††institutetext: Institut des Hautes Études Scientifiques, 91440 Bures-sur-
Yvette, France

# Classifying irreducible fixed points of five scalar fields in perturbation
theory

Junchen Rong  and Slava Rychkov<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Classifying perturbative fixed points near upper critical dimensions plays an
important role in understanding the space of conformal field theories and
critical phases of matter. In this work, we consider perturbative fixed points
of $N=5$ scalar bosons coupled with quartic interactions preserving an
arbitrary subgroup $G\subset{\rm O}(5)$. We perform an exhaustive algorithmic
search over the symmetry groups $G$ which are irreducible and satisfy the
Landau condition, so that the fixed point can be reached by fine-tuning a
single mass term and there is no need to tune the cubic couplings. We also
impose stability of the RG flow in the space of quartic couplings, and
reality. We thus prove that there exist no new stable fixed points in
$d=4-\epsilon$ dimensions beyond the two known ones: namely the ${\rm O}(5)$
invariant fixed point and the Cubic(5) fixed point. This work is a
continuation of the classification of such fixed points with $N=4$ scalars by
Toledano, Michel, Toledano and Brézin in 1985 toledano1985renormalization .

##  1 Introduction

Classification of critical states of matter is an important problem, which can
be approached from many different directions. One possibility is to classify
renormalization group (RG) fixed points in perturbation theory close to the
upper critical dimension. This approach provides interesting information,
since found fixed points often continue down to the physical dimensions.

In this paper we consider RG fixed points for the theory of $N$ scalar fields
in $d=4-\varepsilon$ dimensions, described by the Lagrangian

$\frac{1}{2}(\partial_{\mu}\phi^{i})^{2}+\frac{1}{2}\mu_{ij}\phi^{i}\phi^{j}+\frac{1}{3!}c_{ijk}\phi^{i}\phi^{j}\phi^{k}+\frac{1}{4!}\lambda_{ijkl}\phi^{i}\phi^{j}\phi^{k}\phi^{l}.$
(1)

At the fixed point, the mass term $\mu_{ij}$ and the cubic couplings $c_{ijk}$
have to be tuned to zero, while the tensor of quartic couplings, assumed real,
has to satisfy the beta function equation

$\beta_{ijkl}=-\epsilon\lambda_{ijkl}+\frac{1}{16\pi^{2}}\left(\lambda_{ijmn}\lambda_{mnkl}+\text{2
permutations}\right)+O(\lambda^{3}).$ (2)

The most important characteristic of the fixed point is its global symmetry
group $G$, which is defined as the maximal subgroup of ${\rm O}(N)$ leaving
(1) invariant. Excluding the Gaussian fixed point $\lambda=0$, the most
symmetric fixed point is the Wilson-Fisher fixed point wilson1972critical
which preserves the full ${\rm O}(N)$. For other symmetries, the most
interesting case arises when the group $G$ is large enough so that three
conditions are satisfied:

I. (Irreducibility) There is a single mass term compatible with $G$, namely
the fully isotropic one $\mu_{ij}=\mu\delta_{ij}$;

L. (Landau condition) No cubic term is allowed by symmetry $G$;

S. (Stability) All quartic couplings allowed by symmetry $G$ are irrelevant,
in the RG sense, at the fixed point.

If these conditions are satisfied, the fixed point can describe a continuous
phase transition (critical point) obtained by tuning a single relevant
coupling consistent with the symmetry (the isotropic mass). Violating any of
these conditions would imply multiple tunings necessary to reach the fixed
point. This would correspond to multi-critical behavior, making it more
difficult to realize such fixed points experimentally.

The irreducibility condition is called so because mathematically, a single
mass term means that the vector representation of ${\rm O}(N)$ remains
irreducible over the real numbers under $G$.111An $N$-dimensional
representation irreducible over the reals maybe be reducible over complex
numbers. It then becomes a pair of complex conjugate irreps with dimension
$N/2$. Clearly, this may happen only if $N$ is even. In this paper, we focus
on $N=5$ so for us this never happens. Unless otherwise specified,
“irreducible” in our paper stands for irreducible over the real numbers. Such
subgroups of $G$ are called irreducible. The Landau condition is important
because phase transitions violating this condition are expected to be first
order, unless of course extra tuning is performed to tune the cubic couplings
to zero.222In low dimensions, the Landau condition may be relaxed because
cubic couplings may become irrelevant, as happens famously for the 3-state
Potts model in two dimensions. Stability condition is similarly motivated -
phase transition will be first order if the microscopic theory does not belong
to the basin of attraction of a stable fixed point, in particular, if no
stable fixed point exists.

Note that conditions I and L depend only on the symmetry group $G$, while
verifying condition S requires solving the beta function equations. Fixed
points satisfying all three conditions will be called ILS fixed points. It is
interesting to classify, for each $N$, such fixed points.

For $N=2,3$ the only ILS fixed point is the ${\rm O}(N)$ fixed point. For
$N=4$ there are four ILS fixed points which were classified many years ago by
Toledano, Michel, Toledano and Brézin toledano1985renormalization , using the
mathematical knowledge of finite subgroups of $O(4)$. It turns out that
classification of finite subgroups of O(5) is also available mecchia2011finite
. Using this result, in this paper we will classify ILS fixed points for
$N=5$.

We will start in Section 2 by describing the classification of
toledano1985renormalization  as well as the results available for other $N$.
Then in Section 3, we introduce the general strategy to study general Landau
theories with $N$ scalars. Certain group theory notions are introduced for
later convenience. In Section 4, we discuss the maximal irreducible subgroups
of ${\rm O}(5)$, drawing on various mathematical results. In Section 5 we
present an algorithm which generates all the irreducible subgroups of ${\rm
O}(N)$ and their group-subgroup relations starting from a list that is known
to contain all the maximal irreducible subgroups of ${\rm O}(N)$. The results
of applying this algorithm for $N=5$ are summarized in Section 6. These
results allow us to greatly reduce the number of Landau theories we need to
consider. We describe these Landau theories in Section 7 and then study their
RG flows in Section 8. Our results and future directions are discussed in
Section 9. Group theory computations of this paper are done using GAPGAP4 .

##  2 Review of known results

In this section we review what is known about ILS fixed points for various
$N$. All statements about the existence and stability of fixed points are at
one loop unless otherwise specified.

The ${\rm O}(N)$ fixed point is stable with respect to ${\rm O}(N)$-symmetric
perturbations for any $N$. In $N=2,3$ the ${\rm O}(N)$ fixed point is globally
stable, while any other fixed point is unstable brezin1974discussion . This is
in accord with a theorem of Michel Michel:1983in  (see Rychkov:2018vya  for a
review): at one-loop order, for any symmetry $G$ the number of stable fixed
points preserving symmetry $G$ or larger is at most one.

######  Remark 2.1.

In this paper, all statements about fixed point stability refer to
$d=4-\varepsilon$ dimensions, unless stated otherwise. It does sometime happen
that fixed points which are stable near 4 dimensions become unstable in 3
dimensions, and vice versa. This may happen because two fixed points collide
in some intermediate dimension and exchange their stability properties. Here
are two famous examples of this phenomenon:

  * •

In $N=3$, the $O(3)$ fixed point is stable and the cubic fixed point, see
below, is unstable in $d=4-\varepsilon$, but the stability properties are
reversed in $d=3$ Pelissetto:2000ek ; Chester:2020iyt ; Hasenbusch:2022zur .

  * •

In $N=4$, the fixed point of two decoupled $O(2)$ models is unstable, while
the $O(2)\times O(2)$ fixed point, see below, is (two-loop) stable. In $d=3$
the stability is reversed.

For $N=4$, the $O(4)$ fixed point is, at one loop, marginal with respect to
$O(4)$ breaking perturbations brezin1974discussion . This implies that any
fixed point differing from $O(4)$ at one loop is unstable brezin1974discussion
. To resolve the degeneracy of $O(4)$ breaking directions requires two-loop
analysis toledano1985renormalization , which we discuss below.

For $N\geqslant 5$ the ${\rm O}(N)$ fixed point is one-loop unstable in ${\rm
O}(N)$ breaking directions, which opens the possibility to having one-loop
stable fixed points with symmetry smaller than ${\rm O}(N)$.

Before discussing full classification results in $N=4$
toledano1985renormalization  and in $N=5$ (this work), as well as partial
results available for $N=6$ PhysRevB.32.7624 ; PhysRevB.33.1774 ;
PhysRevB.33.6196 , let us mention some infinite series of fixed points.

There are several infinite families of I&L symmetries $G$ which have only two
quartic invariants, so that the quartic Landau theory contains only two
couplings. In this case there are three fixed points: ${\rm O}(N)$ and two
$G$-invariant ones. For $N\geqslant 5$, one of the two $G$-invariant fixed
points is stable, the other unstable brezin1974discussion ; Osborn:2017ucf
(also reviewed in Rychkov:2018vya ). Two-loop analysis is needed if these two
fixed points are one-loop degenerate. Symmetries falling into this category
are:

  * •

cubic: $\operatorname{Cubic}(N)=(\mathbb{Z}_{2})^{N}\rtimes S_{N}$
aharony1973critical . There is a one-loop stable cubic fixed point for all
$N\geqslant 5$. For $N=4$ it is degenerate with the $O(4)$ fixed point, but
becomes distinct from it, and stable, from two loops on
toledano1985renormalization . For $N=3$ the cubic fixed point is one-loop
unstable (see however Remark 2.1).

  * •

tetrahedral: $S_{N+1}\times\mathbb{Z}_{2}$ zia1975critical ; Osborn:2017ucf
(also known as “restricted Potts model”). There is a one-loop stable fixed
point for all $N\geqslant 6$. For $N=4$ it is stable, after resolving
degeneracy from $O(4)$ at two loops toledano1985renormalization . At $N=5$ the
fixed point has complex couplings (at two loops), see Section 8.3.

  * •

bifundamental: $O(p)\times O(q)/\mathbb{Z}_{2}$, $N=pq$. There is a real fixed
point at one loop if Kawamura ; Osborn:2017ucf

$R_{pq}=p^{2}+q^{2}-10pq-4(p+q)+52\geqslant 0.$ (3)

Restricting to $p\geqslant q$ this condition is satisfied for $p=q=2$ and for
$p\geqslant 5q+2+2\sqrt{6(q-1)(q+2)}$ which requires large values of $p$ e.g.
$p\geqslant 22$ for $q=2$. The $p=q=2$ fixed point coincides with the $O(4)$
fixed point. It becomes distinct, and stable, at two loops
toledano1985renormalization  (see however Remark 2.1).

  * •

“MN”: $\Gamma_{p,q}=O(p)^{q}\rtimes S_{q}$, $N=pq$ Mukamel ; Osborn:2017ucf .
The case $p=1$ is equivalent to the cubic, and $p=q=2$ to the bifundamental.
There are two fixed points, one fully interacting and another factorized ($q$
copies of $O(p)$ fixed points). The factorized fixed point is one-loop stable
for $p>5$. For $p=4$ the two fixed points coincide at one loop so they are
marginally stable. For $p=2,3$ the fully interacting fixed point is stable.

There are very few examples of stable fixed points having symmetry groups with
3 or more quartic invariants. A series of examples was constructed in MichelFP
(see also Grinstein1982StableFP ; Osborn:2017ucf ). His smallest example has
$N=12$, $G=\Gamma_{2,6}\cap\Gamma_{6,2}$, and three quartic invariants.

Finally, let us discuss classifications for $N=4$ and $N=5$ and partial
results for $N=6$.

In $N=4$, Toledano et al toledano1985renormalization  classified all I&L
subgroups of $O(4)$, their quartic Landau theories and the fixed points. Using
two-loop analysis when needed, they proved that there exist exactly four ILS
fixed points, having symmetries $O(4)$, Cubic(4),
$S_{N+1}\times\mathbb{Z}_{2}$, $O(2)^{2}\rtimes\mathbb{Z}_{2}$. All of them
are members of the above families.333 In toledano1985renormalization , the
tetrahedral symmetry is called di-icosahedral, and
$O(2)^{2}\rtimes\mathbb{Z}_{2}$ bi-cylindrical.

In this work, we will perform a similar full classification in $N=5$. Our
result is that, apart from ${\rm O}(5)$ and Cubic(5), there are no new stable
fixed points.

In $N=6$, a partial classification was carried out in PhysRevB.32.7624 ;
PhysRevB.33.1774 ; PhysRevB.33.6196 . These authors consider subgroups of
$O(6)$ which can be realized as images of 6-dimensional representations of one
of 230 space groups in three dimensions. Their work is thus relevant to the
study of second-order structural phase transitions. They identify 11 such
images, which are irreducible subgroups of $O(6)$. It’s a large family of
subgroups, but the only stable fixed point they identified was of the “MN”
type, namely $O(2)^{3}\rtimes S_{3}$.

##  3 General strategy

Given any symmetry group $G\subset O(N)$, it is in principle straightforward
to classify ILS fixed points having this symmetry. First one checks whether
conditions I and L are satisfied. These condition say that $I_{2}=1$,
$I_{3}=0$, where $I_{2}$ and $I_{3}$ are the numbers of linearly independent
quadratic and cubic invariant polynomials. They may be computed e.g. using the
Molien function. (See Eq. (23) for more details.) To verify the condition S,
one first needs to find the basis of quartic invariant polynomials, call it
$P_{4,a}(\phi^{i})$, $a=1,\ldots,I_{4}$. Then one writes down the quartic
Landau theory consistent with the symmetry (setting the quadratic term to
zero)

$L(\phi)=\frac{1}{2}(\partial_{\mu}\phi^{i})^{2}+\sum_{a=1}^{I_{4}}\lambda_{a}P_{4,a}(\phi^{i}),$
(4)

computes beta functions for the quartic couplings $\lambda_{a}$, finds fixed
points, and selects those of them, if any, which are real and RG stable.

In general, the group ${\rm O}(N)$ has infinitely many subgroups. Our goal is
to find all ILS fixed points. One way to approach this goal would be to first
find all subgroups satisfying conditions I and L, and then verify condition S
for each of them as described above. This is still a bit tedious since the
number of I&L subgroups is significant. In this paper we will speed up the
procedure further by using group-subgroup relations. Our procedure will
consist of three steps:

1\. Generate a list containing all maximal irreducible subgroups of ${\rm
O}(N)$. In our case ($N=5$), this will be done with the help of mathematical
literature classifying finite subgroups of ${\rm O}(5)$.

2\. Given the list from Step 1, generate all irreducible subgroups of $O(N)$,
and their group-subgroup relations. In practice, we have an algorithm which
deals with finite irreducible subgroups. The irreducible subgroups which are
Lie groups have to be dealt with separately. For the $N=5$ case this turns out
to be easy, since there are very few of them.

3\. Study the Landau theories of the minimal I&L subgroups, to find their
fixed points and select the stable ones. Here “minimal I&L” means not having
any subgroups satisfying I&L conditions.

Step 3 is the crucial simplification - instead of having to study many Landau
theories, we will have to study only a few. Of course this is only possible if
one knows which subgroups are minimal - something we will know thanks to Step
2.

###  3.1 Matrix groups vs abstract groups

One subtlety of the discussion below is that we will have to talk about two
different kinds of groups. The most important for us are symmetry groups of
Landau theories, which are matrix groups, subgroups of ${\rm O}(N)$. A matrix
group is a set of matrices with group multiplication given by matrix
multiplication.444Notice that since we use “set” and not “list”, all matrices
are distinct.  Two matrix groups $G$ and $G^{\prime}$ are considered
equivalent for our purposes (written $G\cong G^{\prime}$) if they are
conjugate within ${\rm O}(N)$, i.e. there is a similarity transformation
$g\rightarrow SgS^{-1}$, where $S\in O(N)$ is a fixed matrix, which maps the
set of matrices forming $G$ onto the matrices forming $G^{\prime}$. We will
use “conjugate within ${\rm O}(N)$” and “equivalent” interchangeably.

We will also have to talk about abstract groups, i.e. about sets with a
multiplication operation, not necessarily realized as matrices. To avoid
confusion, we use Latin letters $G,H,\ldots$ to denote matrix groups, and
Gothic letters $\mathfrak{G},\mathfrak{H},\ldots$ to denote abstract groups.

Two abstract groups $\mathfrak{G}$ and $\mathfrak{G}^{\prime}$ are called
isomorphic if there is a one-to-one map from one to the other which preserves
the multiplication rule.

Every matrix group $G$ gives rise to the corresponding abstract group
$\mathfrak{G}$ which is obtained from $G$ by extracting its multiplication
table. Obviously, if $G$ and $G^{\prime}$ are conjugate within ${\rm O}(N)$,
then the corresponding abstract groups $\mathfrak{G}$ and
$\mathfrak{G}^{\prime}$ are isomorphic. But the converse is not necessarily
true: it may happen that two matrix groups $G_{1}$ and $G_{2}$, while not
conjugate to each other, are both isomorphic to the same abstract group
$\mathfrak{G}$. Such matrix groups will be considered as different groups for
our purposes, and will be denoted accordingly.

We want to discuss the opposite operation: how given an abstract group
$\mathfrak{G}$ to obtain all inequivalent matrix groups $G\subset O(N)$
isomorphic to $\mathfrak{G}$.

Clearly, we can obtain a matrix group $G\subset O(N)$ from an abstract group
$\mathfrak{G}$ by using any real $N$-dimensional representation $\rho$ of
$\mathfrak{G}$, and defining $G$ as a set of matrices representing the
elements of $\mathfrak{G}$ in this representation. In addition, if the
representation is faithful, the so obtained matrix group $G$ will be
isomorphic to $\mathfrak{G}$ as an abstract group. The faithfulness condition
is important: if it is violated, then some elements of $\mathfrak{G}$ will be
mapped to identical matrices. The set of non-identical matrices obtained this
way will still form a matrix group, but it will not be isomorphic to
$\mathfrak{G}$.

If two representations $\rho_{1}$ and $\rho_{2}$ are isomorphic in the sense
of representation theory, then the two matrix groups $G_{1}$ and $G_{2}$
obtained this way will clearly be conjugate within ${\rm O}(N)$.

There is however another subtlety. Sometimes two matrix groups $G_{1}$ and
$G_{2}$ will be conjugate within ${\rm O}(N)$ even if $\rho_{1}$ and
$\rho_{2}$ are not isomorphic representations. Such cases are related to outer
automorphisms of the abstract group $\mathfrak{G}$.

An automorphism is an isomorphism from a group to itself, which respects the
multiplication table of the group. The group of automorphisms,
Aut($\mathfrak{G}$), acts on representations of the group. If
$\rho:\mathfrak{G}\rightarrow\operatorname{End}(V)$ is a representation, and
$f:\mathfrak{G}\rightarrow\mathfrak{G}$ is an automorphism, then
$\rho^{f}=\rho\circ f$ is another representation. Automorphisms can be inner
and outer. Inner automorphisms, forming the group Inn($\mathfrak{G}$), are
(abstract) conjugations

$g\rightarrow h^{-1}gh,\qquad h\in\mathfrak{G}.$ (5)

For inner automorphisms, representation $\rho^{f}$ is isomorphic to $\rho$.
Outer automorphism group of group $\mathfrak{G}$ is defined as
Aut($\mathfrak{G}$)/Inn($\mathfrak{G}$). For outer automorphisms, $\rho^{f}$
may not be isomorphic to $\rho$. However, the matrix groups constructed from
$\mathfrak{G}$ using $\rho$ and $\rho^{f}$ clearly will be equivalent, since
they will consist of the same matrices, up to reshuffling.

The bottomline of this discussion is the following basic fact.

######  Lemma 3.1.

We will obtain all inequivalent matrix groups $G\subset O(N)$ from an abstract
group $\mathfrak{G}$ by using all real $N$-dimensional faithful representation
$\rho$ of $\mathfrak{G}$, up to equivalence by outer automorphisms.

To apply this criterion we have to know when two group irreps are equivalent
by an outer automorphism. The crucial property of an outer automorphism is
that it acts as a permutation on conjugacy classes of the group, and on
characters of representations of the group. We will have to identify
permutations corresponding to all outer automorphisms, and identify characters
related by these permutations.

######  Example 3.1.

The abstract group $S_{6}$ has 4 different faithful 5-dimensional irreps, but
only two of them are non-equivalent, up to outer automorphisms. This means
that there are two different matrix groups $G_{1},G_{2}\subset{\rm O}(5)$
which are both isomorphic to $S_{6}$ as abstract groups, which we will denote
as $S_{6}\color[rgb]{1,0,0}{-A}$ and $S_{6}\color[rgb]{1,0,0}{-B}$.

Let us next discuss group-subgroup relations. If $G_{1}\subset O(N)$ and
$G_{2}\subset O(N)$ are two matrix groups, we say that $G_{1}\subset G_{2}$ if
$G_{1}$ is conjugate to a subgroup of $G_{2}$. Note that if $G_{1}\subset
G_{2}$ in this sense, then we also have
$\mathfrak{G}_{1}\subset\mathfrak{G}_{2}$ at the abstract group level, where
$\mathfrak{G}_{1},\mathfrak{G}_{2}$ are the corresponding abstract groups. But
the converse is not necessarily true: if we take two matrix groups $G_{1}$ and
$G_{2}$ constructed from $\mathfrak{G}_{1},\rho_{1}$ and
$\mathfrak{G}_{2},\rho_{2}$, with $\mathfrak{G}_{1}\subset\mathfrak{G}_{2}$,
then not necessarily $G_{1}\subset G_{2}$ as matrix groups.

###  3.2 Steps 2&3

With definitions out of the way, let us give more details about Steps 2&3 of
the above strategy. Below we will present an algorithm to generate the list of
irreducible subgroups of ${\rm O}(N)$ from a list containing all maximal
irreducible subgroups, and simultaneously figure out group-subgroup relations
among all of them. Applying this algorithm, we will organize the irreducible
subgroups in a directed graph, schematically shown in Fig. 1, where on each
level we put subgroups of the groups from the previous level. Furthermore, for
each subgroup in the graph, we can check whether the Landau condition is
satisfied or not (denoted by a star). Clearly, if this condition holds for any
subgroup $\mathcal{G}$, then it holds for all larger subgroups $G^{\prime}$,
$G\subset G^{\prime}\subset{\rm O}(N)$.

Figure 1: Graph encoding group-subgroup relations, see the text.

Given such a graph, we consider minimal I&L subgroups $H_{1},\ldots,H_{r}$
($r=2$ in Fig.1). We will study Landau theories only for these minimal I&L
subgroups. Indeed, when we study the RG flow of these Landau theories and
search for fixed points, we are guaranteed to find all the fixed points whose
symmetry is larger or equal to $H_{i}$, which by definition includes all I&L
subgroups.

Another feature of our analysis will be the phenomenon of symmetry
enhancement, discussed in Section 7.2.1 below.

##  4 Maximal irreducible subgroups of ${\rm O}(5)$

We will now carry out the above strategy for $N=5$. The first step is to make
a list containing all maximal irreducible subgroups of ${\rm O}(5)$. We first
discuss finite subgroups of ${\rm O}(5)$, and then Lie subgroups of ${\rm
O}(5)$.

###  4.1 Finite subgroups of ${\rm O}(5)$

The finite subgroups $G\subset{\rm O}(5)$ were classified in mecchia2011finite
where they were shown to fall into the following three classes

  * •

$G$ is (conjugate to) a subgroup of $(\mathbb{Z}_{2})^{5}\rtimes S_{5}$,
called Cubic(5).

  * •

$G$ is, as an abstract group, isomorphic to $A_{5}$, $S_{5}$, $A_{6}$ or
$S_{6}$, or to the direct product of one of them with $\mathbb{Z}_{2}$, where,
in the matrix group realization, the $\mathbb{Z}_{2}$ factor is represented as

$\mathbb{Z}^{{\rm O}(5)}_{2}=\\{I,-I\\},$ (6)

where $I$ is the $5\times 5$ identity matrix.

  * •

$G$ is (conjugate to) a subgroup of ${\rm O}(3)\times{\rm O}(2)$ or ${\rm
O}(4)\times\mathbb{Z}_{2}$.

In the last class, the ${\rm O}(3)\times{\rm O}(2)$ or ${\rm
O}(4)\times\mathbb{Z}_{2}$ subgroups are embedded in ${\rm O}(5)$ in the
natural way, or in other words, the 5 dimensional vector irrep of ${\rm O}(5)$
decomposes as $3+2$ and 4+1. This means that none of these subgroups are
irreducible, and we can ignore them.555As a side comment, we note that this
class of subgroups can be further specified with the help of the knowledge of
finite subgroups of $O(4)$ du1964homographies  and ${\rm O}(3)$
klein1884vorlesungen . The finite subgroups of ${\rm O}(3)$ famously follow an
ADE classification.

This mathematical result gives us one matrix group and four abstract groups:

Cubic(5)(matrix group) $\displaystyle
A_{5}\times\mathbb{Z}_{2},S_{5}\times\mathbb{Z}_{2},A_{6}\times\mathbb{Z}_{2},S_{6}\times\mathbb{Z}_{2}\qquad\text{(abstract
groups)}.$ (7)

Maximal irreducible subgroups of ${\rm O}(5)$ should be Cubic(5) or
isomorphic, as abstract groups, to one or more of the abstract groups in Eq.
(7), although at this stage we can’t yet say to which ones.

As a next step we have to find matrix groups corresponding to the abstract
groups in (7). By Lemma 3.1, this amounts to finding their real 5-dimensional
faithful representations which are irreducible over real numbers, irreducible
since we are interested in irreducible subgroups, and up to outer automorphism
equivalence.

Let us discuss how this is done for the abstract group
$S_{5}\times\mathbb{Z}_{2}$, other cases being similar. All group theory
calculations are done using GAP GAP4 , see App. C. In GAP, we specify
$S_{5}\times\mathbb{Z}_{2}$ group by

g:=DirectProduct(SymmetricGroup(5),CyclicGroup(2)); (8)

Once the group is specified, we compute its character table:

tbl:=CharacterTable(g); Display(tbl); (9)

and look up the faithful 5-dimensional irreps. Recall that the dimension $n$
of the representation equals to $\chi(e)$, character at the identity. A simple
criterion for detecting unfaithful representations is this:

######  Lemma 4.1 (ludl2010finite , p.5).

An $n$-dimensional representations is unfaithful if and only if its character
$\chi$ takes value $n$ more than once.

These representation are not necessarily real representations. To select the
real representations, we can calculate their (second) Frobenius–Schur
indicators by

Indicator(tbl,2); (10)

which equals to 1, 0 and $-1$ for real, complex and quaternionic
representations respectively. Here tbl is the character table calculated in
(9).666 If $N$ were an even number, one also need keep complex irreps with
dimension $N/2$.

For $S_{5}\times\mathbb{Z}_{2}$, we find that there are 2 non-isomorphic
faithful 5-dim irreps, which are the 11th and 12th irreps of the group. We
next have to check whether these irreps are equivalent by an outer
automorphism. Automorphism groups can be easily constructed in GAP:777Here
RightCosets(G,H) calculates the right cosets of $H$ in $G$. The command
Representative(...) gives the representative elements of the coset, which in
our case will be an outer automorphism (excluding the trivial coset). The
function List(ojbects, map) lists the objects after applying the specified
map.

AutG:=AutomorphismGroup(g); InnG:=InnerAutomorphismsAutomorphismGroup(AutG);
outerautomorphisms:=List(RightCosets(AutG,InnG),c->Representative(c)); (11)

For $S_{5}\times\mathbb{Z}_{2}$, we thus discover that there is just one
nontrivial outer automorphism. We first calcluate a matrix representation of
11th irrep by:

psi:=Irr(tbl);; irrepmap:=IrreducibleRepresentationsDixon(g,psi[11]); (12)

In GAP, an irrep is given by a map from abstract group elements to matrices.
To calculate the character of the 11th irrep, we use:

List(ConjugacyClasses(g),c->TraceMat(Image(irrepmap,Representative(c)))); (13)

To calculate the character of the 11th irrep after applying the non-trivial
automorphism, we use

List(ConjugacyClasses(g),c->TraceMat(
Image(irrepmap,Image(outerautomorphisms[2],Representative(c))))); (14)

The result turns out to be equal to the characters of the 12th irrep.
Therefore, the two 5-dimensional faithful irreps are related by an outer
automorphism. We conclusion of this discussion that there exists a single
matrix embedding of $S_{5}\times\mathbb{Z}_{2}$ into ${\rm O}(5)$.

Similarly, we can show that for each of the remaining groups in the list (7)
there is a single embedding to consider. As a remark, these groups are easiest
specified in GAP as an abstract group by using the “small group ID” (see App.
B).

To summarize, we have identified a list of matrix subgroups of ${\rm O}(5)$
which for sure contains all maximal irreducible subgroups (although not all
subgroups in the list are actually maximal irreducible, as we will see).
Having such a list is enough to run the group-subgroup relation algorithm to
be described in Section 5.

###  4.2 Irreducible Lie subgroups of ${\rm O}(5)$

Since we are interested in symmetries of Landau theories, we consider only
closed subgroups of ${\rm O}(N)$, so any subgroup of interest to us is either
finite or a Lie subgroup. We need a separate discussion about irreducible
subgroups of ${\rm O}(5)$ which are Lie subgroups.

The result of the discussion below can be simply stated as follows. There are
only three irreducible Lie subgroups of ${\rm O}(5)$, namely
$\operatorname{SO}(5),$ $\operatorname{SO}(3)_{T}$ and
SO(3)${}_{T}\times\mathbb{Z}^{{\rm O}(5)}_{2}$, where $\mathbb{Z}^{{\rm
O}(5)}_{2}$ is defined in (6), and $\operatorname{SO}(3)_{T}$ is a subgroup
obtained by embedding $\operatorname{SO}(3)$ into $\operatorname{SO}(5)$ using
the symmetric traceless 5-dimensional irrep of $\operatorname{SO}(3)$, see
App. A.

This is argued as follows. First of all since ${\rm
O}(5)=\operatorname{SO}(5)\times\mathbb{Z}^{{\rm O}(5)}_{2}$, any irreducible
Lie subgroup of O(5) must have the form $G$ or $G\times\mathbb{Z}^{{\rm
O}(5)}_{2}$ where $G$ is an irreducible Lie subgroup of
$\operatorname{SO}(5)$.888Note that an analogous statement is not true for
finite subgroups of ${\rm O}(5)$. Namely there exist finite subgroups of ${\rm
O}(5)$ which are irreducible yet they are not of the form $G$ or
$G\times\mathbb{Z}_{2}^{{\rm O}(5)}$ where $G$ is a subgroup of
$\operatorname{SO}(5)$. An example in $S_{6}\color[rgb]{1,0,0}{-A}$ in Fig. 4.
We are thus reduced to showing that the only irreducible Lie subgroup of SO(5)
is $\operatorname{SO}(3)_{T}$.

Let $H$ be an irreducible Lie subgroup of SO(5), and let $G$ be a maximal Lie
subgroup of SO(5) containing $H$. Of course $G$ is also irreducible.

$G$ may be non-simple or simple - we consider these two cases in turn.

1) $G$ is non-simple. Ref. antoneli2006maximal  classified all maximal non-
simple Lie subgroup of $\operatorname{SO}(n)$ with $n\geq 5$. According to
antoneli2006maximal , $\operatorname{SO}(5)$ has only two such subgroups:

$\qquad S({\rm O}(4)\times{\rm O}(1))\quad\operatorname{and}\quad S({\rm
O}(3)\times{\rm O}(2)),$ (15)

where

$S({\rm O}(m)\times{\rm O}(n)):=\left\\{(A,B)\in{\rm O}(m)\times{\rm
O}(n),\quad\operatorname{with}\quad\det(A)\det(B)=1\right\\},$ (16)

which can be naturally embedded in ${\rm O}(m+n)$. Both subgroups (15) are not
irreducible (the 5-dimensional representation is reducible). Thus this case is
not of interest to us.

2) $G$ is simple. Its Lie algebra should be a simple subalgebra of so(5),
which can only be (a compact real form of) so(4), so(3) and so(2). Of these,
as a Lie algebra, only so(3) has a 5-dimensional faithful irrep. Thus $G$ must
have so(3) as its Lie algebra, and have a 5-dimensional faithful irrep. This
singles out SO(3), embedded into SO(5) as the subgroup SO(3)T, see App. A.

So far we proved that $H$ must be $\operatorname{SO}(3)_{T}$ or a subgroup of
$\operatorname{SO}(3)_{T}$. All subgroups of $\operatorname{SO}(3)$ being
known antoneli2006maximal ; klein1884vorlesungen , the only Lie subgroups of
SO(3) are SO(2) and O(2), and neither of these has a faithful 5-dimensional
representation, so this case is excluded. This finishes the proof.

##  5 Group-subgroup relations algorithm

In this section, we discuss an algorithm which given a seed list
groups_initial of finite irreducible subgroups of O(N), outputs all the
irreducible finite subgroups of ${\rm O}(N)$, together with their group-
subgroup relations. The seed list must satisfy the following property:

$\text{{\bf groups\\_initial} contains all finite maximal irreducible
subgroups}.$ (17)

In our case $N=5$ such a list was obtained in Section 4.

The algorithm is as follows:

Input:

groups_initial={$G_{1}$, $G_{2}$,…$G_{k}$} - a list of finite irreducible
subgroups of O(N) known to contain all maximal such subgroups.

Output:

groups \- the list of all finite irreducible subgroups of O(N).

relations \- group-subgroup relations between the groups.

Algorithm:

groups:=groups_initial;

tocheck:=groups_initial;

checked:={};

relations:={};

While[tocheck$\neq${},

$G$:=tocheck[[1]];

subgroups := all maximal subgroups of $G$;

subgroups := Select[subgroups, irreducible subgroups];

For[$n:=1$, $n\leq$ Length[subgroups], $n$++,

$H$:=subgroups[[$n$]];

relations:=relations $\cup$ {$G\supset H$};

If[ $H\notin$ groups, groups:=groups $\cup$ {$H$}; ];

If[ $H\notin$ checked, tocheck:=tocheck $\cup$ {$H$}; ];

tocheck:=DeleteElements[tocheck, $G$];

];

];

Print[groups,relations];

The list tocheck stores the groups whose maximal subgroups we still need to
find, while the list checked stores the groups whose maximal subgroups have
been calculated. The code terminates when tocheck is empty.

The two lines,

subgroups := all maximal subgroups of $G$;

subgroups := Select[subgroups, irreducible subgroups];

are meant to find all the finite irreducible subgroups of group $G$ (up to
conjugation). In GAP, this is done as follows. After specifying a matrix group
$G$ in GAP (see Eq. (8)), the following command creates a list of maximal
matrix subgroups of $G$ (up to conjugation),

l:=ConjugacyClassesMaximalSubgroups(G); (18)

The above command generates a list of conjugacy classes. Each of these classes
contains a set of maximal subgroups which are conjugate to each other. We then
check whether these subgroups are irreducible. To do this we compute the
character $\chi$ of the $N$-dimensional representation, and its inner product
with itself. The following command does this for the first maximal subgroup in
class number i:

conjugacyclassofsubgroups=l[i]; grp:=conjugacyclassofsubgroups[1];
chi:=Character(grp,List(ConjugacyClasses(grp),
c->TraceMat(Representative(c)))); ScalarProduct(chi,chi); (19)

The command ScalarProduct(chi,chi) calculates the number $n$ of irreducible
representations contained in this 5-dimensional representation by calculating

$n=\frac{1}{|G|}\sum_{C\in\operatorname{conjugacy}\operatorname{classes}}N_{C}\overline{\chi(C)}\chi(C).$
(20)

Here $N_{C}$ is the number of elements in the conjugacy class $C$, and $|G|$
is the order of the group. We get $n=1$ only if the $N$-dimensional
representation remains irreducible over complex numbers. When $N$ is an odd
number, this is equivalent to being irreducible over reals, which is what we
need. When $N$ is an even number, on the other hand, it may happen that an
$N$-dimensional representation which are reducible over complex numbers is
irreducible over the reals. In that case, one should instead calculate the
Molien function and then check that the coefficient of the $z^{2}$ term is 1.
We will discuss the Molien function in Section 7.1. In this paper we have
$N=5$ so we can use the simpler check based on the characters.

Next, we check that $H\notin$ groups in the pseudo code. When we do this, we
need to check that $H$ is not conjugate in ${\rm O}(N)$ to any members of the
list groups.

Let us explain how such a comparison can be done for any two matrix groups
$G_{1}$ and $G_{2}$. First we check whether the two matrix groups are
isomorphic to each other as abstract groups. This is done as follows. After
specifying the groups $G_{1}$ and $G_{2}$ using their generators (in terms of
$N\times N$ matrices), we try to build an isomorphism between them as abstract
groups:

iso:=IsomorphismGroups(G1,G2); (21)

If an isomorphism exist, the output is an isomorphism map, if not, the output
just gives “fail”. Let us assume an isomorphism
$\operatorname{iso}:\mathfrak{G}_{1}\rightarrow\mathfrak{G}_{2}$ exists. We
also have two representations $\rho_{1}:\mathfrak{G}_{1}\rightarrow{\rm O}(N)$
and $\rho_{2}:\mathfrak{G}_{2}\rightarrow O(N)$, which are nothing but the
matrix groups $G_{1}$ and $G_{2}$. Composing the isomorphism and $\rho_{2}$ we
have a second representation $\mathfrak{G}_{1}$ given by
$\rho_{2}\circ\operatorname{iso}$. By Lemma 3.1, the two matrix groups $G_{1}$
and $G_{2}$ will be conjugate if and only if the representations $\rho_{1}$
and $\rho_{2}\circ\operatorname{iso}$ of $\mathfrak{G}_{1}$ are equivalent up
to outer automorphisms. As we already explained, this can be done by comparing
their characters up to the action of the outer automorphism group of
$\mathfrak{G}_{1}$, which can all be computed in GAP.

For example, the characters of $\rho_{1}$ and
$\rho_{2}\circ\operatorname{iso}$ are computed by

List(ConjugacyClasses(G1),c->TraceMat(Representative(c)));
List(ConjugacyClasses(G1),c->TraceMat(Image(iso,Representative(c))));

##  6 Results I: Irreducible subgroups of ${\rm O}(5)$

Let us explain how we run the above algorithm for ${\rm O}(5)$. In Section 4,
we generated the list of five matrix subgroups of $O(5)$ which contains all
maximal irreducible finite subgroups of ${\rm O}(5)$. As abstract groups, they
are isomorphic to groups listed in (7), and we discussed how to obtain their
matrix embeddings. In addition, as discussed in Section 4.2, ${\rm O}(5)$ has
exactly three irreducible Lie subgroups, namely $\operatorname{SO}(5)$,
$\operatorname{SO}(3)_{T}$ and
$\operatorname{SO}(3)_{T}\times\mathbb{Z}_{2}^{{\rm O}(5)}$.

We first run the algorithm from Section 5 with groups_initial set to five
matrix subgroups corresponding to the list (7). This generates all finite
irreducible subgroups of ${\rm O}(5)$, with their group-subgroup relations.
The result is shown in Figs. 2, 3 and 4.

Let us comment on these results starting from Fig. 2. In this figure we see
that ${\rm O}(5)$ has four maximal irreducible subgroups, namely Cubic(5),
$\operatorname{SO}(5)$, $S_{6}\times\mathbb{Z}_{2}$ and
$\operatorname{SO}(3)_{T}\times\mathbb{Z}_{2}$. To avoid clutter, subgroups of
these groups are shown in Figs. 3 and 4. (We will see below how
$\operatorname{SO}(5)$, $\operatorname{SO}(3)_{T}$ and
SO(3)${}_{T}\times\mathbb{Z}^{{\rm O}(5)}_{2}$ were positioned in Figs. 3 and
4.)

Let us look at Fig. 3. We see that our algorithm discovered a plethora of
finite irreducible subgroups of Cubic(5) which were not in the original list
(7), together with their group-subgroup relations.

Groups in Fig. 3 are named according to the structure description in GAP of
the corresponding abstract group, which can be easily checked using
StructureDescription(G). In Fig. 3 (and also in Appendix B), to be in accord
with GAP, we use the notation “$G=N:H$” to denote the (inner) semi-direct
product between group $N$ and $H$. In particular, $N$ is a normal subgroup of
$G$. This is usually denoted as $N\rtimes H$. Note that the semi-direct
product notation does not always fully specify the group; we need to know a
homomorphism $\varphi:H\rightarrow\operatorname{Aut}(N)$ which may be not
unique. The “small group ID” is of course unique.

Another interesting thing to note is that there are three pairs of matrix
groups in Fig. 3 which are isomorphic as abstract groups but not conjugate
within ${\rm O}(5)$. E.g. $(\mathbb{Z}_{2})^{4}:S_{5}\color[rgb]{1,0,0}{-A}$
and $(\mathbb{Z}_{2})^{4}:S_{5}\color[rgb]{1,0,0}{-B}$ form such a pair.

Black stars and red/blue dots in Figs. 2, 3 and 4 refer to the Landau
condition and RG stability to be discussed in Section 7.

Figure 2: Maximal irreducible subgroups of ${\rm O}(5)$.

Figure 3: Irreducible subgroups of ${\rm O}(5)$. The arrows indicates group-
subgroup relations: if there is an arrow from $G$ to $H$, this means $H\subset
G$ in the sense of matrix groups. Only maximal subgroup relations are
indicated by arrows, but note that group-subgroup relations are transitive.
The GAP semidirect product notation $N:H$ is equivalent to $N\rtimes H$ more
common in physics literature. The numbers in the “[ ]” are “small group ID” of
the corresponding abstract group. We use black star to denote that the matrix
group satisfies the Landau condition. The groups denoted with red/blue dots
are the groups whose Landau theory has a stable fixed point, the color
indicating the two different fixed points \- see Section 7. The generators and
Molien functions of all these groups are given in Appendix B. Note that
$\mathbb{Z}_{2}$ factors for these subgroups do not necessarily act as
$\mathbb{Z}_{2}^{{\rm O}(5)}$.

Let us have a similar discussion for Fig. 4. Here, all finite groups are
simply related to the groups in the list (7), excluding Cubic(5). This is not
too surprising, because the classification of finite subgroups of ${\rm O}(5)$
used in Section 4.1 treats these subgroups specially.

Figure 4: Irreducible subgroups of ${\rm O}(5)$ (continued), and their group-
subgroup relations. Same notation as in Fig. 3.

Let us finally discuss how we inserted $\operatorname{SO}(5)$,
$\operatorname{SO}(3)_{T}\times\mathbb{Z}_{2}$ and $\operatorname{SO}(3)_{T}$
in these figures. To position $\operatorname{SO}(5)$, we simply check the
determinant of the matrix group generators. To position
$\operatorname{SO}(3)_{T}\times\mathbb{Z}_{2}$ and $\operatorname{SO}(3)_{T}$,
we use the fact that while $\operatorname{SO}(3)_{T}$ does not have
irreducible Lie subgroups, it has exactly one irreducible finite subgroup,
namely $A_{5}$ (as can be checked using the list of all subgroups of
$\operatorname{SO}(3)$ antoneli2006maximal ; klein1884vorlesungen  plus some
other checks). An explicit embedding of $A_{5}\subset\operatorname{SO}(3)_{T}$
can be found in Appendix A. This implies unambiguously the group-subgroup
relations between $\operatorname{SO}(3)_{T}\times\mathbb{Z}_{2}$,
$\operatorname{SO}(3)_{T}$ and other groups shown in Fig. 4.

##  7 Results II: Landau actions

It remains to analyze the Landau theories corresponding to the irreducible
subgroups listed in Figs. 3, 4. As we discussed, we can focus on the Landau
theories of the minimal I&L subgroups. Fixed points with larger symmetries, if
they exist, will be found as a part of the RG analysis. In this section we
work out the Landau theories. RG analysis is performed in the next section.

###  7.1 Landau condition

First we determine which irreducible subgroups satisfy the Landau condition
(the absence of cubic invariants). All subgroups containing
$\mathbb{Z}_{2}^{{\rm O}(5)}$ clearly do not allow cubic invariants. For other
groups, this check is done by examining the Taylor expansion of the Molien
function, defined as

$M_{\rho}(z)=\frac{1}{|G|}\sum_{g\in G}\frac{1}{\det(1-z\rho(g))}.$ (23)

Here $\rho(g)$ denotes the matrix representation of the group element $g$. In
the Taylor expansion of the Molien function, the coefficients of the monomial
$z^{n}$ counts the number of degree-$n$ invariant polynomials. The Landau
condition is satisfied if the coefficient of $z^{3}$ vanishes. The Molien
functions of finite groups can be easily calculated using the GAP command

MolienSeries(chi); (24)

The command takes as an input the character and calculates the Molien function
of the corresponding irrep.

The above covers all cases except for $\operatorname{SO}(3)_{T}$. For Lie
groups, the role of Molien function is played by the Hilbert series, see for
example feng2007counting ; hanany2010hilbert . In our case we don’t need this
technology, since $\operatorname{SO}(3)_{T}$ clearly has a cubic invariant,
given by $t_{ij}t_{jk}t_{ki}$ where $t_{ij}$ is the symmetric traceless tensor
of O(3) used to construct the embedding $\operatorname{SO}(3)_{T}\subset{\rm
O}(5)$ in App. A.

We mark the subgroups satisfying the Landau condition by a black star in Figs.
3,4.

We can now identify three minimal I&L subgroups. These are
$(\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}$ in Fig. 3 and
$S_{6}\color[rgb]{1,0,0}{-}\color[rgb]{1,0,0}{B}$ and
$A_{5}\times\mathbb{Z}^{{\rm O}(5)}_{2}$ in Fig. 4.

###  7.2 Quartic Landau theories

We will now construct Landau theory for the three minimal I&L subgroups. Let
us discuss how to do this for a given matrix subgroup $G\subset{\rm O}(N)$
explicitly. The quadratic, cubic, and quartic invariant polynomials can be
constructed from symmetric tensors $T^{(2)},T^{(3)},T^{(4)}$, which are
invariant, i.e. satisfy the following equations:

$\displaystyle g^{j_{1}}_{i_{1}}g^{j_{2}}_{i_{2}}T^{(2)}_{j_{1}j_{2}}$
$\displaystyle=T^{(2)}_{i_{1}i_{2}},$ $\displaystyle
g^{j_{1}}_{i_{1}}g^{j_{2}}_{i_{2}}g^{j_{3}}_{i_{3}}T^{(3)}_{j_{1}j_{2}j_{3}}$
$\displaystyle=T^{(3)}_{j_{1}j_{2}j_{3}},$ $\displaystyle
g^{j_{1}}_{i_{1}}g^{j_{2}}_{i_{2}}g^{j_{3}}_{i_{3}}g^{j_{4}}_{i_{4}}T^{(4)}_{j_{1}j_{2}j_{3}j_{4}}$
$\displaystyle=T^{(4)}_{j_{1}j_{2}j_{3}j_{4}},$ (25)

which should be satisfied for all $g\in G$. Since we care about fixed point in
$4-\epsilon$ dimension, we truncate the Landau theories to the quartic order
and do not consider higher-order invariants. If $G$ is finite, it’s enough to
check the above equations for a finite list of generators $g_{1},\ldots
g_{k}$. We get a finite list of linear equations, solving which we get a basis
of invariants.

Since our subgroup is irreducible, the $T^{(2)}$ equation allows a single
solution $T_{ij}^{(2)}=\delta_{ij}$. Because of the Landau condition, the
second equation has no nontrivial solutions. Solving the third equation, we
construct the quartic Landau theory. The number of solutions to this equation,
$I_{4}$, can be determined from the $z^{4}$ coefficient of the Molien
function, but to find the actual invariant tensors we have to solve this
equation.

So we have to do this for $(\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}$ in Fig. 3, and
for $S_{6}\color[rgb]{1,0,0}{-}\color[rgb]{1,0,0}{B}$ and
$A_{5}\times\mathbb{Z}^{{\rm O}(5)}_{2}$ in Fig. 4. The RG analysis will be
discussed in the next section.

####  7.2.1 $(\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}$

For $(\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}$, from the Molien series, we know
that $I_{4}=3$ \- there are three independent invariant quartic polynomials.
Using the generators listed in Appendix B, we can calculate them:

$\displaystyle P_{1}(x)$ $\displaystyle=$
$\displaystyle(x^{2}_{1}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2})^{2},$
$\displaystyle P_{2}(x)$ $\displaystyle=$ $\displaystyle
x^{4}_{1}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4}+x_{5}^{4},$ $\displaystyle P_{3}(x)$
$\displaystyle=$ $\displaystyle
x_{1}^{2}x_{2}^{2}+x_{2}^{2}x_{3}^{2}+x_{3}^{2}x_{4}^{2}+x_{4}^{2}x_{5}^{2}+x_{5}^{2}x_{1}^{2}.$
(26)

We parametrize the Landau potential as

$V(\phi)=\frac{1}{4!}\lambda_{1}P_{1}(\phi)+\frac{1}{4!}\lambda_{2}P_{2}(\phi)+\frac{1}{4!}\lambda_{3}P_{3}(\phi),$
(27)

At this point we can ask the following question. All polynomials in (26) are,
by construction, invariant under $(\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}$. But
can it be that they are actually invariant under some larger subgroup $G$
containing $(\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}$? As a matter of fact, this
indeed happens in our situation. To see that this happens, it is enough to
compute the number of quartic invariants $I_{4}$ for subgroups $G$ in Fig. 4
containing $(\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}$. Doing so one discovers that
the three subgroups “one level up”, i.e.
$((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2}$,
$(((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2})\color[rgb]{1,0,0}{-A}$
and
$(((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2})\color[rgb]{1,0,0}{-B}$
also have $I_{4}=3$. Going one level further up, the group
$(((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2})\times\mathbb{Z}_{2}^{O(5)}$
still has $I_{4}=3$, while other subgroups at this level have $I_{4}=2$. On
the next level up, all subgroups have $I_{4}<3$.

This computation implies that all polynomials in (26) are also invariant
polynomials of the largest of the above-mentioned $I_{4}=3$ subgroups, namely
$(((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2})\times\mathbb{Z}_{2}^{{\rm
O}(5)}$. Following Ref. toledano1985renormalization , we call centralizer the
maximal symmetry group of a quartic Landau theory. Any fixed point, if it
exists, will have symmetry at least as large as the centralizer, which in the
case at hand is strictly larger than the minimal irreducible subgroup we
started with. We call this phenomenon “centralizer enhancement”.

It’s good to have some idea how the discussed groups actually act. The group
$(((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2})\times\mathbb{Z}_{2}^{O(5)}$
can be generated by three generators, two of them being permutations of
coordinate axes

$(1,2,3,4,5)\quad\operatorname{and}\quad(1,5)(2,4).$ (28)

Here we are following the cycle notation to denote permutation of the five
coordinates. The third generator is given by

$X:\quad x_{1}\rightarrow-x_{1},\quad\text{and}\quad x_{i}\rightarrow
x_{i}\quad\operatorname{for}\quad i\neq 1.$ (29)

On the other hand the group $(\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}$ is generated
by the axis permutation (1,2,3,4,5) and the following second generator

$Y:\quad x_{1}\rightarrow-x_{1},\quad x_{2}\rightarrow-x_{2},\quad
x_{3}\rightarrow-x_{3},\quad x_{4}\rightarrow-
x_{4},\quad\operatorname{and}\quad x_{5}\rightarrow x_{5}$ (30)

(one checks easily that
$Y\in(((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2})\times\mathbb{Z}_{2}^{{\rm
O}(5)}$). Now, it is easy to see that polynomials (26) are invariant under
$(\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}$ and that they are also invariant under
the bigger group
$(((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2})\times\mathbb{Z}_{2}^{O(5)}$.
This provides a check of the above reasoning based on the Molien series.

The phenomenon of centralizer enhancement can be defined generally. If
$H\subset G$, and $H$ and $G$ have the same truncated Molien series up to
$z^{4}$, their quartic Landau theories will be equivalent. In other words,
there is a basis such that the quartic Landau theories for $H$ and $G$ look
exactly the same. Groups related by group-subgroup relations with the same
truncated Molien series form a directed graph; they all have equivalent
quartic Landau theories. In fact, the symmetry of the quartic Landau action
will be given by the maximal vertex of this directed graph.

It turns out that the quartic Landau theories corresponding to all the groups
in Fig. 3 all reduce to the following Landau theories (we have included the
O(5) group for demonstration purposes)

$\displaystyle{\rm O}(5)$ $\displaystyle:$
$\displaystyle\frac{1}{4!}\lambda_{1}P_{1}(x)$
$\displaystyle\operatorname{Cubic}(5)$ $\displaystyle:$
$\displaystyle\frac{1}{4!}\lambda_{1}P_{1}(x)+\frac{1}{4!}\lambda_{2}P_{2}(x)$
$\displaystyle(((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2})\times\mathbb{Z}_{2}^{O(5)}$
$\displaystyle:$
$\displaystyle\frac{1}{4!}\lambda_{1}P_{1}(x)+\frac{1}{4!}\lambda_{2}P_{2}(x)+\frac{1}{4!}\lambda_{3}P_{3}(x)$
(31)

In this table, we label the quartic Landau theories by their maximal symmetry
groups i.e. the centralizer. In fact, when we wrote down the invariant
polynomials (26), we have taken into account the above structure. In other
words, the polynomial $P_{1}(x)$ preserves the ${\rm O}(5)$ group, the
polynomial $P_{2}(x)$ breaks the $O(5)$ group to $\operatorname{Cubic}(5)$,
the polynomial $P_{3}(x)$ further breaks the Cubic(5) group to
$(((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2})\times\mathbb{Z}_{2}^{{\rm
O}(5)}$.

####  7.2.2 $S_{6}\color[rgb]{1,0,0}{-}\color[rgb]{1,0,0}{B}$ and
$A_{5}\times\mathbb{Z}^{{\rm O}(5)}_{2}$

Due to the centralizer enhancement (see Section 7.2.1), the quartic Landau
theory for $S_{6}\color[rgb]{1,0,0}{-B}$, $A_{5}\times\mathbb{Z}^{O(5)}_{2}$,
and their parents $S_{6}\times\mathbb{Z}_{2}^{{\rm O}(5)}$ and
$A_{6}\times\mathbb{Z}_{2}^{{\rm O}(5)}$ are all equivalent. This can be seen
by computing $I_{4}$ for these groups. It turns out that $I_{4}=2$ for all
these groups, which implies the above statement. So we discuss the Landau
theory for $S_{6}\times\mathbb{Z}^{{\rm O}(5)}_{2}$.

This Landau theory belongs to a family of Landau theories with the tetrahedral
symmetry $S_{N+1}\times\mathbb{Z}^{{\rm O}(N)}_{2}$ mentioned in Section 2.
The Landau theory is constructed by taking $N+1$ vectors
$e^{\alpha}_{i}\in\mathbb{R}^{N}$, $\alpha=1\ldots N+1$ such that
zia1975critical

$e^{\alpha}_{i}e^{\beta}_{i}=\frac{N+1}{N}\delta^{\alpha\beta}-\frac{1}{N}\,.$
(32)

The endpoints of these $N+1$ vectors are vertices of a hypertetrahedron in an
$N$-dimensional Euclidean space. In addition to the ${\rm O}(N)$ invariant
quartic coupling, the only other quartic coupling allowed by the symmetry is
$\left(\sum_{\alpha}e^{\alpha}_{i}e^{\alpha}_{j}e^{\alpha}_{k}e^{\alpha}_{l}\right)\phi^{i}\phi^{j}\phi^{k}\phi^{l}$.
The cubic term
$\left(\sum_{\alpha}e^{\alpha}_{i}e^{\alpha}_{j}e^{\alpha}_{k}\right)\phi_{i}\phi_{j}\phi_{k}$
is forbidden by $\mathbb{Z}^{{\rm O}(N)}_{2}$.

Applying this construction for $N=5$, we get the
$S_{6}\times\mathbb{Z}_{2}^{O(5)}$ Landau theory

$\displaystyle S_{6}\times\mathbb{Z}_{2}^{{\rm O}(5)}$ $\displaystyle:$
$\displaystyle\frac{1}{4!}\lambda_{1}P_{1}(\phi)+\frac{1}{4!}\lambda_{4}P_{4}(\phi),$
(33)

with $P_{1}(x)$ defined in (26) and
$P_{4}(\phi)=\left(\sum_{\alpha}e^{\alpha}_{i}e^{\alpha}_{j}e^{\alpha}_{k}e^{\alpha}_{l}\right)\phi^{i}\phi^{j}\phi^{k}\phi^{l}$.999This
polynomial is not particularly beautiful so we don’t report it.

To summarize, the Landau theories in Eq. (31) and Eq. (33) exhaust all the
quartic Landau theories that can potentially give us an irreducible fixed
point for $N=5$ in $d=4-\epsilon$ dimensions. We will now proceed to discuss
their RG flows.

##  8 Results III: RG analysis

At the beginning of this paper we stated our goal as classifying all real
fixed points with $N=5$ having symmetry $G$ and satisfying the conditions I
(irreducibility), L (Landau) and S (stability).

Our RG analysis will allow us to solve this goal, but at the same time to
provide a more nuanced information concerning symmetry enhancement, which we
will now describe.

###  8.1 Stability of fixed points

In the introduction we described stability S as irrelevance of all quartic
perturbations preserving the fixed point symmetry $G$. More generally, we will
say that a fixed point is $H$-stable, if all quartic perturbations preserving
symmetry $H$ are irrelevant. Otherwise we call it $H$-unstable. Here $H$ may
be a subgroup of the fixed point symmetry $G$.

This general definition is interesting, because it may happen that an RG flow
having smaller symmetry $H$ leads to a stable fixed point having larger
symmetry $G$. This is called symmetry enhancement. For this to happen without
additional fine-tuning, the fixed point has to be $H$-stable.

Having this application in mind, we will only consider $H$ which are
themselves irreducible and satisfy the Landau condition.

In general, to discuss $H$-stability of a fixed point, one has to realize it
within the Landau theory having symmetry $H$, and compute the eigenvalues of
the stability matrix

$\frac{\partial\beta_{i}}{\partial\lambda_{j}}$ (34)

at the fixed point, where $\beta_{i}$ are the beta functions and $\lambda_{i}$
are the independent couplings. The fixed point is $H$-stable if all the
eigenvalues are positive.

As we have seen in Section 7.2.1, often two groups $H_{1}\subset H_{2}$ have
the same Landau theories (centralizer enhancement). If so, stability
properties of a fixed point with respect to $H_{2}$ and $H_{1}$ will be
exactly the same. For us this means that we need to analyze $H$-stability only
with respect to 4 groups which are the symmetry groups of the four Landau
theories listed in Eqs. (31) and Eq. (33).

Given that some fixed points will occur repeatedly below, let us discuss their
stability here. The first fixed point is the free theory, which has $G=O(N)$
invariance. This fixed point is ${\rm O}(N)$-unstable (and hence $H$-unstable
for any $H\subset{\rm O}(N)$). This follows from considering the beta function
for the single ${\rm O}(N)$-invariant coupling.

Then, there is the ${\rm O}(N)$ Wilson-Fisher fixed point, which also has
$G={\rm O}(N)$. This fixed point is ${\rm O}(N)$-stable, the eigenvalue given
by

$\omega=\varepsilon+{\rm O}(\varepsilon^{2}).$ (35)

For $N\geqslant 5$, it is unstable with respect to any smaller $H$. This was
first shown in brezin1974discussion . The proof is as follows. Let us
decompose all quartic perturbations of the ${\rm O}(N)$ Wilson-Fisher fixed
point with respect to the fixed point symmetry ${\rm O}(N)$. In addition to
the ${\rm O}(N)$ invariant perturbation, there are perturbations in symmetric
traceless representations with 2 and 4 indices. 2-index perturbations are not
allowed by symmetry $H$ if it is irreducible, as we assume. Some 4-index
perturbations may be allowed by $H$, but whatever $H$ is, all of them will
have exactly the same stability matrix eigenvalue at the fixed point. This
eigenvalue can be computed using any symmetric traceless 4-index perturbation,
and it is given by brezin1974discussion  (see also henriksson2023critical ,
Table 25, line 1),

$\omega^{\prime}=\frac{N-4}{N+8}\varepsilon+{\rm O}(\varepsilon^{2}).$ (36)

###  8.2
$(((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2})\times\mathbb{Z}_{2}^{{\rm
O}(5)}$ invariant quartic Landau theory

We now proceed to the RG analysis of the
$(((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2})\times\mathbb{Z}_{2}^{{\rm
O}(5)}$ invariant quartic Landau theory given in Eq. (31). Recall that this
theory was originally derived for the symmetry group
$(\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}$, but because of the centralizer
enhancement, the symmetry of the quartic Landau theory turned out to be
larger.

Using the general one-loop beta functions (2), we get beta functions for the
couplings $\lambda_{1},\lambda_{2},\lambda_{3}$:

$\displaystyle\beta_{1}$ $\displaystyle=$
$\displaystyle-\epsilon\lambda_{1}+\frac{13\lambda_{1}^{2}}{48\pi^{2}}+\frac{\lambda_{2}\lambda_{1}}{8\pi^{2}}+\frac{\lambda_{3}\lambda_{1}}{24\pi^{2}}+\frac{\lambda_{3}^{2}}{192\pi^{2}},$
$\displaystyle\beta_{2}$ $\displaystyle=$
$\displaystyle-\epsilon\lambda_{2}+\frac{3\lambda_{2}^{2}}{16\pi^{2}}+\frac{\lambda_{1}\lambda_{2}}{4\pi^{2}}+\frac{\lambda_{3}^{2}}{192\pi^{2}},$
(37) $\displaystyle\beta_{3}$ $\displaystyle=$
$\displaystyle-\epsilon\lambda_{3}+\frac{\lambda_{3}^{2}}{32\pi^{2}}+\frac{\lambda_{2}\lambda_{3}}{8\pi^{2}}+\frac{\lambda_{1}\lambda_{3}}{4\pi^{2}}.$

They have in total 4 fixed points with real couplings:

$\displaystyle F_{1}:\qquad\lambda_{1}=\lambda_{2}=\lambda_{3}=0\,,$
$\displaystyle
F_{2}:\qquad\lambda_{1}=\frac{48\pi^{2}}{13}\epsilon,\lambda_{2}=0,\lambda_{3}=0\,,$
$\displaystyle
F_{3}:\qquad\lambda_{1}=0,\lambda_{2}=\frac{16\pi^{2}}{3}\epsilon,\lambda_{3}=0\,,$
(38) $\displaystyle
F_{4}:\qquad\lambda_{1}=\frac{16\pi^{2}}{5}\epsilon,\lambda_{3}=\frac{16\pi^{2}}{15}\epsilon,\lambda_{3}=0\,.$

Let us discuss the physical interpretation. Fixed points $F_{1}$ and $F_{2}$
preserve full ${\rm O}(5)$, these are the free theory and the ${\rm O}(5)$
Wilson-Fisher fixed point, at the lowest order in $\epsilon$. Fixed points
$F_{3}$ and $F_{4}$ preserve the Cubic(5) subgroup. Fixed point $F_{3}$ is
interpreted as 5 decoupled copies of the Ising model fixed point. Fixed point
$F_{4}$ is the $N=5$ case of the $\operatorname{Cubic}(N)$ fixed point first
mentioned in Section 2.

In addition to the above 4 real fixed points, we have 4 fixed points with
complex couplings:

$\displaystyle
F_{5}:\qquad\lambda_{1}=\frac{8}{7}\left(3-i\sqrt{3}\right)\pi^{2}\epsilon,\quad\lambda_{2}=\frac{8}{7}\left(1+i\sqrt{3}\right)\pi^{2}\epsilon,\quad\lambda_{3}=\frac{32}{7}i\sqrt{3}\pi^{2}\epsilon,$
(39) $\displaystyle
F_{7}:\qquad\lambda_{1}=\frac{8}{5}\left(3-i\sqrt{3}\right)\pi^{2}\epsilon,\quad\lambda_{2}=\frac{8}{5}i\left(\sqrt{3}+i\right)\pi^{2}\epsilon,\quad\lambda_{3}=\frac{32}{5}i\sqrt{3}\pi^{2}\epsilon,$
(40)

and their complex conjugated partners $F_{6}=F_{5}^{*}$ and $F_{8}=F_{7}^{*}$.
These fixed points have symmetry
$(((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2})\times\mathbb{Z}_{2}^{{\rm
O}(5)}$.101010Considering a finite set of polynomials $\\{P_{i}\\}$, the
symmetry group of a specific linear combination of such polynomials
$\sum_{i}\lambda_{i}P_{i}$ is defined as the group that leaves it invariant.
This is also sometimes referred to as the little group. The symmetry group, as
defined, may depend on the coefficients $\lambda_{i}$. The centralizer group,
on the other hand, is the intersection of the symmetry groups of all such
polynomials. Usually the symmetry group of a generic polynomial coincides with
the centralizer group. In rare cases this is violated and a generic polynomial
has a strictly larger symmetry group. If this happens, one expects to finds
continuous families of fixed points rather than isolated fixed points
toledano1985renormalization . In our case there are only isolated fixed
points, and so the symmetry group of all fixed points coincides with the
centralizer of the corresponding polynomial family. Since the above mentioned
fixed points are complex they are not of our primary physical interest.

Let us verify fixed point counting. We are solving 3 quadratic polynomial
equations $\beta_{i}=0$, with $i=1,2,3$. Generically the number of solutions
equals the product of degrees of the polynomials Bezout , and that’s what we
have here, which means that every solution has multiplicity one.
Mathematically this means that the Jacobian matrix
$\frac{\partial\beta_{i}}{\partial\lambda_{j}}$ is non-degenerate (the
determinant is nonzero) for every of the above solutions, as one can easily
check. This matrix is the same as the stability matrix mentioned above. Its
eigenvalues for fixed points $F_{1},\ldots,F_{4}$ will be reported below, and
we will see that they are all nonzero.

The above analysis was based on the one-loop beta function equations, and one
may ask what will happen with the above solutions when we add further loops.
In the case at hand, higher loops will not change the picture. The above
solutions will be corrected by higher integer powers of $\epsilon$, but the
number of solutions will not change and their symmetry will remain the same.
This is so precisely because all one-loop solutions have multiplicity one, and
the Jacobian is non-degenerate.111111Mathematically, this is easy to see by
applying the implicit function theorem after rescaling the couplings
$g\rightarrow\epsilon g$. For an explicit discussion, see Michel:1983in .

We note that we are not interested here in the solutions of higher-order beta
functions equations whose couplings do not go to zero as $\epsilon\rightarrow
0$.

For every fixed point in the above list, we will now examine if it is
$H$-stable with respect to some $H\subset G$ where $G$ is the fixed point
symmetry. As discussed in Section 8.1, we may restrict our attention to $H$
being one of the four groups which are symmetries of the Landau theories in
Eq. (31) and Eq. (33). Any other $H$ will have Landau theory equivalent to one
of these four theories, and the stability properties will be the same. Eq.
(33) will be studied in the next subsection; here we are interested in Eq.
(31).

At $F_{1}$ (free theory), $G_{\text{}}={\rm O}(5)$, the matrix has three
degenerate eigenvalues with $\omega_{1}=\omega_{2}=\omega_{3}=-\epsilon$. One
of the eigenvectors is ${\rm O}(5)$ invariant. This means that this fixed
point is unstable even if we take the largest $H=G_{\text{}}={\rm O}(5)$. We
have already discussed this instability generally in Section 8.1.

At $F_{2}$ (${\rm O}(5)$ Wilson-Fisher fixed point), we have

$\displaystyle\omega_{1}=\epsilon,$ $\displaystyle P_{1}(\phi),$
$\displaystyle{\rm O}(5),$ $\displaystyle\omega_{2}=-\frac{1}{13}\epsilon,$
$\displaystyle P_{1}(\phi)-7P_{2}(\phi),$
$\displaystyle\operatorname{Cubic}(5),$ (41)
$\displaystyle\omega_{3}=-\frac{1}{13}\epsilon,$ $\displaystyle
3P_{1}(\phi)-7P_{3}(\phi),$
$\displaystyle(((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2})\times\mathbb{Z}_{2}^{O(5)}.$

where the second column shows the eigenvectors and the third columns shows the
largest $H$ which allows this perturbation. We see that this fixed point is
$H$-stable only if $H=G_{\text{}}={\rm O}(5)$. This agrees with the general
discussion in Section 8.1. The degeneracy of $\omega_{2}$ and $\omega_{3}$ has
a simple reason, also discussed there — the corresponding eigenvectors belong
to the same irrep of ${\rm O}(5)$, that is the 4-index symmetric traceless
tensor irrep.

At $F_{3}$ (decoupled Ising models), the symmetry group of the fixed point is
$G_{\text{}}=\operatorname{Cubic}(5)$. We have

$\displaystyle\omega_{1}=\epsilon,$ $\displaystyle P_{2}(\phi),$
$\displaystyle\operatorname{Cubic}(5),$
$\displaystyle\omega_{2}=-\frac{1}{3}\epsilon,$ $\displaystyle
P_{1}(\phi)-P_{2}(\phi),$ $\displaystyle\operatorname{Cubic}(5),$ (42)
$\displaystyle\omega_{2}=-\frac{1}{3}\epsilon,$ $\displaystyle P_{3}(\phi),$
$\displaystyle(((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2})\times\mathbb{Z}_{2}^{{\rm
O}(5)}.$

We conclude that this fixed point is unstable even if
$H=G=\operatorname{Cubic}(5)$.

At $F_{4}$ (the cubic fixed point) we also have
$G_{\text{}}=\operatorname{Cubic}(5)$ and

$\displaystyle\omega_{1}=\epsilon,$ $\displaystyle 3P_{1}(\phi)+P_{2}(\phi),$
$\displaystyle\operatorname{Cubic}(5),$
$\displaystyle\omega_{2}=\frac{1}{15}\epsilon,$ $\displaystyle
P_{1}(\phi)-2P_{2}(\phi),$ $\displaystyle\operatorname{Cubic}(5),$ (43)
$\displaystyle\omega_{2}=-\frac{1}{13}\epsilon,$ $\displaystyle
P_{1}(\phi)-P_{2}(\phi)-4P_{3}(\phi),$
$\displaystyle(((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2})\times\mathbb{Z}_{2}^{{\rm
O}(5)}.$

Clearly this fixed point is $H$-stable for $H=G=\operatorname{Cubic}(5)$, and
is unstable for smaller $H$. This fixed point belongs to the family of fixed
points with $\operatorname{Cubic}(N)$ symmetry discussed in Section 2.

###  8.3 $S_{6}\times\mathbb{Z}_{2}^{{\rm O}(5)}$ invariant quartic Landau
theory

We now move to the Landau theory (33). The one-loop beta functions are

$\displaystyle\beta_{1}(\lambda_{1},\lambda_{2})=-\lambda_{1}\epsilon+\frac{13\lambda_{1}\text{}^{2}}{48\pi^{2}}+\frac{3\lambda_{1}\lambda_{2}\text{}}{20\pi^{2}}+\frac{27\lambda_{2}\text{}^{2}}{2500\pi^{2}},$
$\displaystyle\beta_{2}(\lambda_{1},\lambda_{2})=-\lambda_{2}\epsilon+\frac{\lambda_{1}\lambda_{2}\text{}}{4\pi^{2}}+\frac{9\lambda_{2}^{2}}{50\pi^{2}}.$
(44)

Solving the beta function equations we have three real fixed points at one-
loop:

$\displaystyle F_{1}:\qquad$ $\displaystyle\lambda_{1}=\lambda_{2}=0,$
$\displaystyle F_{2}:\qquad$
$\displaystyle\lambda_{1}=\frac{48\pi^{2}}{13}\epsilon,\lambda_{2}=0,$
$\displaystyle F_{9},F_{10}:\qquad$
$\displaystyle\lambda_{1}=2\pi^{2}\epsilon,\lambda_{2}=\frac{25\pi^{2}}{9}\epsilon.$
(45)

The first two fixed points are the free theory and the ${\rm O}(5)$ fixed
point discussed in Section 8.2. The third fixed point is a new fixed point
having $S_{6}\times\mathbb{Z}_{2}^{{\rm O}(5)}$ symmetry. It’s actually a pair
of fixed points which are degenerate at this order, since we expect four fixed
points by Bezout’s theorem. To resolve their degeneracy we go to the two-loop
order. The two-loop beta functions for the general quartic coupling
$\frac{1}{4!}\lambda_{ijkl}\phi^{i}\phi^{j}\phi^{k}\phi^{l}$ takes the form
brezin1974discussion ; wallace1975gradient ; fei2016yukawa 121212We note that
the beta function for the general quartic has recently been obtained up to six
loops bednyakov2021six .

$\beta^{\rm
2nd}_{ijkl}(\lambda)=\frac{1}{(4\pi)^{4}}\left(\frac{1}{12}\lambda_{ijkl}(H_{ii}+H_{jj}+H_{kk}+H_{ll})-\frac{1}{4}\sum_{\operatorname{perms}}\sum_{mnpr}\lambda_{ijmn}\lambda_{kmpr}\lambda_{lnpr}\right),$
(46)

where

$H_{ij}=\sum_{mnp}\lambda_{imnp}\lambda_{jmnp}.$ (47)

We do not assume the Einstein summation convention in the above formulas. The
summation $\sum_{\operatorname{perms}}$ sums over the 24 permutations of
$i,j,k,l$ indices.

In our case we get

$\displaystyle\beta_{1}^{\rm
2nd}=-\frac{29\lambda_{1}^{3}}{768\pi^{4}}-\frac{11\lambda_{2}\lambda_{1}^{2}}{320\pi^{4}}-\frac{423\lambda_{2}^{2}\lambda_{1}}{40000\pi^{4}}-\frac{81\lambda_{2}^{3}}{62500\pi^{4}},$
$\displaystyle\beta_{2}^{\rm
2nd}=-\frac{171\lambda_{2}^{3}}{8000\pi^{4}}-\frac{97\lambda_{1}\lambda_{2}^{2}}{1600\pi^{4}}-\frac{107\lambda_{1}^{2}\lambda_{2}}{2304\pi^{4}}.$
(48)

Adding this correction to the one-loop beta function, we find that the pair
$F_{9},F_{10}$ becomes a pair of complex conjugate fixed points

$\displaystyle F_{9},F_{10}:\quad$ $\displaystyle\lambda_{1}=$ $\displaystyle
2\pi^{2}\epsilon\mp\sqrt{6}\pi^{2}i\epsilon^{3/2}+O(\epsilon^{2}),\quad\lambda_{2}=\frac{25\pi^{2}}{9}\epsilon\pm\frac{25\pi^{2}}{3\sqrt{6}}i\epsilon^{3/2}+O(\epsilon^{2}).$
(49)

The same fractional power of $\epsilon$ appears in the eigenvalues of the
stability matrix.

This discussion leaves us only two real fixed points, the free theory fixed
point and the ${\rm O}(5)$ Wilson-Fisher. Their stability properties have
already been discussed in the previous sections.

##  9 Discussion

In this paper we classified all fixed points with $N=5$ scalar fields in
$4-\varepsilon$ dimensions whose symmetry group $G\subset{\rm O}(N)$ is
irreducible (thus allowing a single mass term) and satisfies the Landau
condition (thus forbidding the cubic terms). In addition, we impose the
constraints of RG stability and reality. In the end we found only two fixed
points satisfying all these conditions: the Wilson-Fisher ${\rm O}(5)$ and the
cubic fixed point having Cubic(5) symmetry. Both of these fixed points are of
course well known. So our main result is that no further fixed points of this
type exist for $N=5$.

It’s interesting to compare our analysis to the work of Toledano et al
toledano1985renormalization , who in 1985 solved the analogous problem for
$N=4$. They had to work much harder than us, for several reasons. Their list
of irreducible subgroups, group-subgroup relations, and Landau theories turned
to be much longer than for us. Many of their fixed points were degenerate at
one-loop, necessitating to go to the two-loop order. The results of their
beautiful analysis were somewhat disappointing, as for us, since they
discovered no new stable fixed points - all 4 stable fixed points of their
final list have already been known before their work.

It is important however not to despair and to continue this program to higher
values of $N$. The $N=6$ case appears within reach, because finite subgroups
of $\operatorname{SU}(4)$ have been classified blichfeldt1917finite ;
hanany2001monograph . It would be very exciting if exhaustive analysis yielded
a new stable fixed point missed by prior studies.

One may also consider an analogous problem for more general theories,
including fermions and gauge fields. Recall that the two-loop beta functions
for the most general renormalizable quantum field theories in four dimension
(quartic scalar, Yukawa, and gauge couplings) are known since the 1980’s
machacek1983two ; machacek1984two ; machacek1985two . It will be interesting
to systematically study fixed points of these theories in $d=4-\varepsilon$,
with the help of group theory results. Notice that the classification of
finite subgroups of many Lie groups are also finished, including SU(3)
miller1916theory ; blichfeldt1917finite  and SU(4) mentioned above.

Another variation of our problem would be to try to classify tri-critical
fixed points, which means fixed points with two relevant couplings (which may
be two mass terms, one mass and one cubic, or one mass and one quartic).

In this paper we only worked in perturbation theory around $d=4$. It is
interesting to inquire what may happen when the fixed points we discussed are
continued to $\epsilon=O(1)$. Some stability results we mentioned are known to
be robust with respect to such continuation. For example, the ${\rm O}(5)$
Wilson-Fisher fixed point is stable in any $2<d<4$. However sometimes two
fixed points collide in some intermediate dimension and exchange their
stability properties. Two such examples were mentioned in Remark 2.1.

There is also another way how physics near 4 dimensions may be different from
the physics in 3 dimensions. We found several fixed points, with interesting
symmetries, which had complex couplings at small $\epsilon$. More precisely
these were pairs of complex-conjugate fixed points. (Similar fixed points were
also found in toledano1985renormalization .) One may wonder if at
$\epsilon=O(1)$ some of these pairs can merge and come to the real plane.

Finally, we mention that although in this paper we focused on the ILS fixed
points, as the most important ones for physics applications, it’s also
mathematically interesting to classify all fixed points for a given $N$,
independently of their symmetry and stability properties. For recent work in
this direction see Osborn:2017ucf ; Rychkov:2018vya ; Codello:2020lta ;
Hogervorst:2020gtc ; Osborn:2020cnf . One open problem is whether, for some
$N$, there exists a scalar fixed point with smallest possible symmetry, which
is $\mathbb{Z}_{2}$. A Dom Pérignon champagne bottle offered for solving this
problem in Rychkov:2018vya  remains so far unclaimed.

###### Acknowledgements.

This work is supported by the Simons Foundation grant 733758 (Simons Bootstrap
Collaboration).

##  Appendix A Embedding SO(3) in SO(5)

In this section we discuss an embedding of SO(3) into SO(5) which gives an
irreducible subgroup, which we call $\operatorname{SO}(3)_{T}$. SO(3) has a
five dimensional representation which is the symmetric traceless two-index
tensor representation. The action is

$t\rightarrow t^{\prime}=R^{T}.t.R$ (50)

where $R\in\operatorname{SO}(3)$ and $t$ is a $3\times 3$ symmetric traceless
matrix. Parametrizing it as

$t_{ij}=\left(\begin{array}[]{ccc}x_{1}&x_{3}&x_{4}\\\ x_{3}&x_{2}&x_{5}\\\
x_{4}&x_{5}&-x_{1}-x_{2}\end{array}\right).$ (51)

We get, for each $R$, a 5-dimensional matrix

$\tilde{R}=\frac{\partial x^{\prime}_{i}}{\partial x_{j}}.$ (52)

This is not yet an embedding into $\operatorname{SO}(5)$ because the matrix
$\tilde{R}$ is not orthogonal.

The bilinear form which is preserved by $\tilde{R}$ is

$t_{ij}t_{ij}=x^{t}gx,\quad g=\left(\begin{array}[]{ccccc}4&2&0&0&0\\\
2&4&0&0&0\\\ 0&0&4&0&0\\\ 0&0&0&4&0\\\ 0&0&0&0&4\end{array}\right)\,.$ (53)

In other words, we have

$\tilde{R}.g.\tilde{R}^{T}=g\,.$ (54)

Let us perform the Cholesky decomposition:

$g=L^{t}.L,\qquad\operatorname{with}\quad
L=\left(\begin{array}[]{ccccc}2&1&0&0&0\\\ 0&\sqrt{3}&0&0&0\\\ 0&0&2&0&0\\\
0&0&0&2&0\\\ 0&0&0&0&2\end{array}\right).$ (55)

We define the new $5\times 5$ matrices by a change of basis:

$R^{{\rm O}(5)}=(L^{-1})^{t}.\tilde{R}.L^{t}\,.$ (56)

The map $R\rightarrow R^{{\rm O}(5)}$ is still a representation of SO(3). In
addition it is orthogonal – we have $(R^{{\rm O}(5)})^{t}.R^{{\rm
O}(5)}=\operatorname{Id}$. Thus we obtained the embedding of SO(3) into SO(5),
which we call SO(3)T.

It is easy to work out the images of $\operatorname{SO}(3)$ generators under
the above embedding. We have

$\displaystyle J_{1}=\left(\begin{array}[]{ccc}0&1&0\\\ -1&0&0\\\
0&0&0\end{array}\right)$ $\displaystyle\rightarrow$
$\displaystyle\left(\begin{array}[]{ccccc}0&0&-1&0&0\\\ 0&0&\sqrt{3}&0&0\\\
1&-\sqrt{3}&0&0&0\\\ 0&0&0&0&-1\\\ 0&0&0&1&0\end{array}\right),$ (65)
$\displaystyle J_{2}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&1\\\
0&-1&0\end{array}\right)$ $\displaystyle\rightarrow$
$\displaystyle\left(\begin{array}[]{ccccc}0&0&0&0&-1\\\ 0&0&0&0&-\sqrt{3}\\\
0&0&0&-1&0\\\ 0&0&1&0&0\\\ 1&\sqrt{3}&0&0&0\end{array}\right),$ (74)
$\displaystyle J_{3}=\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\
-1&0&0\end{array}\right)$ $\displaystyle\rightarrow$
$\displaystyle\left(\begin{array}[]{ccccc}0&0&0&-2&0\\\ 0&0&0&0&0\\\
0&0&0&0&-1\\\ 2&0&0&0&0\\\ 0&0&1&0&0\end{array}\right).$ (83)

In Section 6, we mentioned that $\operatorname{SO}(3)_{T}$ group has a maximal
subgroup which is isomorphic to $A_{5}$. This subgroup is generated by

$\displaystyle g_{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccc}\frac{1}{16}\left(-3\sqrt{5}-1\right)&-\frac{1}{16}\sqrt{3}\left(\sqrt{5}-1\right)&\frac{\sqrt{5}}{4}&\frac{1}{8}\left(\sqrt{5}+3\right)&\frac{1}{8}\left(3-\sqrt{5}\right)\\\
-\frac{1}{16}\sqrt{3}\left(\sqrt{5}-1\right)&\frac{1}{16}\left(3\sqrt{5}+1\right)&\frac{\sqrt{3}}{4}&-\frac{1}{8}\sqrt{3}\left(\sqrt{5}-1\right)&-\frac{1}{8}\sqrt{3}\left(\sqrt{5}+1\right)\\\
\frac{\sqrt{5}}{4}&\frac{\sqrt{3}}{4}&\frac{1}{2}&0&\frac{1}{2}\\\
\frac{1}{8}\left(\sqrt{5}+3\right)&-\frac{1}{8}\sqrt{3}\left(\sqrt{5}-1\right)&0&\frac{1}{2}&-\frac{1}{2}\\\
\frac{1}{8}\left(3-\sqrt{5}\right)&-\frac{1}{8}\sqrt{3}\left(\sqrt{5}+1\right)&\frac{1}{2}&-\frac{1}{2}&0\end{array}\right),$
(89) $\displaystyle g_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccc}\frac{1}{16}\left(3\sqrt{5}-1\right)&-\frac{1}{16}\sqrt{3}\left(\sqrt{5}+1\right)&\frac{1}{8}\left(\sqrt{5}+3\right)&\frac{1}{8}\left(3-\sqrt{5}\right)&\frac{\sqrt{5}}{4}\\\
-\frac{1}{16}\sqrt{3}\left(\sqrt{5}+1\right)&\frac{1}{16}\left(1-3\sqrt{5}\right)&-\frac{1}{8}\sqrt{3}\left(\sqrt{5}-1\right)&-\frac{1}{8}\sqrt{3}\left(\sqrt{5}+1\right)&\frac{\sqrt{3}}{4}\\\
\frac{1}{8}\left(\sqrt{5}+3\right)&-\frac{1}{8}\sqrt{3}\left(\sqrt{5}-1\right)&0&-\frac{1}{2}&-\frac{1}{2}\\\
\frac{1}{8}\left(\sqrt{5}-3\right)&\frac{1}{8}\sqrt{3}\left(\sqrt{5}+1\right)&\frac{1}{2}&-\frac{1}{2}&0\\\
-\frac{\sqrt{5}}{4}&-\frac{\sqrt{3}}{4}&\frac{1}{2}&0&-\frac{1}{2}\end{array}\right).$
(95)

##  Appendix B Generators and Molien functions

We list here the generators and Molien functions (see (23)) for all the
subgroups listed in Fig. 3 and Fig. 4. Some of the generators below are
calculated using GAP, asking for generators of an abstract group in a
5-dimensional irrep. GAP gives generators which are not yet orthogonal
matrices. This does not affect the character calculation. If we wish to
explicitly construct the invariant tensors of the group, as in Section 7, we
need to change to the orthogonal basis. To achieve that, on need to first
calculate the quadratic invariant tensor using the first equation in (25), and
then perform the Cholesky decomposition

$T^{(2)}=L^{t}.L.$

The new generators $g_{a}^{\prime}=L^{-1,t}.g_{a}.L^{t}$ are orthogonal
matrices.

We also give “small group ID” of the groups, if they have one. The Small
Groups library is a library of GAP that allows us to work with groups of small
order conveniently. The “small group ID” is a list of two integers: the first
integer equals the order of the group, while the second integer serves to
distinguish groups with the same order. Given the ID, one can easily specify
the abstract group in GAP by a command like

G1:=SmallGroup(1920,240996); (97)

To find the generators of an abstract group, with their relations, we use

gens:=SmallGeneratingSet(G1); (98)

(These are not yet matrix group generators. These may be abstract generators
with their relations, or a list of cycles if the group was defined as a group
of permutations.)

We have explained how to calculate the character table of the group in (9). To
calculate a matrix representation, one needs to first extract the group
characters from the character table:

irreps:=Irr(tbl); (99)

The following comment then calculates (generators of) the 7th irreducible
representations of the abstract group $\mathfrak{G}_{1}$,

hom:=IrreducibleRepresentationsDixon(G1, irreps[7]); (100)

The output of the above command is an isomorphism from the abstract group
$\mathfrak{G}_{1}$ to a matrix group. To find the actual matrix representation
of the generators we use

List(gens, c->Image(hom, c)); (101)

After this preamble, we give the generators and the Molien series. We start
with the finite irreducible subgroups, and consider the Lie subgroups below.

  1. 1.

$\operatorname{Cubic}(5)\text{{\color[rgb]{0.5,0,0}\color[rgb]{1,0,0}{}}}$:

$\left(\begin{array}[]{ccccc}-1&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&1&0&0\\\
0&0&0&1&0\\\
0&0&0&0&1\end{array}\right),\left(\begin{array}[]{ccccc}0&0&0&0&1\\\
1&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&1&0&0\\\
0&0&0&1&0\end{array}\right),\left(\begin{array}[]{ccccc}0&1&0&0&0\\\
1&0&0&0&0\\\ 0&0&1&0&0\\\ 0&0&0&1&0\\\ 0&0&0&0&1\end{array}\right),$ (102)
$M_{5}(z)=\frac{1}{(1-z^{2})(1-z^{4})(1-z^{6})(1-z^{8})(1-z^{10})}.$ (103)

  2. 2.

$(\mathbb{Z}_{2})^{4}:S_{5}\color[rgb]{1,0,0}{-A}$, [1920,240996]:

$\left(\begin{array}[]{ccccc}0&0&0&0&1\\\ 0&0&0&-1&0\\\ 0&0&-1&0&0\\\
-1&0&0&0&0\\\
0&-1&0&0&0\end{array}\right),\left(\begin{array}[]{ccccc}0&0&0&-1&0\\\
0&-1&0&0&0\\\ -1&0&0&0&0\\\ 0&0&1&0&0\\\ 0&0&0&0&-1\end{array}\right),$ (104)
$M_{5}(z)=\frac{1}{(1-z^{2})(1-z^{4})(1-z^{5})(1-z^{6})(1-z^{8})}.$ (105)

  3. 3.

$(\mathbb{Z}_{2})^{4}:S_{5}\color[rgb]{1,0,0}{-B}$, [1920,240996]:

$\left(\begin{array}[]{ccccc}0&0&0&0&-1\\\ 0&1&0&0&0\\\ 0&0&-1&0&0\\\
0&0&0&-1&0\\\
1&0&0&0&0\end{array}\right),\left(\begin{array}[]{ccccc}0&0&0&0&1\\\
0&0&1&0&0\\\ 0&0&0&-1&0\\\ 1&0&0&0&0\\\ 0&-1&0&0&0\end{array}\right),$ (106)
$M_{5}(z)=\frac{z^{20}-z^{15}+z^{10}-z^{5}+1}{(1-z^{2})(1-z^{4})(1-z^{5})(1-z^{6})(1-z^{8})}.$
(107)

  4. 4.

$((\mathbb{Z}_{2})^{4}:A_{5})\times\mathbb{Z}^{{\rm O}(5)}_{2},[1920,240997]:$

$\left(\begin{array}[]{ccccc}0&0&0&0&1\\\ 0&0&0&1&0\\\ 1&0&0&0&0\\\
0&0&1&0&0\\\
0&1&0&0&0\end{array}\right),\left(\begin{array}[]{ccccc}0&-1&0&0&0\\\
0&0&0&1&0\\\ 1&0&0&0&0\\\ 0&0&0&0&-1\\\
0&0&1&0&0\end{array}\right),\left(\begin{array}[]{ccccc}1&0&0&0&0\\\
0&-1&0&0&0\\\ 0&0&1&0&0\\\ 0&0&0&1&0\\\ 0&0&0&0&1\end{array}\right),$ (108)
$M_{5}(z)=\frac{z^{16}-z^{12}+z^{8}-z^{4}+1}{(1-z^{2})(1-z^{4})^{2}(1-z^{6})(1-z^{10})}.$
(109)

  5. 5.

$(((\mathbb{Z}_{2})^{4}:A_{5}):\mathbb{Z}_{4})\times\mathbb{Z}^{O(5)}_{2},[640,21536]:$

$\left(\begin{array}[]{ccccc}0&0&1&0&0\\\ 1&0&0&0&0\\\ 0&0&0&1&0\\\
0&1&0&0&0\\\
0&0&0&0&1\end{array}\right),\left(\begin{array}[]{ccccc}0&0&-1&0&0\\\
0&0&0&0&1\\\ 1&0&0&0&0\\\ 0&0&0&-1&0\\\
0&1&0&0&0\end{array}\right),\left(\begin{array}[]{ccccc}1&0&0&0&0\\\
0&1&0&0&0\\\ 0&0&-1&0&0\\\ 0&0&0&1&0\\\ 0&0&0&0&1\end{array}\right),$ (110)
$M_{5}(z)=\frac{z^{12}+z^{6}-z^{2}+1}{(1-z^{2})^{2}(1-z^{4})(1-z^{8})(1-z^{10})}.$
(111)

  6. 6.

$((\mathbb{Z}_{2})^{4}:A_{5}),[960,11358]:$

$\left(\begin{array}[]{ccccc}0&1&0&0&0\\\ 0&0&-1&0&0\\\ 0&0&0&0&-1\\\
1&0&0&0&0\\\
0&0&0&1&0\end{array}\right),\left(\begin{array}[]{ccccc}0&0&0&0&1\\\
0&0&1&0&0\\\ 0&0&0&-1&0\\\ -1&0&0&0&0\\\ 0&1&0&0&0\end{array}\right),$ (112)
$M_{5}(z)=\frac{z^{16}-z^{12}+z^{8}-z^{4}+1}{(1-z^{2})(1-z^{4})^{2}(1-z^{5})(1-z^{6})}.$
(113)

  7. 7.

$((\mathbb{Z}_{2})^{4}:A_{5}):\mathbb{Z}_{4}{\color[rgb]{1,0,0}-A},[320,1635]:$

$\left(\begin{array}[]{ccccc}0&1&0&0&0\\\ 0&0&0&0&1\\\ 0&0&-1&0&0\\\
-1&0&0&0&0\\\
0&0&0&1&0\end{array}\right),\left(\begin{array}[]{ccccc}0&0&1&0&0\\\
-1&0&0&0&0\\\ 0&0&0&0&-1\\\ 0&-1&0&0&0\\\ 0&0&0&-1&0\end{array}\right),$ (114)
$M_{5}(z)=\frac{z^{12}+z^{6}-z^{2}+1}{(1-z^{2})^{2}(1-z^{4})(1-z^{5})(1-z^{8})}.$
(115)

  8. 8.

$((\mathbb{Z}_{2})^{4}:A_{5}):\mathbb{Z}_{4}{\color[rgb]{1,0,0}{-B}},[320,1635]:$

$\left(\begin{array}[]{ccccc}0&0&0&1&0\\\ 1&0&0&0&0\\\ 0&0&0&0&1\\\
0&0&1&0&0\\\
0&1&0&0&0\end{array}\right),\left(\begin{array}[]{ccccc}0&0&-1&0&0\\\
1&0&0&0&0\\\ 0&0&0&1&0\\\ 0&-1&0&0&0\\\ 0&0&0&0&-1\end{array}\right),$ (116)
$M_{5}(z)=\frac{z^{16}-z^{14}-z^{11}+z^{10}+z^{9}+z^{7}+z^{6}-z^{5}-z^{2}+1}{(1-z^{2})^{2}(1-z^{4})(1-z^{5})(1-z^{8})}.$
(117)

  9. 9.

$(((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2})\times\mathbb{Z}^{{\rm
O}(5)}_{2},[320,1636]:$

$\left(\begin{array}[]{ccccc}0&0&0&0&-1\\\ 0&0&-1&0&0\\\ -1&0&0&0&0\\\
0&-1&0&0&0\\\
0&0&0&-1&0\end{array}\right),\left(\begin{array}[]{ccccc}0&0&0&-1&0\\\
0&0&1&0&0\\\ 0&1&0&0&0\\\ 1&0&0&0&0\\\ 0&0&0&0&-1\end{array}\right),$ (118)
$M_{5}(z)=\frac{z^{12}-z^{10}+2z^{6}-z^{2}+1}{(1-z^{2})^{2}(1-z^{4})^{2}(1-z^{10})}.$
(119)

  10. 10.

$((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2}{\color[rgb]{1,0,0}{-A}},[160,234]$:

$\left(\begin{array}[]{ccccc}0&0&0&-1&0\\\ 0&0&-1&0&0\\\ 0&-1&0&0&0\\\
1&0&0&0&0\\\
0&0&0&0&-1\end{array}\right),\left(\begin{array}[]{ccccc}0&0&0&1&0\\\
1&0&0&0&0\\\ 0&0&0&0&1\\\ 0&0&-1&0&0\\\ 0&-1&0&0&0\end{array}\right),$ (120)
$M_{5}(z)=\frac{z^{12}-z^{10}+2z^{6}-z^{2}+1}{(1-z^{2})^{2}(1-z^{4})^{2}(1-z^{5})}.$
(121)

  11. 11.

$((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}):\mathbb{Z}_{2}{\color[rgb]{1,0,0}{-B}},[160,234]$:

$\left(\begin{array}[]{ccccc}0&0&0&-1&0\\\ 0&0&-1&0&0\\\ 0&-1&0&0&0\\\
1&0&0&0&0\\\
0&0&0&0&1\end{array}\right),\left(\begin{array}[]{ccccc}0&0&0&-1&0\\\
1&0&0&0&0\\\ 0&0&0&0&1\\\ 0&0&-1&0&0\\\ 0&1&0&0&0\end{array}\right),$ (122)
$M_{5}(z)=\frac{z^{7}+2z^{6}-z^{5}-z^{2}+1}{(1-z^{2})^{2}(1-z^{4})^{2}(1-z^{5})}.$
(123)

  12. 12.

$((\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5})\times\mathbb{Z}^{{\rm
O}(5)}_{2},[160,235]:$

$\left(\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&1&0&0\\\
0&0&0&-1&0\\\
0&0&0&0&-1\end{array}\right),\left(\begin{array}[]{ccccc}0&0&0&-1&0\\\
-1&0&0&0&0\\\ 0&0&0&0&-1\\\ 0&0&-1&0&0\\\ 0&-1&0&0&0\end{array}\right),$ (124)
$M_{5}(z)=\frac{z^{8}-3z^{6}+5z^{4}-3z^{2}+1}{(1-z^{2})^{4}(1-z^{10})}.$ (125)

  13. 13.

$(\mathbb{Z}_{2})^{4}:\mathbb{Z}_{5}\text{{\color[rgb]{0.5,0,0}\color[rgb]{1,0,0}{}}}$,
[80,49]:

$\left(\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&1&0&0\\\
0&0&0&-1&0\\\
0&0&0&0&-1\end{array}\right),\left(\begin{array}[]{ccccc}0&0&0&0&1\\\
1&0&0&0&0\\\ 0&1&0&0&0\\\ 0&0&-1&0&0\\\ 0&0&0&-1&0\end{array}\right),$ (126)
$M_{5}(z)=\frac{z^{8}-3z^{6}+5z^{4}-3z^{2}+1}{(1-z^{2})^{4}(1-z^{5})}.$ (127)

  14. 14.

$S_{6}\times\mathbb{Z}^{{\rm O}(5)}_{2},[1440,5842]$:

$\left(\begin{array}[]{ccccc}0&1&1&0&0\\\ -1&1&1&-1&0\\\ 0&0&0&1&0\\\
-1&0&1&0&0\\\
0&1&0&-1&-1\end{array}\right),\left(\begin{array}[]{ccccc}0&-1&0&0&0\\\
0&-1&0&1&1\\\ 0&0&0&-1&0\\\ 0&0&1&0&1\\\ 1&-1&-1&1&0\end{array}\right),$ (128)
$M_{5}(z)=\frac{z^{8}+1}{(1-z^{2})(1-z^{4})(1-z^{6})^{2}(1-z^{10})}.$ (129)

  15. 15.

$S_{6}\color[rgb]{1,0,0}{-A}$, [720, 763]:

$\left(\begin{array}[]{ccccc}0&-1&0&0&0\\\ 0&-1&0&1&1\\\ 0&0&0&-1&0\\\
0&0&1&0&1\\\
1&-1&-1&1&0\end{array}\right),\left(\begin{array}[]{ccccc}0&-1&-1&0&0\\\
1&-1&-1&1&0\\\ 0&0&0&-1&0\\\ 1&0&-1&0&0\\\ 0&-1&0&1&1\end{array}\right),$
(130) $M_{5}(z)=\frac{1}{(1-z^{2})(1-z^{3})(1-z^{4})(1-z^{5})(1-z^{6})}.$
(131)

This irrep of $S_{6}$ is sometimes called the “standard” irrep.

  16. 16.

$S_{6}\color[rgb]{1,0,0}{-B}$, [720, 763]:

$\left(\begin{array}[]{ccccc}0&1&1&0&0\\\ -1&1&1&-1&0\\\ 0&0&0&1&0\\\
-1&0&1&0&0\\\
0&1&0&-1&-1\end{array}\right),\left(\begin{array}[]{ccccc}0&1&0&0&0\\\
0&1&0&-1&-1\\\ 0&0&0&1&0\\\ 0&0&-1&0&-1\\\ -1&1&1&-1&0\end{array}\right),$
(132)
$M_{5}(z)=\frac{-z^{17}+z^{15}+z^{14}-z^{11}-z^{10}-z^{9}+z^{8}+z^{7}+z^{6}-z^{3}-z^{2}+1}{(1-z^{2})^{2}(1-z^{3})(1-z^{4})(1-z^{5})(1-z^{6})}.$
(133)

  17. 17.

$A_{6}\times\mathbb{Z}^{{\rm O}(5)}_{2}$, [720, 766]:

$\left(\begin{array}[]{ccccc}0&-1&0&1&1\\\ -1&1&1&0&0\\\ 0&-1&0&0&1\\\
-1&0&1&0&0\\\
0&1&1&0&0\end{array}\right),\left(\begin{array}[]{ccccc}0&-1&-1&1&0\\\
1&0&-1&0&-1\\\ -1&0&0&0&0\\\ 0&1&0&0&-1\\\ 1&0&-1&0&0\end{array}\right),$
(134)
$M_{5}(z)=\frac{z^{20}+z^{18}+z^{8}+1}{(1-z^{2})(1-z^{4})(1-z^{6})^{2}(1-z^{10})}.$
(135)

  18. 18.

$S_{5}\times\mathbb{Z}^{{\rm O}(5)}_{2}$, [240, 189]:

$\left(\begin{array}[]{ccccc}0&1&0&-1&0\\\ 0&0&1&0&1\\\ 0&0&-1&0&0\\\
-1&0&1&0&1\\\
0&1&1&0&0\end{array}\right),\left(\begin{array}[]{ccccc}0&0&-1&0&0\\\
0&-1&0&1&1\\\ 0&0&-1&0&-1\\\ 1&-1&-1&1&0\\\ 0&-1&0&0&1\end{array}\right),$
(136)
$M_{5}(z)=\frac{z^{20}+z^{18}+z^{16}+2z^{14}+2z^{12}+z^{10}+2z^{8}+z^{6}+1}{(1-z^{2})(1-z^{4})(1-z^{6})^{2}(1-z^{10})}.$
(137)

  19. 19.

$A_{6}\color[rgb]{1,0,0}{}$, [360, 118]:

$\left(\begin{array}[]{ccccc}0&0&-1&0&0\\\ 0&1&0&0&-1\\\ 0&0&0&0&1\\\
-1&1&1&0&0\\\
0&1&0&-1&-1\end{array}\right),\left(\begin{array}[]{ccccc}0&1&1&-1&0\\\
1&-1&-1&0&0\\\ 0&1&1&0&0\\\ 1&0&-1&0&-1\\\ 0&-1&0&0&0\end{array}\right),$
(138)
$M_{5}(z)=\frac{z^{12}-z^{9}+z^{6}-z^{3}+1}{(1-z^{2})(1-z^{3})^{2}(1-z^{4})(1-z^{5})}.$
(139)

  20. 20.

$S_{5}\color[rgb]{1,0,0}{-A}$, [120, 34]:

$\left(\begin{array}[]{ccccc}0&0&0&1&0\\\ 0&0&-1&0&0\\\ 0&-1&0&1&0\\\
-1&0&0&0&0\\\
0&0&0&0&1\end{array}\right),\left(\begin{array}[]{ccccc}0&1&0&0&-1\\\
1&-1&-1&1&0\\\ 0&1&0&-1&-1\\\ 1&0&-1&0&0\\\ 0&-1&-1&1&0\end{array}\right),$
(140)
$M_{5}(z)=\frac{z^{12}+z^{10}+z^{9}+z^{8}+z^{6}+1}{(1-z^{2})(1-z^{3})(1-z^{4})(1-z^{5})(1-z^{6})}.$
(141)

  21. 21.

$S_{5}\color[rgb]{1,0,0}{-B}$, [120,34]:

$\left(\begin{array}[]{ccccc}0&1&0&0&0\\\ -1&0&1&0&1\\\ 1&0&-1&0&0\\\
0&-1&0&1&1\\\
-1&0&0&0&0\end{array}\right),\left(\begin{array}[]{ccccc}0&-1&0&0&1\\\
-1&1&1&-1&0\\\ 0&-1&0&1&1\\\ -1&0&1&0&0\\\ 0&1&1&-1&0\end{array}\right),$
(142)
$M_{5}(z)=\frac{z^{14}-z^{13}+z^{12}-z^{11}+z^{10}-z^{9}+2z^{8}-z^{7}+2z^{6}-z^{5}+z^{4}-z^{3}+z^{2}-z+1}{(1-z)(1-z^{3})(1-z^{4})(1-z^{5})(1-z^{6})}.$
(143)

  22. 22.

$A_{5}\times\mathbb{Z}^{{\rm O}(5)}_{2}$, [120, 35]:

$\left(\begin{array}[]{ccccc}-1&1&1&-1&0\\\ 0&0&-1&0&0\\\ 0&1&1&-1&0\\\
0&0&0&0&1\\\
0&-1&-1&0&0\end{array}\right),\left(\begin{array}[]{ccccc}0&-1&0&1&0\\\
1&-1&-1&0&0\\\ 0&0&0&1&0\\\ 1&0&-1&0&-1\\\
0&-1&-1&0&0\end{array}\right),\left(\begin{array}[]{ccccc}0&0&-1&0&0\\\
1&-1&-1&0&0\\\ -1&0&0&0&0\\\ 0&-1&0&0&1\\\ 1&-1&-1&1&0\end{array}\right),$
(144)
$M_{5}(z)=\frac{2z^{18}+z^{14}+3z^{12}+z^{10}+2z^{8}+2z^{6}+z^{4}-z^{2}+1}{(1-z^{2})^{2}(1-z^{6})^{2}(1-z^{10})}.$
(145)

  23. 23.

$A_{5}$, [60,5]:

$\left(\begin{array}[]{ccccc}0&1&1&-1&0\\\ -1&0&1&0&1\\\ 1&0&0&0&0\\\
0&-1&0&0&1\\\
-1&0&1&0&0\end{array}\right),\left(\begin{array}[]{ccccc}1&-1&-1&1&0\\\
0&0&1&0&0\\\ 0&-1&-1&1&0\\\ 0&0&0&0&-1\\\ 0&1&1&0&0\end{array}\right),$ (146)
$M_{5}(z)=\frac{z^{10}-z^{8}+z^{6}+z^{5}+z^{4}-z^{2}+1}{(1-z^{2})^{2}(1-z^{3})^{2}(1-z^{5})}.$
(147)

We now give the corresponding information about the Lie subgroups. The Hilbert
series $H(z)$ provides in this case information analogous to the Molien series
of finite groups.

  * •

SO(5): The generators are given by the anti-symmetric matrices
$(R_{ij})^{kl}=\delta_{i}^{k}\delta_{j}^{l}-\delta_{i}^{l}\delta_{j}^{k}$,
with $i\neq j$. The Hilbert series is

$H(z)=1/(1-z^{2}).$ (148)

  * •

$\operatorname{SO}(3)_{T}:$ The generators $J_{1},J_{2}$ and $J_{3}$ of
$\operatorname{SO}(3)_{T}\subset\operatorname{SO}(5)$ are given in (83). The
Hilbert series is pinto2017hilbert

$H(z)=\frac{1}{(1-z^{2})(1-z^{3})}.$ (149)

  * •

$\operatorname{SO}(3)_{T}\times\mathbb{Z}^{{\rm O}(5)}_{2}:$ The nontrivial
$\mathbb{Z}^{{\rm O}(5)}_{2}$ element acts as

$\left(\begin{array}[]{ccccc}-1&0&0&0&0\\\ 0&-1&0&0&0\\\ 0&0&-1&0&0\\\
0&0&0&-1&0\\\ 0&0&0&0&-1\end{array}\right).$ (150)

The Hilbert series is

$H(z)=\frac{1}{1-z^{2}-z^{6}+z^{8}}.$ (151)

##  Appendix C GAP basics

In this appendix, we collect some basic usage of GAP. To specify the finite
group that one wish to work with, one can either use the Small Group Id (if
known) as explained in (97), or use the built-in commands if the group has a
commonly known name, such as

g:=SymmetricGroup(5); (152)

for the symmetric group $S_{5}$ and

g:=AlternatingGroup(5); (153)

for the alternating group $A_{5}.$ To find the generators of the group, one
can use

GeneratorsOfGroup(g); (154)

The resulting set is expressed in terms of abstract group elements; in the
case of groups consisting of permutations, the set will be a list of
permutations written in cycle notations.

For two groups $\mathfrak{G}_{1}$ and $\mathfrak{G}_{2}$, one can check
whether they are isomorphic, as explained in (21).

To calculate and display the character table, one should use the command in
(9). One can also calculate the (second) Frobenius–Schur indicator the irreps
as in (10).

To figure out a matrix representation of a irreps, one need to use (100) and
(101). Before that, one need to extract characters of the irreps from the
character table, as in (99). One can also easily find the Molien function of
an irrep using (24).

For a group $\mathfrak{G}$, one can find all its maximal subgroups, up to
conjugation, as was explained in (18).

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