\oplus 6\,\mathbf{3}^{(1)}\oplus 6\,\mathbf{3}^{(3)}$ | $2\,\mathbf{3}^{(0)}\oplus 2\,\mathbf{3}^{(1)}\oplus 2\,\mathbf{3}^{(2)}\oplus 2\,\mathbf{3}^{(3)}\oplus 6\,\mathbf{6}^{(0)}$ | $\mathbf{3}^{(0)}\oplus\mathbf{3}^{(1)}\oplus 2\,\mathbf{6}^{(0)}\oplus 2\,\mathbf{6}^{(1)}\oplus 2\,\mathbf{6}^{(2)}\oplus 2\,\mathbf{9}^{(0)}$ | $2\,\mathbf{15}^{(0)}\oplus 2\,\mathbf{6}^{(0)*}\oplus 2\,\mathbf{9}^{(1)*}$
(3,3) | $2\,\mathbf{1}^{(0)}\oplus 2\,\mathbf{1}^{(1)}\oplus 2\,\mathbf{1}^{(2)}\oplus 2\,\mathbf{1}^{(3)}\oplus 7\,\mathbf{4}^{(0)}\oplus 7\,\mathbf{4}^{(0)^{\prime}}$ | $\mathbf{1}^{(0)}\oplus\mathbf{1}^{(1)}\oplus\mathbf{1}^{(2)}\oplus\mathbf{1}^{(3)}\oplus 2\,\mathbf{2}^{(0)}\oplus 7\,\mathbf{8}^{(0)}$ | $\mathbf{1}^{(0)}\oplus\mathbf{2}^{(1)}\oplus\mathbf{2}^{(2)}\oplus\mathbf{3}^{(a)}\oplus 2\,\mathbf{8}^{(1)}\oplus 2\,\mathbf{8}^{(2)}\oplus 3\,\mathbf{8}^{(0)}$ | $\mathbf{5}^{(0)^{\prime}}\oplus\mathbf{5}^{(0)}\oplus\mathbf{8}^{(0)^{\prime}}\oplus\mathbf{8}^{(0)}\oplus 2\,\mathbf{10}^{(0)}\oplus 2\,\mathbf{9}^{(0)}$
(2,4) | $4\,\mathbf{3}^{(0)*}\oplus 4\,\mathbf{3}^{(2)*}\oplus 6\,\mathbf{3}^{(1)*}\oplus 6\,\mathbf{3}^{(3)*}$ | $2\,\mathbf{3}^{(0)*}\oplus 2\,\mathbf{3}^{(1)*}\oplus 2\,\mathbf{3}^{(2)*}\oplus 2\,\mathbf{3}^{(3)*}\oplus 6\,\mathbf{6}^{(0)*}$ | $\mathbf{3}^{(0)*}\oplus\mathbf{3}^{(1)*}\oplus 2\,\mathbf{6}^{(0)*}\oplus 2\,\mathbf{6}^{(1)*}\oplus 2\,\mathbf{6}^{(2)*}\oplus 2\,\mathbf{9}^{(0)*}$ | $2\,\mathbf{15}^{(0)*}\oplus 2\,\mathbf{6}^{(0)}\oplus 2\,\mathbf{9}^{(1)}$
(1,5) | $4\,\mathbf{3}^{(0)}\oplus 4\,\mathbf{3}^{(1)}\oplus 4\,\mathbf{3}^{(2)}\oplus 4\,\mathbf{3}^{(3)}$ | $2\,\mathbf{3}^{(0)}\oplus 2\,\mathbf{3}^{(1)}\oplus 2\,\mathbf{3}^{(2)}\oplus 2\,\mathbf{3}^{(3)}\oplus 4\,\mathbf{6}^{(0)}$ | $\mathbf{3}^{(1)}\oplus\mathbf{3}^{(2)}\oplus\mathbf{6}^{(0)}\oplus\mathbf{6}^{(2)}\oplus 2\,\mathbf{6}^{(1)}\oplus 2\,\mathbf{9}^{(0)}$ | $\mathbf{3}^{(0)}\oplus\mathbf{6}^{(0)*}\oplus\mathbf{9}^{(1)*}\oplus 2\,\mathbf{15}^{(0)}$
(0,6) | $2\,\mathbf{1}^{(0)}\oplus 2\,\mathbf{1}^{(2)}\oplus 3\,\mathbf{4}^{(0)}\oplus 3\,\mathbf{4}^{(0)^{\prime}}$ | $\mathbf{1}^{(0)}\oplus\mathbf{1}^{(1)}\oplus\mathbf{1}^{(2)}\oplus\mathbf{1}^{(3)}\oplus 3\,\mathbf{8}^{(0)}$ | $\mathbf{1}^{(1)}\oplus\mathbf{3}^{(a)}\oplus\mathbf{8}^{(2)}\oplus 2\,\mathbf{8}^{(1)}$ | $\mathbf{1}^{(0)}\oplus\mathbf{5}^{(0)^{\prime}}\oplus\mathbf{5}^{(0)}\oplus\mathbf{8}^{(0)}\oplus\mathbf{9}^{(0)}$
(7,0) | $2\,\mathbf{3}^{(1)*}\oplus 2\,\mathbf{3}^{(3)*}\oplus 4\,\mathbf{3}^{(0)*}\oplus 4\,\mathbf{3}^{(2)*}$ | $2\,\mathbf{3}^{(0)*}\oplus 2\,\mathbf{3}^{(1)*}\oplus 2\,\mathbf{3}^{(2)*}\oplus 2\,\mathbf{3}^{(3)*}\oplus 2\,\mathbf{6}^{(0)*}$ | $\mathbf{6}^{(0)*}\oplus\mathbf{6}^{(1)*}\oplus 2\,\mathbf{3}^{(1)*}\oplus 2\,\mathbf{9}^{(0)*}$ | $\mathbf{3}^{(0)*}\oplus\mathbf{3}^{(1)*}\oplus 2\,\mathbf{15}^{(0)*}$
(6,1) | $5\,\mathbf{3}^{(0)}\oplus 5\,\mathbf{3}^{(1)}\oplus 5\,\mathbf{3}^{(3)}\oplus 6\,\mathbf{3}^{(2)}$ | $2\,\mathbf{3}^{(0)}\oplus 3\,\mathbf{3}^{(1)}\oplus 3\,\mathbf{3}^{(2)}\oplus 3\,\mathbf{3}^{(3)}\oplus 5\,\mathbf{6}^{(0)}$ | $\mathbf{6}^{(0)}\oplus 2\,\mathbf{3}^{(2)}\oplus 2\,\mathbf{6}^{(1)}\oplus 2\,\mathbf{6}^{(2)}\oplus 3\,\mathbf{9}^{(0)}$ | $\mathbf{3}^{(0)}\oplus\mathbf{6}^{(0)*}\oplus\mathbf{9}^{(1)*}\oplus 3\,\mathbf{15}^{(0)}$
(5,2) | $\mathbf{1}^{(0)}\oplus 2\,\mathbf{1}^{(2)}\oplus 3\,\mathbf{1}^{(1)}\oplus 3\,\mathbf{1}^{(3)}\oplus 9\,\mathbf{4}^{(0)}\oplus 9\,\mathbf{4}^{(0)^{\prime}}$ | $\mathbf{1}^{(1)}\oplus\mathbf{1}^{(2)}\oplus\mathbf{1}^{(3)}\oplus 3\,\mathbf{2}^{(0)}\oplus 9\,\mathbf{8}^{(0)}$ | $\mathbf{2}^{(0)}\oplus\mathbf{2}^{(1)}\oplus\mathbf{2}^{(2)}\oplus\mathbf{3}^{(a)}\oplus 3\,\mathbf{8}^{(0)}\oplus 3\,\mathbf{8}^{(1)}\oplus 3\,\mathbf{8}^{(2)}$ | $\mathbf{5}^{(0)^{\prime}}\oplus\mathbf{5}^{(0)}\oplus\mathbf{9}^{(0)}\oplus 2\,\mathbf{8}^{(0)^{\prime}}\oplus 2\,\mathbf{8}^{(0)}\oplus 3\,\mathbf{10}^{(0)}$
(4,3) | $7\,\mathbf{3}^{(1)*}\oplus 7\,\mathbf{3}^{(3)*}\oplus 8\,\mathbf{3}^{(0)*}\oplus 8\,\mathbf{3}^{(2)*}$ | $4\,\mathbf{3}^{(0)*}\oplus 4\,\mathbf{3}^{(1)*}\oplus 4\,\mathbf{3}^{(2)*}\oplus 4\,\mathbf{3}^{(3)*}\oplus 7\,\mathbf{6}^{(0)*}$ | $\mathbf{3}^{(1)*}\oplus\mathbf{3}^{(2)*}\oplus\mathbf{6}^{(1)*}\oplus 2\,\mathbf{3}^{(0)*}\oplus 3\,\mathbf{6}^{(0)*}\oplus 3\,\mathbf{6}^{(2)*}\oplus 4\,\mathbf{9}^{(0)*}$ | $\mathbf{3}^{(0)*}\oplus\mathbf{3}^{(1)*}\oplus\mathbf{6}^{(0)}\oplus 2\,\mathbf{9}^{(1)}\oplus 4\,\mathbf{15}^{(0)*}$
(3,4) | $7\,\mathbf{3}^{(1)}\oplus 7\,\mathbf{3}^{(3)}\oplus 8\,\mathbf{3}^{(0)}\oplus 8\,\mathbf{3}^{(2)}$ | $4\,\mathbf{3}^{(0)}\oplus 4\,\mathbf{3}^{(1)}\oplus 4\,\mathbf{3}^{(2)}\oplus 4\,\mathbf{3}^{(3)}\oplus 7\,\mathbf{6}^{(0)}$ | $\mathbf{3}^{(1)}\oplus\mathbf{3}^{(2)}\oplus\mathbf{6}^{(1)}\oplus 2\,\mathbf{3}^{(0)}\oplus 3\,\mathbf{6}^{(0)}\oplus 3\,\mathbf{6}^{(2)}\oplus 4\,\mathbf{9}^{(0)}$ | $\mathbf{3}^{(0)}\oplus\mathbf{3}^{(1)}\oplus\mathbf{6}^{(0)*}\oplus 2\,\mathbf{9}^{(1)*}\oplus 4\,\mathbf{15}^{(0)}$
(2,5) | $\mathbf{1}^{(0)}\oplus 2\,\mathbf{1}^{(2)}\oplus 3\,\mathbf{1}^{(1)}\oplus 3\,\mathbf{1}^{(3)}\oplus 9\,\mathbf{4}^{(0)}\oplus 9\,\mathbf{4}^{(0)^{\prime}}$ | $\mathbf{1}^{(1)}\oplus\mathbf{1}^{(2)}\oplus\mathbf{1}^{(3)}\oplus 3\,\mathbf{2}^{(0)}\oplus 9\,\mathbf{8}^{(0)}$ | $\mathbf{2}^{(0)}\oplus\mathbf{2}^{(1)}\oplus\mathbf{2}^{(2)}\oplus\mathbf{3}^{(a)}\oplus 3\,\mathbf{8}^{(0)}\oplus 3\,\mathbf{8}^{(1)}\oplus 3\,\mathbf{8}^{(2)}$ | $\mathbf{5}^{(0)^{\prime}}\oplus\mathbf{5}^{(0)}\oplus\mathbf{9}^{(0)}\oplus 2\,\mathbf{8}^{(0)^{\prime}}\oplus 2\,\mathbf{8}^{(0)}\oplus 3\,\mathbf{10}^{(0)}$
(1,6) | $5\,\mathbf{3}^{(0)*}\oplus 5\,\mathbf{3}^{(1)*}\oplus 5\,\mathbf{3}^{(3)*}\oplus 6\,\mathbf{3}^{(2)*}$ | $2\,\mathbf{3}^{(0)*}\oplus 3\,\mathbf{3}^{(1)*}\oplus 3\,\mathbf{3}^{(2)*}\oplus 3\,\mathbf{3}^{(3)*}\oplus 5\,\mathbf{6}^{(0)*}$ | $\mathbf{6}^{(0)*}\oplus 2\,\mathbf{3}^{(2)*}\oplus 2\,\mathbf{6}^{(1)*}\oplus 2\,\mathbf{6}^{(2)*}\oplus 3\,\mathbf{9}^{(0)*}$ | $\mathbf{3}^{(0)*}\oplus\mathbf{6}^{(0)}\oplus\mathbf{9}^{(1)}\oplus 3\,\mathbf{15}^{(0)*}$
(0,7) | $2\,\mathbf{3}^{(1)}\oplus 2\,\mathbf{3}^{(3)}\oplus 4\,\mathbf{3}^{(0)}\oplus 4\,\mathbf{3}^{(2)}$ | $2\,\mathbf{3}^{(0)}\oplus 2\,\mathbf{3}^{(1)}\oplus 2\,\mathbf{3}^{(2)}\oplus 2\,\mathbf{3}^{(3)}\oplus 2\,\mathbf{6}^{(0)}$ | $\mathbf{6}^{(0)}\oplus\mathbf{6}^{(1)}\oplus 2\,\mathbf{3}^{(1)}\oplus 2\,\mathbf{9}^{(0)}$ | $\mathbf{3}^{(0)}\oplus\mathbf{3}^{(1)}\oplus 2\,\mathbf{15}^{(0)}$

The subduction of $SU(3)$ IRs into those of its finite subgroups are presented
in Table 9. Each table row correlates a specific $SU(3)$ IR, characterized by
the quantum numbers $(p,q)$, with a direct sum of the IRs of the finite
subgroups. Given that $d_{r}$ increases with $(p,q)$, satisfying Eq.
4necessitates larger direct sums. Further, in contrast to $SU(2)$, where there
is only 1 IR of a given dimension, $SU(3)$ can have 0,1 or 2 IRs per
dimension. To compare to the subductions, it is useful to note that $SU(3)$
has one 1d and 8d IR; two 3d, 6d, 10d, 21d IRs; and four 15d IRs. Thus,
looking at the character tables, we can establish that discrete group IRs of
other dimensions like $\mathbf{2}$ or $\mathbf{5}^{(0)}$ are subduced only in
a direct sum.

From Table 9, we observe that while increasing the group size generically
reduces the complexity of the subductions, the number of one-to-one
subductions is not always increasing. Instead, we find that
$\operatorname{\Sigma(216)}$ and $\operatorname{\Sigma(648)}$ both have six
such subductions. Further, we observe that for the smallest group
$\operatorname{\Sigma(108)}$, the only nontrivial IR that subduces one-to-one
is the fundamental.

To further explore the relationship between $SU(3)$ and its subgroups, we
again present the inductions. For the sake of conciseness, for pairs of
complex IRs we only enumerate the induction for one, e.g. $\mathbf{3}^{(0)}$,
while the other in the pair ($\mathbf{3}^{(0)*}$) can be computed by
exchanging $p\leftrightarrow q$. For $\operatorname{\Sigma(108)}$ the
induction up to $p+q=5$ is

$\displaystyle\mathbf{1}^{(0)}$
$\displaystyle\rightarrow{\\{0,0\\}|_{1},\\{2,2\\}|_{1},\\{1,4\\}|_{1},\\{4,1\\}|_{1},\ldots}$
$\displaystyle\mathbf{1}^{(1)},\mathbf{1}^{(3)}$
$\displaystyle\rightarrow{\\{0,3\\}|_{1},\\{3,0\\}|_{1},\\{1,4\\}|_{1},\\{4,1\\}|_{1},\ldots}$
$\displaystyle\mathbf{1}^{(2)}$
$\displaystyle\rightarrow{\\{2,2\\}|_{2},\ldots}$
$\displaystyle\mathbf{3}^{(0)}$
$\displaystyle\rightarrow\\{0,1\\}|_{1},\\{1,2\\}|_{1},\\{0,4\\}|_{1},\\{3,1\\}|_{2},$
$\displaystyle\phantom{xxxxxxxxx}\\{2,3\\}|_{4},\\{5,0\\}|_{3},\ldots$
$\displaystyle\mathbf{3}^{(1)},\mathbf{3}^{(3)}$
$\displaystyle\rightarrow\\{2,0\\}|_{1},\\{1,2\\}|_{1},\\{0,4\\}|_{2},\\{3,1\\}|_{2},$
$\displaystyle\phantom{xxxxxxxxx}\\{2,3\\}|_{3},\\{5,0\\}|_{1},\ldots$
$\displaystyle\mathbf{3}^{(2)}$
$\displaystyle\rightarrow{\\{1,2\\}|_{2},\\{3,1\\}|_{2},\\{2,3\\}|_{4},\\{5,0\\}|_{2},\ldots}$
$\displaystyle\mathbf{4}^{(0)},\mathbf{4}^{(0)^{\prime}}$
$\displaystyle\rightarrow\\{1,1\\}|_{1},\\{0,3\\}|_{1},\\{3,0\\}|_{1},\\{2,2\\}|_{3},$
$\displaystyle\phantom{xxxxxxxxx}\\{1,4\\}|_{4},\\{4,1\\}|_{4},\ldots$

The $\operatorname{\Sigma(108)}$ character table in Tab. 4 shows that every
dimension has more IRs than it should to match $SU(3)$ and that the largest
pair is 4d. From the inductions, one observes that the pairs
$\mathbf{1}^{(1)}$ & $\mathbf{1}^{(3)}$, $\mathbf{3}^{(1)}$ &
$\mathbf{3}^{(3)}$, and $\mathbf{4}^{(0)}$ & $\mathbf{4}^{(0)^{\prime}}$
always appear together, indication that even when the subduction is not one-
to-one that one-to-two subductions can have residual symmetry. In particular
the pair $\mathbf{4}^{(0)}$ & $\mathbf{4}^{(0)^{\prime}}$ correspond to the
subduction of the adjoint, 8d, IR of $SU(3)$; this IR is commonly used in
modified actions for discrete and continuous groups because it arises first in
strong-coupling and character expansions [86, 111, 112, 113, 114, 49, 115]. In
Sec. V, we investigate the effect of including the 4d IRs into a modified
action.

Another metric to consider is mixing between IRs of $SU(3)$ in the fundamental
IR of $\operatorname{\Sigma(108)}$. Since this IR is typically used in lattice
actions, if its mixing under induction represents systematic errors in
approximating $SU(3)$ by the group. Assuming dimensionality is a proxy for
energy along with the 3d $\\{1,0\\}$, the next largest contributors to
$\mathbf{3}^{(0)*}$ are the 15d $\\{2,1\\}$ and $\\{4,0\\}$ IRs.

Moving onto $\operatorname{\Sigma(216)}$, for $p+q\leq 5$ the inductions are

$\displaystyle\mathbf{1}^{(0)}$
$\displaystyle\rightarrow{\\{0,0\\}|_{1},\\{1,4\\}|_{1},\\{4,1\\}|_{1},\ldots}$
$\displaystyle\mathbf{1}^{(1)},\mathbf{1}^{(2)},\mathbf{1}^{(3)}$
$\displaystyle\rightarrow{\\{2,2\\}|_{1},\ldots}$
$\displaystyle\mathbf{2}^{(0)}$
$\displaystyle\rightarrow{\\{0,3\\}|_{1},\\{3,0\\}|_{1},\\{1,4\\}|_{1},\\{4,1\\}|_{1},\ldots}$
$\displaystyle\mathbf{3}^{(0)}$
$\displaystyle\rightarrow\\{0,1\\}|_{1},\\{0,4\\}|_{1},\\{3,1\\}|_{1},$
$\displaystyle\phantom{xxxxxxxxx}\\{2,3\\}|_{2},\\{5,0\\}|_{2},\ldots$
$\displaystyle\mathbf{3}^{(1)},\mathbf{3}^{(2)},\mathbf{3}^{(3)}$
$\displaystyle\rightarrow{\\{1,2\\}|_{1},\\{3,1\\}|_{1},\\{2,3\\}|_{2},\\{5,0\\}|_{1},\ldots}$
$\displaystyle\mathbf{6}^{(0)}$
$\displaystyle\rightarrow\\{2,0\\}|_{1},\\{1,2\\}|_{1},\\{0,4\\}|_{2},\\{3,1\\}|_{2},$
$\displaystyle\phantom{xxxxxxxxx}\\{2,3\\}|_{3},\\{5,0\\}|_{1},\ldots$
$\displaystyle\mathbf{8}^{(0)}$
$\displaystyle\rightarrow\\{1,1\\}|_{1},\\{0,3\\}|_{1},\\{3,0\\}|_{1},\\{2,2\\}|_{3},$
$\displaystyle\phantom{xxxxxxxxx}\\{1,4\\}|_{4},\\{4,1\\}|_{4},\ldots$

Given that $\operatorname{\Sigma(108)}$ is a subgroup of
$\operatorname{\Sigma(216)}$, it is interesting to compare their inductions.
First, we observe that the pairs of IRs have merged:
$\mathbf{1}^{(1)}\oplus\mathbf{1}^{(3)}\rightarrow\mathbf{2}$,
$\mathbf{3}^{(1)}\oplus\mathbf{3}^{(3)}\rightarrow\mathbf{6}^{(0)}$, and
$\mathbf{4}^{(0)}\oplus\mathbf{4}^{(0)^{\prime}}\rightarrow\mathbf{8}^{(0)}$.
We also see that triplets of
$\mathbf{1}^{(1)},\mathbf{1}^{(2)},\mathbf{1}^{(3)}$ and
$\mathbf{3}^{(1)},\mathbf{3}^{(2)},\mathbf{3}^{(3)}$ have arisen. Further, the
gaps between $SU(3)$ IRs that mix under induction have grown. For 3d
$\\{1,0\\}\rightarrow\mathbf{3}^{(0)*}$, the nearest IR is now a single 15d
one – $\\{4,0\\}$.

For $\operatorname{\Sigma(648)}$ which has $\operatorname{\Sigma(216)}$ as a
subgroup, the inductions up to $p+q=5$ are

$\displaystyle\mathbf{1}^{(0)}$
$\displaystyle\rightarrow{\\{0,0\\}|_{1}\ldots}$
$\displaystyle\mathbf{1}^{(1)}$
$\displaystyle\rightarrow{\\{4,1\\}|_{1},\ldots}$
$\displaystyle\mathbf{1}^{(2)}$
$\displaystyle\rightarrow{\\{1,4\\}|_{1},\ldots}$
$\displaystyle\mathbf{2}^{(0)}$
$\displaystyle\rightarrow{\\{1,4\\}|_{1},\\{4,1\\}|_{1},\ldots}$
$\displaystyle\mathbf{2}^{(1)}$
$\displaystyle\rightarrow{\\{0,3\\}|_{1}\ldots}$
$\displaystyle\mathbf{2}^{(2)}$
$\displaystyle\rightarrow{\\{3,0\\}|_{1}\ldots}$
$\displaystyle\mathbf{3}^{(a)}$
$\displaystyle\rightarrow{\\{2,2\\}|_{1},\ldots}$
$\displaystyle\mathbf{3}^{(0)}$
$\displaystyle\rightarrow{\\{0,1\\}|_{1},\\{2,3\\}|_{1},\ldots}$
$\displaystyle\mathbf{3}^{(1)}$
$\displaystyle\rightarrow{\\{3,1\\}|_{1},\\{5,0\\}|_{1}\ldots}$
$\displaystyle\mathbf{3}^{(2)}$
$\displaystyle\rightarrow{\\{0,4\\}|_{1},\\{2,3\\}|_{1},\\{5,0\\}|_{1}\ldots}$
$\displaystyle\mathbf{6}^{(0)}$
$\displaystyle\rightarrow{\\{2,0\\}|_{1},\\{3,1\\}|_{1},\\{2,3\\}|_{1}\ldots}$
$\displaystyle\mathbf{6}^{(1)}$
$\displaystyle\rightarrow\\{0,4\\}|_{1},\\{3,1\\}|_{1},\\{2,3\\}|_{1},\\{5,0\\}|_{1}\ldots$
$\displaystyle\mathbf{6}^{(2)}$
$\displaystyle\rightarrow{\\{1,2\\}|_{1},\\{0,4\\}|_{1},\\{2,3\\}|_{1}\ldots}$
$\displaystyle\mathbf{8}^{(0)}$
$\displaystyle\rightarrow{\\{1,1\\}|_{1},\\{2,2\\}|_{1},\\{1,4\\}|_{1},\\{4,1\\}|_{1},\\{3,3\\}|_{3},\ldots}$
$\displaystyle\mathbf{8}^{(1)}$
$\displaystyle\rightarrow\\{3,0\\}|_{1},\\{2,2\\}|_{1},\\{1,4\\}|_{1},\\{4,1\\}|_{2}\ldots$
$\displaystyle\mathbf{8}^{(2)}$
$\displaystyle\rightarrow\\{0,3\\}|_{1},\\{2,2\\}|_{1},\\{1,4\\}|_{2},\\{4,1\\}|_{1}\ldots$
$\displaystyle\mathbf{9}^{(0)}$
$\displaystyle\rightarrow\\{1,2\\}|_{1},\\{3,1\\}|_{1},\\{2,3\\}|_{2},\\{5,0\\}|_{1}\ldots$

Comparing to $\operatorname{\Sigma(216)}$, we observe that some IRs have
merged in $\operatorname{\Sigma(648)}$:
$\mathbf{1}^{(1)}\oplus\mathbf{1}^{(2)}\oplus\mathbf{1}^{(3)}\rightarrow\mathbf{3}^{(a)}$,
and
$\mathbf{3}^{(1)}\oplus\mathbf{3}^{(2)}\oplus\mathbf{3}^{(3)}\rightarrow\mathbf{9}^{(0)}$.
Unlike the $\operatorname{\Sigma(108)}$ case, some IRs of
$\operatorname{\Sigma(216)}$ subduce into new IRs of
$\operatorname{\Sigma(648)}$ :
$\mathbf{2}^{(0)}\rightarrow\mathbf{2}^{(0)}\oplus\mathbf{2}^{(1)}\oplus\mathbf{2}^{(2)}$,
$\mathbf{6}^{(0)}\rightarrow\mathbf{6}^{(0)}\oplus\mathbf{6}^{(1)}\oplus\mathbf{6}^{(2)}$,
$\mathbf{8}^{(0)}\rightarrow\mathbf{8}^{(0)}\oplus\mathbf{8}^{(1)}\oplus\mathbf{8}^{(2)}$.
The gap between the fundamental IR of $SU(3)$ and the next has grown, first
mixing with the 42-d $\\{3,2\\}$.

Finally for $\operatorname{\Sigma(1080)}$, we have up to $p+q=6$:

$\displaystyle\mathbf{1}^{(0)}$
$\displaystyle\rightarrow{\\{0,0\\}|_{1},\\{0,6\\}|_{1},\\{6,0\\}|_{1},\ldots}$
$\displaystyle\mathbf{3}^{(0)}$
$\displaystyle\rightarrow{\\{0,1\\}|_{1},\\{5,0\\}|_{1},\\{1,5\\}|_{1},\ldots}$
$\displaystyle\mathbf{3}^{(1)}$
$\displaystyle\rightarrow{\\{2,3\\}|_{1},\\{5,0\\}|_{1},\ldots}$
$\displaystyle\mathbf{5}^{(0)}$
$\displaystyle,\mathbf{5}^{(0)^{\prime}}\rightarrow{\\{2,2\\}|_{1},\\{0,6\\}|_{1},\\{3,3\\}|_{1},\\{6,0\\}|_{1},\ldots}$
$\displaystyle\mathbf{6}^{(0)}$
$\displaystyle\rightarrow{\\{0,2\\}|_{1},\\{4,0\\}|_{1},\\{2,4\\}|_{2},\\{5,1\\}|_{1},\ldots}$
$\displaystyle\mathbf{8}^{(0)}$
$\displaystyle\rightarrow\\{1,1\\}|_{1},\\{1,4\\}|_{1},\\{4,1\\}|_{1},$
$\displaystyle\phantom{xxxxxxxxx}\\{0,6\\}|_{1},\\{3,3\\}|_{1},\\{6,0\\}|_{1},\ldots$
$\displaystyle\mathbf{8}^{(0)^{\prime}}$
$\displaystyle\rightarrow{\\{2,2\\}|_{1},\\{1,4\\}|_{1},\\{4,1\\}|_{1},\\{3,3\\}|_{1},\ldots}$
$\displaystyle\mathbf{9}^{(0)}$
$\displaystyle\rightarrow\\{2,2\\}|_{1},\\{1,4\\}|_{1},\\{4,1\\}|_{1},$
$\displaystyle\phantom{xxxxxxxxx}\\{0,6\\}|_{1},\\{3,3\\}|_{2},\\{6,0\\}|_{1},\ldots$
$\displaystyle\mathbf{9}^{(1)}$
$\displaystyle\rightarrow{\\{1,3\\}|_{1},\\{4,0\\}|_{1},\\{3,2\\}|_{1},\\{2,4\\}|_{2},\\{5,1\\}|_{1},\ldots}$
$\displaystyle\mathbf{10}^{(0)}$
$\displaystyle\rightarrow{\\{0,3\\}|_{1},\\{3,0\\}|_{1},\\{1,4\\}|_{1},\\{4,1\\}|_{1},\\{3,3\\}|_{2},\ldots}$
$\displaystyle\mathbf{15}^{(0)}$
$\displaystyle\rightarrow\\{1,2\\}|_{1},\\{3,1\\}|_{1},\\{2,3\\}|_{2},\\{5,0\\}|_{1},$
$\displaystyle\phantom{xxxxxxxxx}\\{1,5\\}|_{2},\\{4,2\\}|_{2},\ldots$

Since $\operatorname{\Sigma(108)}$ is a subgroup of
$\operatorname{\Sigma(1080)}$, we can compare their inductions, and we observe
some of the IRs of $\operatorname{\Sigma(108)}$ have merged in
$\operatorname{\Sigma(1080)}$: $5\,\mathbf{1}^{(1)}\oplus
5\,\mathbf{1}^{(3)}\rightarrow\mathbf{10}^{(0)}$,
$\mathbf{4}^{(0)}\oplus\mathbf{4}^{(0)^{\prime}}\rightarrow\mathbf{8}^{(0)}$,
$5\,\mathbf{3}^{(2)}\rightarrow\mathbf{15}^{(0)}$,
$\mathbf{3}^{(1)*}\oplus\mathbf{3}^{(2)*}\rightarrow\mathbf{6}^{(0)}$. While
generically, the gaps between $SU(3)$ IRs that mix under induction have
increased and multiplicities decreased compared to the smaller
$\operatorname{\Sigma(648)}$, the fundamental 3d $\\{1,0\\}$ mixes with the
21-d $\\{0,5\\}$ rather than the much larger 42-d $\\{3,2\\}$. The physical
consequences of the difference should be investigated.

##  V Numerical Results

To study the extent to which discrete subgroups can approximate continuous
groups, one can compare expectation values of observables weighted by a
lattice action. In general, single-plaquette lattice actions can be
constructed from the characters of IRs $\rho$,

$S_{\rho}=-\sum_{p,\rho}\beta_{\rho}\operatorname{Re}\chi^{(\rho)}(U_{p}),$
(5)

where $U_{p}$ is a plaquette built from group-valued links. The commonly-used
Wilson action for $SU(3)$ depends only on the fundamental 3d IR:

$S_{W}(\beta)=-\frac{\beta_{\mathbf{3}^{(0)}}}{6}\sum_{p}\operatorname{Re}\chi^{\mathbf{3}^{(0)}}(U_{p}),$
(6)

while actions with additional IRs are referred to as _modified actions_. For
continuous groups which are asymptotically free, there are typically two
”phases” for a lattice action: a lattice phase for which lattice artifacts
dominate separated at $\beta_{s}$ (which in a modified action is a plane of
couplings) from a _scaling regime_ where these effects are sufficiently small
that extrapolation to the continuum can be performed. For finite subgroups,
there is an additional Higgs or so-called frozen phase at some subgroup-
dependent $\beta_{f}$ where the discrete subgroup approximation breaks down.
Thus, for a lattice action of a discrete subgroup to be useful for simulations
$\beta_{s}<\beta_{f}$, i.e., it must support a sufficiently large scaling
regime. From the subductions, we gain a different perspective on how these
phases can arise from mixing eigenstates of $SU(3)$ and gain insight into
which IRs of the discrete subgroup to add to the action to attempt to improve
the agreement with the continuous group.

Using modified actions has been demonstrated to introduce or enlarge the
scaling regime of discrete subgroups [86, 116, 25, 117, 49, 115]. The most
commonly considered modified action adds the 8d adjoint IR to the Wilson
action [118, 111, 112, 114], and has been demonstrated to reduced the lattice
artifacts if the couplings are appropriately chosen.

One useful observable is the expectation value of the plaquette, which
corresponds to the average energy density,

$E(\beta)=\left\langle
1-\frac{1}{6}\operatorname{Re}\chi^{\mathbf{3}^{(0)}}(U_{p})\right\rangle.$
(7)

Comparing to the continuous group $SU(3)$, $E(\beta)$ with the Wilson action
is known to agree analytically up to $\mathcal{O}(\beta^{4})$ for
$\Sigma(1080)$ and $\operatorname{\Sigma(648)}$, while only
$\mathcal{O}(\beta^{2})$ for the smaller groups $\Sigma(648)$ and
$\Sigma(216)$, respectively [84]. If instead of the Wilson action, an
appropriately chosen modified action could be used to improve this strong-
coupling expansion – smoothing left for future work. Instead, we will consider
the nonperturbative calculation of $E(\beta)$ from Monte Carlo simulations.

Motivated by prior experience that a modified action with the adjoint
representation can lead to better agreement with the continuous group, the
subgroup $\operatorname{\Sigma(108)}$, which does not have a one-to-one
subduction of the adjoint is particularly interesting. In Fig. 1, we plot
$E(\beta)$ as a function of $\beta_{\mathbf{3}^{(0)}}$ for different lattice
actions. At any given $\beta_{\mathbf{3}^{(0)}}$, the lattice spacing $a$ for
different modified actions vary, but the smallness of $E(\beta)$ can be used
as a proxy for the discrete subgroup having a larger $\beta_{f}$. Further, for
$SU(3)$, $E(\beta)\sim 0.6$ corresponds to the scaling regime, so a good
discrete subgroup approximation works to get at least that far. The freezing
transition can be recognized by a rapid drop of $E(\beta_{f})\rightarrow 0$.
In the thermodynamic limit, this will turn into a finite gap at
$E(\beta_{f})$.

For $\operatorname{\Sigma(108)}$ with the Wilson action,
$\beta_{\mathbf{4}^{(0)}}=\beta_{\mathbf{4}^{(0)^{\prime}}}=0$ we observed the
freezing transition occurs when $E(\beta_{f})\approx 0.8$. If we wanted to
follow the wisdom of adding something like adjoint representation, we have a
choice as to how. We investigated the relative improvement from adding only
$\mathbf{4}^{(0)}$ or $\mathbf{4}^{(0)^{\prime}}$ or both of them with equal
coupling along coupling trajectories:

$\displaystyle\beta_{\mathbf{4}^{(0)}}$
$\displaystyle=-\kappa\beta_{\mathbf{3}^{(0)}},\qquad\beta_{\mathbf{4}^{(0)^{\prime}}}=0$
(8) $\displaystyle\beta_{\mathbf{4}^{(0)}}$
$\displaystyle=\beta_{\mathbf{4}^{(0)^{\prime}}}=-\kappa\beta_{\mathbf{3}^{(0)}}$
(9)

for $\kappa=-\frac{5}{32},-\frac{5}{16}$. We see in Fig. 1 that for both
$\kappa$ including both IRs leads to a smaller value of $E(\beta_{f})$. It is
particularly striking for $\kappa=-\frac{5}{16}$ where including both IRs
$E(\beta_{f})$ never drops to zero. The mechanism for this is related to how
$\mathbf{4}^{(0)}$ and $\mathbf{4}^{(0)^{\prime}}$ give different weights to
different $\operatorname{\Sigma(108)}$ group elements.

Figure 1: Euclidean calculations of lattice energy density $\langle
E_{0}\rangle$ of $\operatorname{\Sigma(108)}$ as measured by the expectation
value of the plaquette for different modified actions as a function of
fundamental IR coupling $\beta_{\mathbf{3}^{(0)}}$ on $2^{4}$ lattices

A deeper investigation of the effects of subduction can be undertaken by
looking at the static potentials. Given that $\operatorname{\Sigma(1080)}$ is
the best discrete approximation to $SU(3)$, we will focus on this subgroup.
The lattice static potential $\hat{V}_{\rho}$ in a representation is defined
with respect to the expectation value of the rectangular Wilson loop in the
same representation:

$\langle
W_{\rho}(r,t)\rangle=c_{\rho}(r,a)\text{exp}\left[-\hat{V}_{\rho}(r,a)t\right]\qquad
T\rightarrow\infty.$ (10)

where $r,t$ are the spatial and temporal extent of the Wilson loop. The
prefactor, $c_{\rho}(r,a)$, is an additional fitting parameter that captures
aspects of the Wilson loop’s behavior at small $r$ that are not related to the
linear rise of the potential with $r$; it is extracted when analyzing Wilson
loop data to extract the potential.

The static potentials in pure gauge $SU(3)$ theory are expected to following
_Casimir scaling_ where they are equal for large $r$ up to a rescaling by
their quadratic Casimirs [119, 120]

$C_{2}(p,q)=\frac{p^{2}+q^{2}+3p+3q+pq}{3}.$ (11)

Numerical results at both finite lattice spacing and the continuum have
confirmed this scaling up to violations smaller than $5\%$. Here, we compute
the static potentials for all unique IRs of $\operatorname{\Sigma(1080)}$ and
compare their numerical ratios to
$R_{C}=\hat{V}_{\rho}(r/a)/\hat{V}_{\mathbf{3}^{(0)}}(r/a)$ to
$C_{2}(p,q)/C_{2}(1,0)$ of $SU(3)$. Given our subduction table, we can check
to what extent subducing to a direct sum of IRs differs from subducing to a
single IR and how the different IRs of the direct sum behave.

In order to reach the scaling regime, we simulated
$\operatorname{\Sigma(1080)}$ using a modified action that includes both the
fundamental and adjoint IRs

$S_{fa}=-\sum_{p}\left(\beta_{\textbf{3}^{(0)}}\operatorname{Re}\chi^{\textbf{3}^{(0)}}(U_{p})+\beta_{\textbf{8}^{(0)}}\operatorname{Re}\chi^{\textbf{8}^{(0)}}(U_{p})\right)\,,$
(12)

With this action, we used $\beta_{0}=8.407$ and $\beta_{1}=-1.65$ on a
$32^{4}$ lattice which corresponds to $a=0.085(4)$ fm as determined by using
the static potential $\hat{V}_{\mathbf{3}^{(0)}}(r/a)$ with the Sommer
parameter $r_{0}\sim 0.49$ fm defined by [121]

$r^{2}\frac{\partial V}{\partial r}|_{r=r_{0}}=1.65$ (13)

The extrapolation of $a(N_{t})$ in the temporal extent of the loop
$N_{t}^{-1}$ to zero is presented in Fig. 2.

Figure 2: Lattice spacing extracted from the static potential as a function of
the temporal extent of the loop $N_{t}^{-1}$. The $N_{t}^{-1}\rightarrow 0$
limit is obtained from a quadratic fit, finding $a=0.085(4)$ fm.

At this lattice spacing, we computed the unique IR static potentials. Because
of the nuances involved in adequately accounting for the systematics of
smearing discrete subgroups, we have opted not to smear. As a consequence, the
signal-to-noise prevents large $r\times t$ from being probed. Luckily, we
observe that similar to prior studies in $SU(3)$ [119, 120], $R_{C}$ appears
to plateau rapidly. The results for temporal extents $t=1,2,3$ are presented
in Fig. 3 for our ensemble of 32k configurations.

Figure 3: The ratio of static potentials of IRs of
$\operatorname{\Sigma(1080)}$ to the fundamental $\mathbf{3}^{(0)}$ IR,
labelled $R_{C}$, for all measured representations. The colored bands
correspond to the $SU(3)$ results for all $R_{C}<10$.

For the lowest IR, clear signals were obtained for $t/a$ up to 7, and in Tab.
10, we present $R_{C}(t)$ used to extract infinite limit results. For IRs
where $R_{C}(t>3)$ were obtained, we extrapolated to the $t\rightarrow\infty$
with a quadratic fit in $t$. For other IRs where only 2 or 3 values were
obtained, for our estimate of $R_{C}(\infty)$, we take the average and include
an additional systematic error, which is the difference between the highest
and lowest values.

Table 10: Ratios of static potentials of IRs of $\operatorname{\Sigma(1080)}$ to the $\mathbf{3}^{(0)}$ IR, denoted $R_{C}(\tau)$, where $\tau=t/a$. $\rho$ | $R_{C}(1)$ | $R_{C}(2)$ | $R_{C}(3)$ | $R_{C}(4)$ | $R_{C}(5)$ | $R_{C}(6)$ | $R_{C}(7)$ | $R_{C}(\infty)$
---|---|---|---|---|---|---|---|---
$\mathbf{8}^{(0)}$ | 2.2478(10) | 2.2192(14) | 2.2068(28) | 2.205(8) | 2.157(16) | 2.15(7) | 2.01(15) | 2.186(14)
$\mathbf{6}^{(0)*}$ | 2.5045(12) | 2.4856(16) | 2.473(4) | 2.478(11) | 2.48(5) | — | — | 2.463(26)
$\mathbf{15}^{(0)*}$ | 3.978(10) | 4.004(12) | 3.97(9) | — | — | — | — | 3.984(30)(30)
$\mathbf{10}^{(0)*}$ | 4.1522(25) | 4.457(16) | — | — | — | — | — | 4.305(8)(305)
$\mathbf{5}^{(0)}$ | 4.85(5) | 5.84(31) | 5.163(33) | — | — | — | — | 5.28(11)(99)
$\mathbf{8}^{(0)^{\prime}}$ | 5.82(7) | 6.44(34) | — | — | — | — | — | 6.13(17)(62)
$\mathbf{9}^{(0)}$ | 5.26(9) | 5.80(22) | — | — | — | — | — | 5.53(12)(54)
$\mathbf{9}^{(1)}$ | 5.89(17) | 6.3(4) | — | — | — | — | — | 6.09(21)(40)
$\mathbf{3}^{(1)*}$ | $\lesssim 8.6$ | $\lesssim 9$ | — | — | — | — | — | $\lesssim 9$

The results for $\operatorname{\Sigma(1080)}$ can then be compared to those of
the theoretical expectations of $SU(3)=C_{2}(p,q)/C_{2}(1,0)$. Numerical
results for $SU(3)$ in [119, 120] have bounded the violation of this scaling
to be less than $5\%$, which we take a metric for agreement from the discrete
group approximation. The comparison is shown in Tab. 11, where we include the
fractional deviation $\delta R$ between the expected scaling and
$\operatorname{\Sigma(1080)}$. We observe that for the four IRs of
$\operatorname{\Sigma(1080)}$, which are subduced one-to-one from $SU(3)$ is
$\delta R<5\%$ with two consistent with zero. In the $\\{2,2\\}$ case, which
subduces into four new IRs, we see mild tension with the $5\%$ violations for
the $\mathbb{5}$ IRs, while the $\mathbb{8}^{(0)^{\prime}}$ and
$\mathbb{9}^{(0)}$ are found to be in agreement. At still higher IRs, we see
$\\{3,1\\}$ and $\\{3,2\\}$ subduce to direct sums of IRs that include ones
for the first time, e.g. $\mathbb{9}^{(1)*}$ and $\mathbb{3}^{(1)*}$,
alongside ones that are subduced into lower IRs. In both cases, we observe
that the new IRs are consistent with the anticipated scaling of the $SU(3)$
IR. Together, these results are suggestive of how the eigenstates of $SU(3)$
break into the IRs of $\operatorname{\Sigma(1080)}$ – and that for the lowest
IRs, good agreement between observables should be anticipated.

Table 11: Comparison of the Casimir scaling of the static potentials of $SU(3)$ with to their subducted IRs in $\operatorname{\Sigma(1080)}$. In the last column, we present the fraction deviation $\delta R$. $SU(3)$ | $\operatorname{\Sigma(1080)}$ | $R_{C}^{SU(3)}$ | $R_{C}^{\operatorname{\Sigma(1080)}}$ | $\delta R$
---|---|---|---|---
{1,1} | $\textbf{8}^{(0)}$ | 2.25 | $2.186(14)$ | 0.028(6)
{2,0} | $\textbf{6}^{(0)*}$ | 2.5 | $2.46(3)$ | 0.016(12)
{2,1} | $\textbf{15}^{(0)*}$ | 4 | $3.98(3)(3)$ | 0.005(8)(8)
{3,0} | $\textbf{10}^{(0)}$ | 4.5 | $4.305(8)(305)$ | 0.0433(18)(700)
{2,2} | $\textbf{5}^{(0)}$ | 6 | $5.28(11)(99)$ | 0.120(18)(70)
| $\textbf{5}^{(0)^{\prime}}$ |  | $5.28(11)(99)$ | 0.120(18)(70)
| $\textbf{8}^{(0)^{\prime}}$ |  | $6.13(17)(62)$ | 0.022(28)(100)
| $\textbf{9}^{(0)}$ |  | $5.53(12)(54)$ | 0.078(20)(90)
{3,1} | $\textbf{9}^{(1)*}$ | 6.25 | $6.09(21)(40)$ | 0.026(34)(60)
| $\textbf{15}^{(0)}$ |  |  $3.98(3)(3)$333 IR present in lower subduction | 0.363(5)(5)
{4,0} | $\textbf{6}^{(0)}$ | 7 |  $2.46(3)$3 | 0.649(4)
| $\textbf{9}^{(1)}$ |  |  $6.09(21)(40)$3 | 0.130(30)(60)
{3,2} | $\mathbf{3}^{(1)*}$ | 8.5 | $\lesssim 9$ | $\lesssim 0.06$
| $\mathbf{9}^{(1)}$ |  |  $6.09(21)(40)$3 | 0.284(25)(50)
| $\mathbf{15}^{(0)*}$ |  |  $3.98(3)(3)$3 | 0.5318(35)(35)

##  VI Conclusion

We provided the subduction of $SU(N)$ groups of interest in lattice gauge
theory to their crystal-like subgroups up to a high dimension. The subduction
tables provide new insight into systematically understanding the limitations
of the discrete subgroup approximation and give guidance on how to improve the
approximation. It was demonstrated how the subduction patterns can be used to
determine effective modified actions and how violations of Casimir scaling are
signals for the discrete subgroup breakdown.

For future work, examining how the subduction of continuous groups to their
discrete subgroups interacts with the Higgs mechanism could provide valuable
insights into symmetry-breaking processes in lattice gauge theories. Further,
developing a systematic understanding of the errors in gauge-smearing
techniques optimized for discrete subgroups could help reduce signal-to-noise
to probe the scaling breakdown more precisely. Additionally, a dedicated
analysis of the excited state glueball spectrum within these approximations
could provide deeper insights into the stability and spectrum of gauge
theories.

###### Acknowledgements.

HL is supported by the Department of Energy through the Fermilab QuantiSED
program in the area of “Intersections of QIS and Theoretical Particle
Physics”. This work was supported by the Fermi National Accelerator Laboratory
(Fermilab), a U.S. Department of Energy, Office of Science, HEP User Facility.
Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under
Contract No. DE–AC02–07CH11359.

## References

  * Aarts [2016] G. Aarts, Introductory lectures on lattice qcd at nonzero baryon number, Journal of Physics: Conference Series 706, 022004 (2016).
  * Philipsen [2007] O. Philipsen, Lattice qcd at finite temperature and density, The European Physical Journal Special Topics 152, 29 (2007).
  * de Forcrand [2010] P. de Forcrand, Simulating qcd at finite density (2010), arXiv:1005.0539 [hep-lat] .
  * Muroya  _et al._ [2003] S. Muroya, A. Nakamura, C. Nonaka, and T. Takaishi, Lattice QCD at Finite Density*): An Introductory Review, Progress of Theoretical Physics 110, 615 (2003), https://academic.oup.com/ptp/article-pdf/110/4/615/5301553/110-4-615.pdf .
  * Gattringer and Langfeld [2016] C. Gattringer and K. Langfeld, Approaches to the sign problem in lattice field theory, Int. J. Mod. Phys. A 31, 1643007 (2016), arXiv:1603.09517 [hep-lat] .
  * Alexandru  _et al._ [2018] A. Alexandru, P. F. Bedaque, H. Lamm, S. Lawrence, and N. C. Warrington, Fermions at Finite Density in 2+1 Dimensions with Sign-Optimized Manifolds, Phys. Rev. Lett. 121, 191602 (2018), arXiv:1808.09799 [hep-lat] .
  * Alexandru  _et al._ [2022] A. Alexandru, G. m. c. Başar, P. F. Bedaque, and N. C. Warrington, Complex paths around the sign problem, Rev. Mod. Phys. 94, 015006 (2022).
  * Troyer and Wiese [2005] M. Troyer and U.-J. Wiese, Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations, Phys. Rev. Lett. 94, 170201 (2005), arXiv:cond-mat/0408370 [cond-mat] .
  * Creutz  _et al._ [1979] M. Creutz, L. Jacobs, and C. Rebbi, Monte Carlo Study of Abelian Lattice Gauge Theories, Phys. Rev. D20, 1915 (1979).
  * Kogut [1979] J. B. Kogut, An Introduction to Lattice Gauge Theory and Spin Systems, Rev. Mod. Phys. 51, 659 (1979).
  * Lisboa and Michael [1982a] P. Lisboa and C. Michael, Lattices in Group Manifolds: Applications to Lattice Gauge Theory, Nucl. Phys. B210, 15 (1982a).
  * Lisboa and Michael [1982b] P. Lisboa and C. Michael, Discrete Subsets of SU(3) for Lattice Gauge Theory, Phys. Lett. B 113, 303 (1982b).
  * Giusti  _et al._ [2001] L. Giusti, M. L. Paciello, C. Parrinello, S. Petrarca, and B. Taglienti, Problems on lattice gauge fixing, Int. J. Mod. Phys. A 16, 3487 (2001), arXiv:hep-lat/0104012 .
  * Zohar  _et al._ [2012] E. Zohar, J. I. Cirac, and B. Reznik, Simulating Compact Quantum Electrodynamics with ultracold atoms: Probing confinement and nonperturbative effects, Phys. Rev. Lett. 109, 125302 (2012), arXiv:1204.6574 [quant-ph] .
  * Zohar  _et al._ [2013a] E. Zohar, J. I. Cirac, and B. Reznik, Cold-Atom Quantum Simulator for SU(2) Yang-Mills Lattice Gauge Theory, Phys. Rev. Lett. 110, 125304 (2013a), arXiv:1211.2241 [quant-ph] .
  * Zohar  _et al._ [2013b] E. Zohar, J. I. Cirac, and B. Reznik, Quantum simulations of gauge theories with ultracold atoms: local gauge invariance from angular momentum conservation, Phys. Rev. A88, 023617 (2013b), arXiv:1303.5040 [quant-ph] .
  * Zohar and Burrello [2015] E. Zohar and M. Burrello, Formulation of lattice gauge theories for quantum simulations, Phys. Rev. D91, 054506 (2015), arXiv:1409.3085 [quant-ph] .
  * Zohar  _et al._ [2016] E. Zohar, J. I. Cirac, and B. Reznik, Quantum Simulations of Lattice Gauge Theories using Ultracold Atoms in Optical Lattices, Rept. Prog. Phys. 79, 014401 (2016), arXiv:1503.02312 [quant-ph] .
  * Zohar  _et al._ [2017] E. Zohar, A. Farace, B. Reznik, and J. I. Cirac, Digital lattice gauge theories, Phys. Rev. A95, 023604 (2017), arXiv:1607.08121 [quant-ph] .
  * Klco  _et al._ [2020] N. Klco, J. R. Stryker, and M. J. Savage, SU(2) non-Abelian gauge field theory in one dimension on digital quantum computers, Phys. Rev. D 101, 074512 (2020), arXiv:1908.06935 [quant-ph] .
  * Ciavarella  _et al._ [2021] A. Ciavarella, N. Klco, and M. J. Savage, A Trailhead for Quantum Simulation of SU(3) Yang-Mills Lattice Gauge Theory in the Local Multiplet Basis (2021), arXiv:2101.10227 [quant-ph] .
  * Bender  _et al._ [2018] J. Bender, E. Zohar, A. Farace, and J. I. Cirac, Digital quantum simulation of lattice gauge theories in three spatial dimensions, New J. Phys. 20, 093001 (2018), arXiv:1804.02082 [quant-ph] .
  * Liu and Xin [2020] J. Liu and Y. Xin, Quantum simulation of quantum field theories as quantum chemistry (2020), arXiv:2004.13234 [hep-th] .
  * Hackett  _et al._ [2019] D. C. Hackett, K. Howe, C. Hughes, W. Jay, E. T. Neil, and J. N. Simone, Digitizing Gauge Fields: Lattice Monte Carlo Results for Future Quantum Computers, Phys. Rev. A 99, 062341 (2019), arXiv:1811.03629 [quant-ph] .
  * Alexandru  _et al._ [2019] A. Alexandru, P. F. Bedaque, S. Harmalkar, H. Lamm, S. Lawrence, and N. C. Warrington (NuQS), Gluon field digitization for quantum computers, Phys.Rev.D 100, 114501 (2019), arXiv:1906.11213 [hep-lat] .
  * Yamamoto [2021] A. Yamamoto, Real-time simulation of (2+1)-dimensional lattice gauge theory on qubits, PTEP 2021, 013B06 (2021), arXiv:2008.11395 [hep-lat] .
  * Haase  _et al._ [2021] J. F. Haase, L. Dellantonio, A. Celi, D. Paulson, A. Kan, K. Jansen, and C. A. Muschik, A resource efficient approach for quantum and classical simulations of gauge theories in particle physics, Quantum 5, 393 (2021), arXiv:2006.14160 [quant-ph] .
  * Armon  _et al._ [2021] T. Armon, S. Ashkenazi, G. García-Moreno, A. González-Tudela, and E. Zohar, Photon-mediated Stroboscopic Quantum Simulation of a $\mathbb{Z}_{2}$ Lattice Gauge Theory (2021), arXiv:2107.13024 [quant-ph] .
  * Bazavov  _et al._ [2019] A. Bazavov, S. Catterall, R. G. Jha, and J. Unmuth-Yockey, Tensor renormalization group study of the non-abelian higgs model in two dimensions, Phys. Rev. D 99, 114507 (2019).
  * Bazavov  _et al._ [2015] A. Bazavov, Y. Meurice, S.-W. Tsai, J. Unmuth-Yockey, and J. Zhang, Gauge-invariant implementation of the Abelian Higgs model on optical lattices, Phys. Rev. D92, 076003 (2015), arXiv:1503.08354 [hep-lat] .
  * Zhang  _et al._ [2018] J. Zhang, J. Unmuth-Yockey, J. Zeiher, A. Bazavov, S. W. Tsai, and Y. Meurice, Quantum simulation of the universal features of the Polyakov loop, Phys. Rev. Lett. 121, 223201 (2018), arXiv:1803.11166 [hep-lat] .
  * Unmuth-Yockey  _et al._ [2018] J. Unmuth-Yockey, J. Zhang, A. Bazavov, Y. Meurice, and S.-W. Tsai, Universal features of the Abelian Polyakov loop in 1+1 dimensions, Phys. Rev. D98, 094511 (2018), arXiv:1807.09186 [hep-lat] .
  * Unmuth-Yockey [2019] J. F. Unmuth-Yockey, Gauge-invariant rotor Hamiltonian from dual variables of 3D $U(1)$ gauge theory, Phys. Rev. D 99, 074502 (2019), arXiv:1811.05884 [hep-lat] .
  * Kreshchuk  _et al._ [2020a] M. Kreshchuk, W. M. Kirby, G. Goldstein, H. Beauchemin, and P. J. Love, Quantum Simulation of Quantum Field Theory in the Light-Front Formulation (2020a), arXiv:2002.04016 [quant-ph] .
  * Kreshchuk  _et al._ [2020b] M. Kreshchuk, S. Jia, W. M. Kirby, G. Goldstein, J. P. Vary, and P. J. Love, Simulating Hadronic Physics on NISQ devices using Basis Light-Front Quantization (2020b), arXiv:2011.13443 [quant-ph] .
  * Raychowdhury and Stryker [2018] I. Raychowdhury and J. R. Stryker, Solving Gauss’s Law on Digital Quantum Computers with Loop-String-Hadron Digitization (2018), arXiv:1812.07554 [hep-lat] .
  * Raychowdhury and Stryker [2020] I. Raychowdhury and J. R. Stryker, Loop, String, and Hadron Dynamics in SU(2) Hamiltonian Lattice Gauge Theories, Phys. Rev. D 101, 114502 (2020), arXiv:1912.06133 [hep-lat] .
  * Davoudi  _et al._ [2020] Z. Davoudi, I. Raychowdhury, and A. Shaw, Search for Efficient Formulations for Hamiltonian Simulation of non-Abelian Lattice Gauge Theories (2020), arXiv:2009.11802 [hep-lat] .
  * Wiese [2014] U.-J. Wiese, Towards Quantum Simulating QCD,  _Proceedings, 24th International Conference on Ultra-Relativistic Nucleus-Nucleus Collisions (Quark Matter 2014): Darmstadt, Germany, May 19-24, 2014_ , Nucl. Phys. A931, 246 (2014), arXiv:1409.7414 [hep-th] .
  * Luo  _et al._ [2019] D. Luo, J. Shen, M. Highman, B. K. Clark, B. DeMarco, A. X. El-Khadra, and B. Gadway, A Framework for Simulating Gauge Theories with Dipolar Spin Systems (2019), arXiv:1912.11488 [quant-ph] .
  * Brower  _et al._ [2019] R. C. Brower, D. Berenstein, and H. Kawai, Lattice Gauge Theory for a Quantum Computer, PoS LATTICE2019, 112 (2019), arXiv:2002.10028 [hep-lat] .
  * Mathis  _et al._ [2020] S. V. Mathis, G. Mazzola, and I. Tavernelli, Toward scalable simulations of Lattice Gauge Theories on quantum computers, Phys. Rev. D 102, 094501 (2020), arXiv:2005.10271 [quant-ph] .
  * Singh [2019] H. Singh, Qubit $O(N)$ nonlinear sigma models (2019), arXiv:1911.12353 [hep-lat] .
  * Singh and Chandrasekharan [2019] H. Singh and S. Chandrasekharan, Qubit regularization of the $O(3)$ sigma model, Phys. Rev. D 100, 054505 (2019), arXiv:1905.13204 [hep-lat] .
  * Buser  _et al._ [2020] A. J. Buser, T. Bhattacharya, L. Cincio, and R. Gupta, Quantum simulation of the qubit-regularized O(3)-sigma model (2020), arXiv:2006.15746 [quant-ph] .
  * Bhattacharya  _et al._ [2020] T. Bhattacharya, A. J. Buser, S. Chandrasekharan, R. Gupta, and H. Singh, Qubit regularization of asymptotic freedom (2020), arXiv:2012.02153 [hep-lat] .
  * Barata  _et al._ [2020] J. a. Barata, N. Mueller, A. Tarasov, and R. Venugopalan, Single-particle digitization strategy for quantum computation of a $\phi^{4}$ scalar field theory (2020), arXiv:2012.00020 [hep-th] .
  * Kreshchuk  _et al._ [2020c] M. Kreshchuk, S. Jia, W. M. Kirby, G. Goldstein, J. P. Vary, and P. J. Love, Light-Front Field Theory on Current Quantum Computers (2020c), arXiv:2009.07885 [quant-ph] .
  * Ji  _et al._ [2020] Y. Ji, H. Lamm, and S. Zhu (NuQS), Gluon Field Digitization via Group Space Decimation for Quantum Computers, Phys. Rev. D 102, 114513 (2020), arXiv:2005.14221 [hep-lat] .
  * Bauer and Grabowska [2021] C. W. Bauer and D. M. Grabowska, Efficient Representation for Simulating U(1) Gauge Theories on Digital Quantum Computers at All Values of the Coupling (2021), arXiv:2111.08015 [hep-ph] .
  * Gustafson [2021] E. Gustafson, Prospects for Simulating a Qudit Based Model of (1+1)d Scalar QED, Phys. Rev. D 103, 114505 (2021), arXiv:2104.10136 [quant-ph] .
  * Halimeh  _et al._ [2021] J. C. Halimeh, H. Lang, and P. Hauke, Gauge protection in non-Abelian lattice gauge theories (2021), arXiv:2106.09032 [cond-mat.quant-gas] .
  * Hartung  _et al._ [2022] T. Hartung, T. Jakobs, K. Jansen, J. Ostmeyer, and C. Urbach, Digitising SU(2) gauge fields and the freezing transition, Eur. Phys. J. C 82, 237 (2022), arXiv:2201.09625 [hep-lat] .
  * Osborne  _et al._ [2022] J. Osborne, I. P. McCulloch, B. Yang, P. Hauke, and J. C. Halimeh, Large-Scale $2+1$D $\mathrm{U}(1)$ Gauge Theory with Dynamical Matter in a Cold-Atom Quantum Simulator (2022), arXiv:2211.01380 [cond-mat.quant-gas] .
  * Grabowska  _et al._ [2022] D. M. Grabowska, C. Kane, B. Nachman, and C. W. Bauer, Overcoming exponential scaling with system size in Trotter-Suzuki implementations of constrained Hamiltonians: 2+1 U(1) lattice gauge theories (2022), arXiv:2208.03333 [quant-ph] .
  * Murairi  _et al._ [2022] E. M. Murairi, M. J. Cervia, H. Kumar, P. F. Bedaque, and A. Alexandru, How many quantum gates do gauge theories require? (2022), arXiv:2208.11789 [hep-lat] .
  * Bauer  _et al._ [2023] C. W. Bauer, I. D’Andrea, M. Freytsis, and D. M. Grabowska, A new basis for Hamiltonian SU(2) simulations (2023), arXiv:2307.11829 [hep-ph] .
  * Gustafson  _et al._ [2024a] E. J. Gustafson, H. Lamm, and F. Lovelace, Primitive quantum gates for an SU(2) discrete subgroup: Binary octahedral, Phys. Rev. D 109, 054503 (2024a), arXiv:2312.10285 [hep-lat] .
  * Alexandru  _et al._ [2023] A. Alexandru, P. F. Bedaque, A. Carosso, M. J. Cervia, E. M. Murairi, and A. Sheng, Fuzzy Gauge Theory for Quantum Computers (2023), arXiv:2308.05253 [hep-lat] .
  * Halimeh  _et al._ [2023] J. C. Halimeh, L. Homeier, A. Bohrdt, and F. Grusdt, Spin exchange-enabled quantum simulator for large-scale non-Abelian gauge theories (2023), arXiv:2305.06373 [cond-mat.quant-gas] .
  * Fromm  _et al._ [2024a] M. Fromm, O. Philipsen, W. Unger, and C. Winterowd, Quantum gate sets for lattice QCD in the strong-coupling limit: N_f=1, EPJ Quant. Technol. 11, 24 (2024a), arXiv:2308.03196 [hep-lat] .
  * Fromm  _et al._ [2024b] M. Fromm, O. Philipsen, M. Spannowsky, and C. Winterowd, Simulating Z_2 lattice gauge theory with the variational quantum thermalizer, EPJ Quant. Technol. 11, 20 (2024b), arXiv:2306.06057 [hep-lat] .
  * Ciavarella and Bauer [2024] A. N. Ciavarella and C. W. Bauer, Quantum Simulation of SU(3) Lattice Yang Mills Theory at Leading Order in Large N (2024), arXiv:2402.10265 [hep-ph] .
  * Bergner  _et al._ [2024] G. Bergner, M. Hanada, E. Rinaldi, and A. Schafer, Toward QCD on Quantum Computer: Orbifold Lattice Approach (2024), arXiv:2401.12045 [hep-th] .
  * Zache  _et al._ [2023] T. V. Zache, D. González-Cuadra, and P. Zoller, Quantum and Classical Spin-Network Algorithms for q-Deformed Kogut-Susskind Gauge Theories, Phys. Rev. Lett. 131, 171902 (2023), arXiv:2304.02527 [quant-ph] .
  * Hasenfratz and Niedermayer [2001a] P. Hasenfratz and F. Niedermayer, Asymptotic freedom with discrete spin variables?,  _Proceedings, 2001 Europhysics Conference on High Energy Physics (EPS-HEP 2001): Budapest, Hungary, July 12-18, 2001_ , PoS HEP2001, 229 (2001a), arXiv:hep-lat/0112003 [hep-lat] .
  * Caracciolo  _et al._ [2001a] S. Caracciolo, A. Montanari, and A. Pelissetto, Asymptotically free models and discrete nonAbelian groups, Phys. Lett. B513, 223 (2001a), arXiv:hep-lat/0103017 [hep-lat] .
  * Hasenfratz and Niedermayer [2001b] P. Hasenfratz and F. Niedermayer, Asymptotically free theories based on discrete subgroups,  _Lattice field theory. Proceedings, 18th International Symposium, Lattice 2000, Bangalore, India, August 17-22, 2000_ , Nucl. Phys. Proc. Suppl. 94, 575 (2001b), arXiv:hep-lat/0011056 [hep-lat] .
  * Patrascioiu and Seiler [1998] A. Patrascioiu and E. Seiler, Continuum limit of two-dimensional spin models with continuous symmetry and conformal quantum field theory, Phys. Rev. E 57, 111 (1998).
  * Krcmar  _et al._ [2016] R. Krcmar, A. Gendiar, and T. Nishino, Phase diagram of a truncated tetrahedral model, Phys. Rev. E 94, 022134 (2016).
  * Caracciolo  _et al._ [2001b] S. Caracciolo, A. Montanari, and A. Pelissetto, Asymptotically free models and discrete non-abelian groups, Physics Letters B 513, 223 (2001b).
  * Zhou  _et al._ [2022] J. Zhou, H. Singh, T. Bhattacharya, S. Chandrasekharan, and R. Gupta, Spacetime symmetric qubit regularization of the asymptotically free two-dimensional O(4) model, Phys. Rev. D 105, 054510 (2022), arXiv:2111.13780 [hep-lat] .
  * Caspar and Singh [2022] S. Caspar and H. Singh, From asymptotic freedom to $\theta$ vacua: Qubit embeddings of the O(3) nonlinear $\sigma$ model (2022), arXiv:2203.15766 [hep-lat] .
  * Jakobs  _et al._ [2023] T. Jakobs, M. Garofalo, T. Hartung, K. Jansen, J. Ostmeyer, D. Rolfes, S. Romiti, and C. Urbach, Canonical momenta in digitized Su(2) lattice gauge theory: definition and free theory, Eur. Phys. J. C 83, 669 (2023), arXiv:2304.02322 [hep-lat] .
  * Zohar [2021] E. Zohar, Quantum Simulation of Lattice Gauge Theories in more than One Space Dimension – Requirements, Challenges, Methods (2021), arXiv:2106.04609 [quant-ph] .
  * Alam  _et al._ [2022] M. S. Alam, S. Hadfield, H. Lamm, and A. C. Y. Li (SQMS), Primitive quantum gates for dihedral gauge theories, Phys. Rev. D 105, 114501 (2022), arXiv:2108.13305 [quant-ph] .
  * Gustafson  _et al._ [2022] E. J. Gustafson, H. Lamm, F. Lovelace, and D. Musk, Primitive quantum gates for an SU(2) discrete subgroup: Binary tetrahedral, Phys. Rev. D 106, 114501 (2022), arXiv:2208.12309 [quant-ph] .
  * Davoudi  _et al._ [2023] Z. Davoudi, A. F. Shaw, and J. R. Stryker, General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory, Quantum 7, 1213 (2023), arXiv:2212.14030 [hep-lat] .
  * Lamm and Lawrence [2018] H. Lamm and S. Lawrence, Simulation of Nonequilibrium Dynamics on a Quantum Computer, Phys. Rev. Lett. 121, 170501 (2018), arXiv:1806.06649 [quant-ph] .
  * Harmalkar  _et al._ [2020] S. Harmalkar, H. Lamm, and S. Lawrence (NuQS), Quantum Simulation of Field Theories Without State Preparation (2020), arXiv:2001.11490 [hep-lat] .
  * Gustafson and Lamm [2021] E. J. Gustafson and H. Lamm, Toward quantum simulations of $\mathbb{Z}_{2}$ gauge theory without state preparation, Phys. Rev. D 103, 054507 (2021), arXiv:2011.11677 [hep-lat] .
  * Saroni  _et al._ [2023] J. Saroni, H. Lamm, P. P. Orth, and T. Iadecola, Reconstructing thermal quantum quench dynamics from pure states, Phys. Rev. B 108, 134301 (2023), arXiv:2307.14508 [quant-ph] .
  * Creutz and Okawa [1983] M. Creutz and M. Okawa, Generalized Actions in $Z(p$) Lattice Gauge Theory, Nucl. Phys. B220, 149 (1983).
  * Bhanot and Rebbi [1981] G. Bhanot and C. Rebbi, Monte Carlo Simulations of Lattice Models With Finite Subgroups of SU(3) as Gauge Groups, Phys. Rev. D24, 3319 (1981).
  * Petcher and Weingarten [1980] D. Petcher and D. H. Weingarten, Monte Carlo Calculations and a Model of the Phase Structure for Gauge Theories on Discrete Subgroups of SU(2), Phys. Rev. D22, 2465 (1980).
  * Bhanot [1982] G. Bhanot, SU(3) Lattice Gauge Theory in Four-dimensions With a Modified Wilson Action, Phys. Lett. 108B, 337 (1982).
  * Weingarten and Petcher [1981] D. H. Weingarten and D. N. Petcher, Monte Carlo Integration for Lattice Gauge Theories with Fermions, Phys. Lett. 99B, 333 (1981).
  * Weingarten [1982] D. Weingarten, Monte Carlo Evaluation of Hadron Masses in Lattice Gauge Theories with Fermions, Phys. Lett. 109B, 57 (1982), [,631(1981)].
  * Carena  _et al._ [2021] M. Carena, H. Lamm, Y.-Y. Li, and W. Liu, Lattice Renormalization of Quantum Simulations (2021), arXiv:2107.01166 [hep-lat] .
  * González-Cuadra  _et al._ [2022] D. González-Cuadra, T. V. Zache, J. Carrasco, B. Kraus, and P. Zoller, Hardware efficient quantum simulation of non-abelian gauge theories with qudits on Rydberg platforms (2022), arXiv:2203.15541 [quant-ph] .
  * Fromm  _et al._ [2022] M. Fromm, O. Philipsen, and C. Winterowd, Dihedral Lattice Gauge Theories on a Quantum Annealer (2022), arXiv:2206.14679 [hep-lat] .
  * Carena  _et al._ [2022] M. Carena, E. J. Gustafson, H. Lamm, Y.-Y. Li, and W. Liu, Gauge Theory Couplings on Anisotropic Lattices (2022), arXiv:2208.10417 [hep-lat] .
  * Charles  _et al._ [2023] C. Charles, E. J. Gustafson, E. Hardt, F. Herren, N. Hogan, H. Lamm, S. Starecheski, R. S. Van de Water, and M. L. Wagman, Simulating $\mathbb{Z}_{2}$ lattice gauge theory on a quantum computer (2023), arXiv:2305.02361 [hep-lat] .
  * Gustafson and Lamm [2023] E. J. Gustafson and H. Lamm, Robustness of Gauge Digitization to Quantum Noise (2023), arXiv:2301.10207 [hep-lat] .
  * Ballini  _et al._ [2024] E. Ballini, G. Clemente, M. D’Elia, L. Maio, and K. Zambello, Quantum computation of thermal averages for a non-Abelian D4 lattice gauge theory via quantum Metropolis sampling, Phys. Rev. D 109, 034510 (2024), arXiv:2309.07090 [quant-ph] .
  * Carena  _et al._ [2024] M. Carena, H. Lamm, Y.-Y. Li, and W. Liu, Quantum error thresholds for gauge-redundant digitizations of lattice field theories (2024), arXiv:2402.16780 [hep-lat] .
  * Gustafson  _et al._ [2024b] E. J. Gustafson, Y. Ji, H. Lamm, E. M. Murairi, and S. Zhu, Primitive Quantum Gates for an SU(3) Discrete Subgroup: $\Sigma(36\times 3)$ (2024b), arXiv:2405.05973 [hep-lat] .
  * Johnson [1982] R. C. Johnson, Angular Momentum on a Lattice, Phys. Lett. B 114, 147 (1982).
  * Moore and Fleming [2006] D. C. Moore and G. T. Fleming, Angular momentum on the lattice: The Case of non-zero linear momentum, Phys. Rev. D 73, 014504 (2006), [Erratum: Phys.Rev.D 74, 079905 (2006)], arXiv:hep-lat/0507018 .
  * Basak  _et al._ [2007] S. Basak, R. G. Edwards, G. T. Fleming, K. J. Juge, A. Lichtl, C. Morningstar, D. G. Richards, I. Sato, and S. J. Wallace, Lattice QCD determination of patterns of excited baryon states, Phys. Rev. D 76, 074504 (2007), arXiv:0709.0008 [hep-lat] .
  * Edwards  _et al._ [2011] R. G. Edwards, J. J. Dudek, D. G. Richards, and S. J. Wallace, Excited state baryon spectroscopy from lattice QCD, Phys. Rev. D 84, 074508 (2011), arXiv:1104.5152 [hep-ph] .
  * Egerer  _et al._ [2021] C. Egerer, R. G. Edwards, C. Kallidonis, K. Orginos, A. V. Radyushkin, D. G. Richards, E. Romero, and S. Zafeiropoulos (HadStruc), Towards high-precision parton distributions from lattice QCD via distillation, JHEP 11, 148, arXiv:2107.05199 [hep-lat] .
  * Detmold  _et al._ [2024] W. Detmold, W. I. Jay, G. Kanwar, P. E. Shanahan, and M. L. Wagman, Multi-particle interpolating operators in quantum field theories with cubic symmetry (2024), arXiv:2403.00672 [hep-lat] .
  * Horn  _et al._ [1979] D. Horn, M. Weinstein, and S. Yankielowicz, Hamiltonian Approach to Z(N) Lattice Gauge Theories, Phys. Rev. D 19, 3715 (1979).
  * Kogut [1980] J. B. Kogut, 1/n Expansions and the Phase Diagram of Discrete Lattice Gauge Theories With Matter Fields, Phys. Rev. D 21, 2316 (1980).
  * Fradkin and Shenker [1979] E. H. Fradkin and S. H. Shenker, Phase Diagrams of Lattice Gauge Theories with Higgs Fields, Phys. Rev. D 19, 3682 (1979).
  * Ovrut [1978] B. A. Ovrut, Isotropy Subgroups of SO(3) and Higgs Potentials, J. Math. Phys. 19, 418 (1978).
  * Merle and Zwicky [2012] A. Merle and R. Zwicky, Explicit and spontaneous breaking of SU(3) into its finite subgroups, JHEP 02, 128, arXiv:1110.4891 [hep-ph] .
  * Luhn [2011] C. Luhn, Spontaneous breaking of SU(3) to finite family symmetries: a pedestrian’s approach, JHEP 03, 108, arXiv:1101.2417 [hep-ph] .
  * Vileta  _et al._ [2023] V. E. Vileta, B. Gavela, R. Houtz, and P. Quilez, Discrete Goldstone bosons, Phys. Rev. D 107, 035009 (2023), arXiv:2205.09131 [hep-ph] .
  * Heller [1995] U. M. Heller, SU(3) lattice gauge theory in the fundamental adjoint plane and scaling along the Wilson axis, Phys. Lett. B362, 123 (1995), arXiv:hep-lat/9508009 [hep-lat] .
  * Heller [1996] U. M. Heller, More on SU(3) lattice gauge theory in the fundamental adjoint plane,  _Lattice ’95. Proceedings, International Symposium on Lattice Field Theory, Melbourne, Australia, July 11-15, 1995_ , Nucl. Phys. Proc. Suppl. 47, 262 (1996), arXiv:hep-lat/9509010 [hep-lat] .
  * Hasenbusch and Necco [2004] M. Hasenbusch and S. Necco, SU(3) lattice gauge theory with a mixed fundamental and adjoint plaquette action: Lattice artifacts, JHEP 08, 005, arXiv:hep-lat/0405012 [hep-lat] .
  * Hasenbusch and Necco [2005] M. Hasenbusch and S. Necco, Lattice artefacts in SU(3) lattice gauge theory with a mixed fundamental and adjoint plaquette action,  _Lattice field theory. Proceedings, 22nd International Symposium, Lattice 2004, Batavia, USA, June 21-26, 2004_ , Nucl. Phys. Proc. Suppl. 140, 743 (2005), arXiv:hep-lat/0409067 [hep-lat] .
  * Ji  _et al._ [2022] Y. Ji, H. Lamm, and S. Zhu, Gluon Digitization via Character Expansion for Quantum Computers (2022), arXiv:2203.02330 [hep-lat] .
  * Fukugita  _et al._ [1983] M. Fukugita, T. Kaneko, and M. Kobayashi, Phase Structure and Duality of $Z(N$) Lattice Gauge Theory With Generalized Actions in Four Space-time Dimensions, Nucl. Phys. B215, 289 (1983).
  * Alexandru  _et al._ [2021] A. Alexandru, P. F. Bedaque, R. Brett, and H. Lamm, The spectrum of qubitized QCD: glueballs in a $S(1080)$ gauge theory (2021), arXiv:2112.08482 [hep-lat] .
  * Blum  _et al._ [1997] T. Blum, C. E. Detar, S. A. Gottlieb, K. Rummukainen, U. M. Heller, J. E. Hetrick, D. Toussaint, R. L. Sugar, and M. Wingate, Improving flavor symmetry in the Kogut-Susskind hadron spectrum, Phys. Rev. D55, R1133 (1997), arXiv:hep-lat/9609036 [hep-lat] .
  * Bali [2000] G. S. Bali, Casimir scaling of SU(3) static potentials, Phys. Rev. D 62, 114503 (2000), arXiv:hep-lat/0006022 .
  * Deldar [2000] S. Deldar, Static SU(3) potentials for sources in various representations, Phys. Rev. D 62, 034509 (2000), arXiv:hep-lat/9911008 .
  * Sommer [1994] R. Sommer, A New way to set the energy scale in lattice gauge theories and its applications to the static force and $\alpha_{s}$ in SU(2) Yang-Mills theory, Nucl. Phys. B 411, 839 (1994), arXiv:hep-lat/9310022 .