#  Generalised Umbral Moonshine

Miranda C. N. Cheng<EMAIL_ADDRESS>(On leave from CNRS, France.) Korteweg-de
Vries Institute for Mathematics, Amsterdam, the Netherlands  Institute of
Physics, University of Amsterdam, Amsterdam, the Netherlands  Paul de Lange
<EMAIL_ADDRESS>Institute of Physics, University of Amsterdam, Amsterdam, the
Netherlands  Department of Physics and Astronomy, University of Kentucky,
Lexington, KY 40506, USA  Daniel P. Z. Whalen<EMAIL_ADDRESS>Korteweg-de
Vries Institute for Mathematics, Amsterdam, the Netherlands  Institute of
Physics, University of Amsterdam, Amsterdam, the Netherlands  Stanford
Institute for Theoretical Physics
Department of Physics and Theory Group, SLAC
Stanford University, Stanford, CA 94305, USA

###### Abstract

Umbral moonshine describes an unexpected relation between 23 finite groups
arising from lattice symmetries and special mock modular forms. It includes
the Mathieu moonshine as a special case and can itself be viewed as an example
of the more general moonshine phenomenon which connects finite groups and
distinguished modular objects. In this paper we introduce the notion of
generalised umbral moonshine, which includes the generalised Mathieu moonshine
[1] as a special case, and provide supporting data for it. A central role is
played by the deformed Drinfel’d (or quantum) double of each umbral finite
group $G$, specified by a cohomology class in $H^{3}(G,U(1))$. We conjecture
that in each of the 23 cases there exists a rule to assign an infinite-
dimensional module for the deformed Drinfel’d double of the umbral finite
group underlying the mock modular forms of umbral moonshine and generalised
umbral moonshine. We also discuss the possible origin of the generalised
umbral moonshine.

###### Contents

  1. 1 Introduction
  2. 2 Background
    1. 2.1 Generalised Monstrous Moonshine
    2. 2.2 Umbral Moonshine
    3. 2.3 Deformed Drinfel’d Double and its Representations
  3. 3 Generalised Umbral Moonshine
  4. 4 Groups, Cohomologies, and Representations
    1. 4.1 Group Cohomology
      1. 4.2 The 3-Cocycles
      2. 4.3 The Rank Two Subgroups
      3. 4.4 Projective Representations
      4. 5 The Generalised Umbral Mock Modular Forms
        1. 5.1 Old Groups
        2. 5.2 New Groups
          1. 6 Summary and Discussion
          2. A Examples
            1. A.1 $\ell$=3, 4B Twist
            2. A.2 $\ell$=4, 2B Twist
            3. A.3 $\ell$=6+3, 3B Twist

##  1 Introduction

Moonshine is a relation between finite groups and modular objects. The study
of this relation started with the so-called monstrous moonshine [2] and has
recently been revived by the discovery of Mathieu moonshine [3, 4, 5, 6, 7]
and umbral moonshine[8, 9], which recovers Mathieu moonshine as a special
case. While in moonshine one considers functions indexed by elements of a
finite group, in generalised moonshine the relevant functions are indexed by
pairs of commuting elements. The latter is a generalisation in that it
recovers the former when the first element of the pair is set to be the
identity element of the group. In this work we introduce generalised umbral
moonshine, recovering the generalised Mathieu moonshine [1] as a special case.
With this we hope to contribute to the understanding of moonshine in the
following two ways: 1) by providing more examples of the moonshine relation,
in particular working towards a complete analysis of umbral moonshine, and 2)
by shedding light on the underlying structure of umbral moonshine which is
still obscure.

The discovery of moonshine was initiated in the late 1970’s with the
realisation [2] that there must be a representation $V^{\natural}_{n}$ of the
monster group responsible for each of the Fourier coefficients of a set of
distinguished modular functions. These modular functions — the Hauptmoduls
$J_{g}$ — are indexed by elements $g$ of the monster group:

$J_{g}=q^{-1}+\sum_{n=1}^{\infty}a_{g}(n)\,q^{n},~{}~{}a_{g}(n)={\rm
Tr}_{V^{\natural}_{n}}g.$ (1.1)

Importantly, it was soon realised that this representation has an underlying
algebraic structure: a vertex operator algebra (VOA), or a 2d chiral conformal
field theory (CFT) in the physics language [10, 11, 12, 13]. In particular,
the monster representation $V^{\natural}_{n}$ is nothing but the space of
quantum states with energy (eigenvalue of ${\hat{H}}=L_{0}-c/24$) $n$ of a
chiral CFT — the so-called monster CFT — that has monster symmetry.

Given a chiral conformal field theory with a discrete symmetry group $G$, one
can also consider orbifolding the theory by subgroups of $G$. This leads to
the concept of twisted sectors, which have symmetries given by elements of the
group $G$ that commute with the twisting subgroup. Considering the action of
this remaining symmetry group on the twisted sector quantum states leads to
the twisted-twined partition functions indexed by two commuting elements — the
twisting and the twining elements — of the original group $G$. At least
heuristically, the existence of these twisted-twined partition functions is
believed to be a conceptual partial explanation for Norton’s generalised
monstrous moonshine [14, 15], which we will review in more detail in §2.1.

The landscape of moonshine has changed dramatically since the observation of
Eguchi, Ooguri, and Tachikawa in 2010 [3]. These authors pointed out a
surprising empirical relation between the elliptic genus of $K3$ surfaces and
the sporadic Mathieu group $M_{24}$. The ensuing development has led to the
discovery of 23 cases of umbral moonshine, which recovers the above-mentioned
$M_{24}$ relation as a special case. The important features of umbral
moonshine include the following facts: 1) the relevant functions are the so-
called mock modular forms which have a modified transformation property under
the modular group, and 2) the 23 cases are organised in terms of 23 special
(Niemeier) lattices of rank 24. In particular, their lattice symmetries
dictate the finite groups featuring in umbral moonshine.

In order to understand umbral moonshine, it is crucial to know what the
underlying algebraic structure is. Despite various recent advances[16, 17, 18,
19, 20, 21], the underlying structure for umbral moonshine is still obscure.
An obvious first guess is that a 2d CFT will again be the relevant structure,
especially given the CFT context in which the first case of umbral moonshine —
the Mathieu moonshine — was uncovered. However, there is a salient
incompatibility between the modular behaviour expected from a CFT partition
function and that displayed in umbral moonshine that we will explain in §2.2.
As a result, new ingredients other than a conventional CFT/VOA are believed to
be necessary for a uniform understanding of umbral moonshine. The quest for
this new structure constitutes one of our motivations to study generalised
umbral moonshine, which establishes the central role played by the deformed
Drinfel’d double. Note that the importance of the third group cohomology in
moonshine has been suggested by Terry Gannon and nicely demonstrated in [1] in
the context of generalised Mathieu moonshine.

The rest of the paper is organised as follows. To describe our results, we
begin in §2 by recalling the (generalised) monstrous moonshine, umbral
moonshine, and reviewing the basic properties of the deformed Drinfel’d double
of a finite group and its representations. In §3 we conjecture that there is a
way to assign an infinite-dimensional module for the deformed Drinfel’d double
of the umbral group underlying each of the 23 cases of generalised umbral
moonshine and unpack this conjecture in a more explicit form in terms of the
six conditions of generalised umbral moonshine. To provide evidence and
explicit data for this conjecture, in §4 we present the group theoretic
underpinnings of the generalised umbral moonshine and exposit our methodology
and results. In §5 we turn our attention to the modular side of moonshine and
present our proposal for the twisted-twined functions of generalised umbral
moonshine. In §6 we close the paper with a summary and discussions. To
illustrate the content of the generalised umbral moonshine conjecture, in the
appendix we provide the expansion of three examples of the $g$-twisted
generalised moonshine functions belonging to three different lambencies, as
well as the decompositions of the first few homogeneous components of the
corresponding (conjectural) generalised umbral moonshine module into
irreducible projective representations.

We also include two files as attachments on the ArXiv; the first describes the
explicit construction of the umbral groups that we employ, and the second
describes the construction of the 3-cocycles.

##  2 Background

In this section we provide the relevant background for generalised umbral
moonshine by reviewing the theories of generalised monstrous moonshine and
umbral moonshine. We also review the basic properties of the deformed
Drinfel’d double of a finite group and its representation theory.

###  2.1 Generalised Monstrous Moonshine

The monster (or the Fischer–Griess monster) group $\mathbb{M}$ is the largest
sporadic group. Monstrous moonshine connects the representation theory of the
monster group and certain distinguished modular functions, namely the
Hauptmoduls for genus zero subgroups of
$\operatorname{PSL}_{2}({\mathbb{R}})$111A discrete subgroup
$\Gamma\subset\operatorname{PSL}_{2}({\mathbb{R}})$ is said to be genus zero
if its fundamental domain on the upper-half plane, when suitabley
compactified, is a genus zero Riemann surface. A function is said to be a
Hauptmodul for $\Gamma$ if it is an isomorphism from the compactified
fundamental domain to the Riemann sphere. . In particular, the Fourier
coefficients of the modular $J$-invariant are equated with the dimensions of
the homogeneous components of a distinguished infinite-dimensional
${\mathbb{Z}}$-graded module for the monster group. This module was
constructed (conjecturally) by Frenkel, Lepowsky, and Meurman [12, 13]. This
module possesses the structure of a VOA and is later proven by Borcherds the
be the monstrous module[22]. In this way, the $q$-powers of the $J$-invariant
acquire the interpretation as the $L_{0}$-eigenvalues of the states in the
corresponding chiral conformal field theory shifted by $-1$. Here $-1=-c/24$,
where $c$ is the central charge of the Virasoro algebra. More precisely, if we
label by $V^{\natural}=\oplus_{n\geq-1}V^{\natural}_{n}$ the Hilbert space of
quantum states of the monstrous chiral conformal field theory, we have

${\rm Tr}_{V^{\natural}}q^{L_{0}-c/24}:=\sum_{n\geq-1}q^{n}{\rm
dim}V^{\natural}_{n}=J(\tau)=q^{-1}+196884\,q+21493760\,q^{2}+\dots,$ (2.1)

where $q:=e^{2\pi i\tau}$. More generally, for any $g\in\mathbb{M}$ the
corresponding twined partition function, given by ${\rm
Tr}_{V^{\natural}}gq^{L_{0}-c/24}$, coincides with the Hauptmodul
$J_{g}(\tau)=q^{-1}+O(q)$ of a genus zero subgroup $\Gamma_{g}$ of ${\rm
PSL}_{2}({\mathbb{R}})$.

An important way to construct new chiral CFTs is to “orbifold” a chiral CFT
that has a discrete symmetry group $G$[13, 23, 24, 25] by combining
$g$-twisted modules for $g\in G$. The $g$-twisted sector Hilbert space,
denoted by ${\cal H}_{g}$, has remaining symmetries given by elements of $G$
commuting with $g$. By considering these symmetries, one can define the
twisted-twined partition function

$Z_{(g,h)}(\tau)={\rm Tr}_{{\cal H}_{g}}h\,q^{L_{0}-c/24}$ (2.2)

for all commuting pairs $(g,h)$ of $G$. Recall that in a (non-chiral) orbifold
conformal field theory the above quantity is associated to a torus path
integral interpretation with $g$-boundary condition for the spatial circle and
$h$-boundary condition for the temporal circle. Hence, for any
$\gamma=\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)\in{\rm
SL}_{2}({\mathbb{Z}})$ the transformation $\tau\mapsto{a\tau+b\over c\tau+d}$
of the torus complex structure is equivalent to the transformation
$(g,h)\mapsto(g^{\prime},h^{\prime})=(g^{a}h^{c},g^{b}h^{d})$ of the twisting-
twining elements. When working with a chiral theory, this consideration then
leads to the expectation that $Z_{(g,h)}(\tfrac{a\tau+b}{c\tau+d})$ and
$Z_{(g^{\prime},h^{\prime})}(\tau)$ coincide up to a phase. Indeed, eight
years after the monstrous moonshine conjecture, Norton proposed the following.

Generalised Monstrous Moonshine Conjecture [14] (revised in [15]): There
exists a rule that assigns to each element $g$ of the monster group
$\mathbb{M}$ a graded projective representation
$V(g)=\bigoplus_{n\in{\mathbb{Q}}}V(g)_{n}$ of the centraliser group
$C_{\mathbb{M}}(g)$, and to each pair $(g,h)$ of commuting elements of
$\mathbb{M}$ a holomorphic function $T_{(g,h)}$ on the upper-half plane
${\mathbb{H}}$, satisfying the following conditions:

  1. I

For any $\left(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\right)\in{\rm
SL}_{2}({\mathbb{Z}})$, $T_{(g^{a}h^{c},g^{b}h^{d})}(\tau)$ is proportional to
$T_{(g,h)}(\tfrac{a\tau+b}{c\tau+d})$.

  2. II

The function $T_{(g,h)}$ is either constant or a Hauptmodul for some genus
zero congruence group.

  3. III

The function $T_{(g,h)}$ is invariant up to constant multiplication under
simultaneous conjugation of the pair $(g,h)$ in ${\mathbb{M}}$.

  4. IV

There is some lift $\tilde{h}$ of $h$ to a linear transformation on $V(g)$
such that

$T_{(g,h)}(\tau)=\sum_{n\in{\mathbb{Q}}}{\rm
Tr}_{V(g)_{n}}\tilde{h}\,q^{n-1}.$ (2.3)

  5. V

The function $T_{(g,h)}$ coincides with the $J$-invariant if and only if
$g=h=e\in\mathbb{M}$ .

The proof of this conjecture has been recently announced [26], building on
previous work [28, 29, 27, 30, 31]. See also [32] for a previous result and
[26] for a more in-depth review of the literature. With the crucial exception
of the genus zero property (II), all the above properties of generalised
moonshine can be understood in the physical framework of holomorphic orbifolds
along the lines discussed above [33].

###  2.2 Umbral Moonshine

Historically, the study of umbral moonshine started with the discovery of
Mathieu moonshine, initiated by a remarkable observation made in the context
of $K3$ conformal field theories in [3]. The Mathieu moonshine can be phrased
as the fact that the Fourier coefficients of a set of mock modular forms can
be equated with the group characters of a distinguished infinite-dimensional
${\mathbb{Z}}$-graded module for $M_{24}$. Later it was realised in [8, 9]
that the Mathieu moonshine is but an instance of a larger structure, which was
named umbral moonshine. Umbral moonshine is labelled by Niemeier lattices, the
unique (up to isomorphism) 23 even self-dual positive-definite lattices in 24
dimensions with a non-trivial root system. Recall that the root system is the
sub-lattice generated by the root verctos, the lattice vectors of norm squared
2. These 23 Niemeier lattices are uniquely specified by their root systems,
which are precisely the 23 unions of simply-laced (ADE) root systems with the
same Coxeter number whose ranks are 24. The relevant finite groups, the so-
called umbral groups, are given by the automorphism of the corresponding
lattice quotiented by the Weyl reflections with respect to the lattice root
vectors.

It turns out to be natural to associate a genus zero subgroup of ${\rm
SL}_{2}({\mathbb{Z}})$ of the form $\Gamma_{0}(m)+e,f,\dots$ to each of the 23
Niemeier lattices [9, 34], where $\Gamma_{0}(m)+e,f,\dots$ denotes the
subgroup of ${\rm SL}_{2}({\mathbb{R}})$ obtained by attaching the
corresponding Atkin–Lenher involutions $W_{e}$, $W_{f}$, $\dots$ to the
congruence subgroup $\Gamma_{0}(m)$. Following [9] we use the shorthand
$\ell=m+e,f,\dots$ for the genus zero group $\Gamma_{0}(m)+e,f,\dots$ and call
$\ell$ the lambency of the corresponding instance of umbral moonshine. The
level $m$ of the group $\Gamma_{0}(m)+e,f,\dots$ coincides with the index of
the corresponding mock Jacobi forms, so we say that $m$ is the index of that
lambency. We also observe that $m$ is the Coxeter number of the root system of
the corresponding Niemeier lattice.

$\boxed{\begin{array}[]{c}\text{ meromorphic}\\\ \text{ Jacobi forms}\\\
~{}\psi^{(\ell)}_{(g,h)}\end{array}}\xrightarrow[\text{the polar
part}]{\text{subtracting}}\boxed{\begin{array}[]{c}\text{mock }\\\
\text{Jacobi forms }\\\
\tilde{\psi}^{(\ell)}_{(g,h)}\end{array}}\xrightarrow[\text{into theta
functions}]{\text{decomposing}}\boxed{\begin{array}[]{c}\text{vector-valued
}\\\ \text{mock modular forms }\\\
H^{(\ell)}_{(g,h)}=\left(H^{(\ell)}_{(g,h),r}\right)\end{array}}$

(2.4)

There are a few ways to view the modular objects of umbral moonshine, which
are summarised in diagram (2.4) and which we will now explain. We start by
defining a number of actions on functions defined on
${\mathbb{H}}\times{\mathbb{C}}$ that will be used in the modularity criteria.
For a given $m\in{\mathbb{Z}}_{+}$ we defined the index $m$ (generalised)
elliptic action by222Note that usually the elliptic action is only defined for
$(\lambda,\mu)\in{\mathbb{Z}}^{2}$, in which case the factor
$\operatorname{e}(m\lambda\mu)$ is unity and is not included in the standard
definition.

$(\phi\big{|}_{m}(\lambda,\mu))(\tau,\zeta):=\phi(\tau,\zeta+\lambda\tau+\mu)\operatorname{e}(m(\lambda^{2}\tau+2\lambda\zeta+\lambda\mu))$
(2.5)

for any $(\lambda,\mu)\in{\mathbb{R}}^{2}$. Here and in the rest of the paper
we will write $\operatorname{e}(x):=e^{2\pi ix}$. Moreover, for
$2k\in{\mathbb{Z}}$ and $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in{\rm SL}_{2}({\mathbb{Z}})$ we define the
modular transformation

$\displaystyle(\phi\big{\lvert}_{k,m}\gamma)(\tau,\zeta)$
$\displaystyle:=\phi\left(\frac{a\tau+b}{c\tau+d},\frac{\zeta}{c\tau+d}\right)\frac{1}{(c\tau+d)^{{k}}}\operatorname{e}\left(-\frac{cm\zeta^{2}}{c\tau+d}\right).$
(2.6)

It is easy to check that the elliptic and modular operators satisfy

$\phi\lvert_{m}(x,y)\lvert_{k,m}\gamma=\phi\lvert_{k,m}\gamma\lvert_{m}(\gamma(x,y)).$
(2.7)

for all $\gamma\in{\rm SL}_{2}({\mathbb{Z}})$ and $(x,y)\in{\mathbb{R}}^{2}$,
where we have defined $\gamma(x,y):=(ax+cy,bx+dy)$.

Associated to each lambency $\ell=m+e,f,\dots$ and each element $g$ of the
corresponding umbral group $G^{(\ell)}$ is a so-called meromorphic Jacobi form
$\psi^{(\ell)}_{g}$. In particular, $\psi^{(\ell)}_{g}$ transforms as a weight
one index $m$ Jacobi form for a congruence subgroup $\Gamma_{g}\subseteq{\rm
SL}_{2}({\mathbb{Z}})$, possibly with a non-trivial multiplier (cf. §3 and
§5.2). Namely, we have

$\psi^{(\ell)}_{g}\big{\lvert}_{m}(\lambda,\mu)=\psi^{(\ell)}_{g}$ (2.8)

and

$\varsigma_{g}(\gamma)\psi^{(\ell)}_{g}\big{\lvert}_{1,m}\gamma=\psi^{(\ell)}_{g}$
(2.9)

for all $(\lambda,\mu)\in{\mathbb{Z}}^{2}$, $\gamma\in\Gamma_{g}$, and for
some $\varsigma_{g}:\Gamma_{g}\to{\mathbb{C}}^{\times}$. Moreover,
$\psi^{(\ell)}_{g}$ is meromorphic as a function
$\zeta\mapsto\psi_{g}^{(\ell)}(\tau,\zeta)$ and can have poles at most at
$2m$-torsion points, ${\zeta\in{1\over 2m}{\mathbb{Z}}+{\tau\over
2m}{\mathbb{Z}}}$.

As explained in [48, 49], meromorphic Jacobi forms such as $\psi_{g}^{(\ell)}$
lead to vector-valued mock modular forms. Recall that a holomorphic function
$f:{\mathbb{H}}\to{\mathbb{C}}$ is said to be a weight $k$ mock modular form
for $\Gamma\subseteq{\rm SL}_{2}({\mathbb{Z}})$ with shadow $g$ if $g$ is a
holomorphic modular form of weight $2-k$ such that the non-holomorphic
function

$\hat{f}(\tau):=f(\tau)+(4i)^{k-1}\int_{-\bar{\tau}}^{\infty}(z+\tau)^{-k}\overline{g(-\bar{\tau})}dz$
(2.10)

transforms as a weight $k$ modular form for $\Gamma$. To explain the relation
between meromorphic Jacobi forms and vector-valued mock modular forms in the
current context, we first define the holomorphic function
$\tilde{\psi}_{g}^{(\ell)}$ from the meromorphic function $\psi_{g}^{(\ell)}$
by subtracting its “polar part”

$\tilde{\psi}_{g}^{(\ell)}(\tau,\zeta):=\psi_{g}^{(\ell)}(\tau,\zeta)-\psi_{g}^{{(\ell)},{\it
P}}(\tau,\zeta).$ (2.11)

The polar part $\psi_{g}^{{(\ell)},{\it P}}$ is

$\psi_{g}^{{(\ell)},{\it
P}}=\sum_{a,b\in{\mathbb{Z}}/2m{\mathbb{Z}}}\chi^{(a,b)}_{g}\mu_{m}\lvert_{m}(\tfrac{a}{2m},\tfrac{b}{2m}),$
(2.12)

with $\chi^{(a,b)}_{g}\in{\mathbb{C}}$ given by the characters of certain
representations of the umbral group $G^{(\ell)}$ [9]. In the above we have
used the generalised Appell-Lerch sum

$\mu_{m}(\tau,\zeta)=-\sum_{k\in{\mathbb{Z}}}q^{mk^{2}}y^{2mk}\frac{1+yq^{k}}{1-yq^{k}},$
(2.13)

with $q:=\operatorname{e}(\tau)$ and $y:=\operatorname{e}(\zeta)$.

It follows from the mock modular property of the Appell–Lerch sum (see, for
instance, [48, 49]) that $\tilde{\psi}_{g}^{(\ell)}$ is an example of the so-
called weak mock Jacobi forms. Moreover, the elliptic transformations of
$\psi_{g}^{(\ell)}$ (cf. equation 3.1) and $\mu_{m}$ we see that
$\tilde{\psi}_{g}^{(\ell)}$ satisfies the same elliptic transformation as
$\psi_{g}^{(\ell)}$, and therefore admits the following theta function
decomposition [49]

$\tilde{\psi}_{g}^{(\ell)}(\tau,\zeta)=\sum_{r\in{\mathbb{Z}}/2m{\mathbb{Z}}}H_{g,r}(\tau)\theta_{m,r}(\tau,\zeta),$
(2.14)

where

$\theta_{m,r}(\tau,\zeta)=\sum_{k=r{\rm\;mod\;}{2m}}q^{k^{2}/4m}y^{k}.$ (2.15)

Recall that, when regarded as a vector-valued function with $2m$ components,
$\theta_{m}=(\theta_{m,r})$ transforms under $\gamma\in{\rm
SL}_{2}({\mathbb{Z}})$ as

$\rho_{m}\left(\gamma\right)\,\theta_{m}\lvert_{{1\over
2},m}\gamma=\theta_{m}$ (2.16)

with a multiplier system $\rho_{m}:{\rm SL}_{2}({\mathbb{Z}})\to
GL(2m,{\mathbb{C}})$ which can be found in, for instance, §5 of [51]. From the
modular property of $\psi_{g}^{(\ell)}$ (2.9) and the mock modular property of
the Appell–Lerch sum [48, 49]), we see that $H_{g}=(H_{g,r})$ is a weight
$1/2$ vector-valued mock modular form333In general $H_{g,r}(\tau)$ grows
exponentially near at most one cusp of the group $\Gamma_{g}$. Hence, it is
the weakly holomorphic vector-valued mock modular forms that we encounter in
generalised umbral moonshine. To avoid clutter we will skip the adjective
“weakly holomorphic” in the remaining of the paper.  for a certain
$\Gamma_{g}\subseteq{\rm SL}_{2}({\mathbb{Z}})$ that we will define in (3.6).

In the rest of the paper, we will generalise the above relation between
meromorphic Jacobi forms, mock Jacobi forms, and vector-valued mock modular
forms, to functions labelled by commuting pairs of elements of $G^{(\ell)}$.
This relation between $\psi_{(g,h)}$, $\tilde{\psi}_{(g,h)}$ and $H_{(g,h)}$
is summarised in diagram (2.4).

Just like the Hauptmoduls in monstrous moonshine, the mock modular forms
$H^{(\ell)}_{g}$ are also special as we shall explain now. Recall that
Cappelli, Itzykson and Zuber found an ADE classification of modular invariant
combinations of $A_{1}$ affine characters [35]. This classification, combined
with the ADE data from the root systems of the Niemeier lattices, leads to a
specification of the modularity property of the mock modular forms of umbral
moonshine as described in [9]. The umbral moonshine conjecture states that
these vector-valued mock modular form $H^{(\ell)}_{g}$ for $g=e$, are in fact
the unique such forms with the slowest possible growth in their Fourier
coefficients given these modularity properties. In particular,
$H_{e}^{(\ell)}$ coincides with the Rademacher sum specified by its modularity
property and its pole at the cusps[9]. See [36, 37, 38, 39] for a discussion
on the cases where $g\neq e$.

After specifying the finite group $G^{(\ell)}$ and the vector-valued mock
modular forms $H^{(\ell)}_{g}=(H_{g,r}^{(\ell)})$ for all elements $g\in
G^{(\ell)}$, the umbral moonshine conjecture states that there exists a
$G^{(\ell)}$-module whose graded characters coincide with the Fourier
expansion of $H^{(\ell)}_{g}$. Namely, there exists an infinite-dimensional
$G^{(\ell)}$-module

$K^{(\ell)}=\bigoplus_{\begin{subarray}{c}r\in{\mathbb{Z}}/2m{\mathbb{Z}}\\\
\alpha\in{\mathbb{Q}}_{>0}\end{subarray}}K^{(\ell)}_{r,\alpha}$ (2.17)

such that for all $g\in G$ and $r\in{\mathbb{Z}}/2m{\mathbb{Z}}$ we have

$\varepsilon_{m}(r)H^{(\ell)}_{g,r}(\tau)=c_{g,r}q^{-1/4m}+\sum_{\alpha\in{\mathbb{Q}}_{>0}}q^{\alpha}\text{Tr}_{K^{(\ell)}_{r,\alpha}}g.$
(2.18)

In the above equation we have defined $\varepsilon_{m}:{\mathbb{Z}}\to\\{0,\pm
1\\}$ by setting $\varepsilon_{m}(r):=\pm 1$ when $\pm
r=r^{\prime}{\rm\;mod\;}2m$ for some $0<r^{\prime}<m$, and
$\varepsilon_{m}(r)=0$ when $r=0{\rm\;mod\;}m$. The coefficient
$c_{g,r}\equiv\mp 2$ whenever $H_{g,r}(\tau)=\pm H_{g,1}(\tau)$ and vanishes
otherwise. Corresponding to the Niemeier lattice with root system $A_{1}^{24}$
is the case of umbral moonshine with $\ell=2$ and $G^{(\ell)}\cong M_{24}$. In
this case, the vector-valued mock modular forms have just a single independent
component since $H^{(2)}_{g,r}=\pm H^{(2)}_{g,1}$ for $r=\pm 1{\rm\;mod\;}{4}$
and $H^{(2)}_{g,r}=0$ for $r=0{\rm\;mod\;}{2}$, and the above conjecture
recovers that of Mathieu moonshine.

The above conjecture was proven in [41, 40], in the sense that the existence
of the module $K^{(\ell)}$ has been established using properties of (mock)
modular forms. However, apart from one case (see [20]) no modules have been
constructed. A uniform construction of the umbral moonshine modules is to the
best of our knowledge not yet in sight and is expected to be the key to a true
understanding of this new moonshine phenomenon. One of the most puzzling
features in this regard is the non-vanishing weight of the umbral moonshine
function: if one assumes that the umbral moonshine module, just like the
monstrous moonshine module, possesses the structure of a conventional chiral
CFT, than the usual physics lore mentioned in the last subsection as well as
mathematical results on VOAs [42] lead to the expectation that one should be
able to attach a weight zero modular object to the moonshine module. This
process can be done in a straightforward manner for the eight cases of umbral
moonshine which correspond to the eight Niemeier lattices with $A$-type root
systems, but not for the remaining fifteen cases involving $D$\- and/or
$E$-type root systems. Moreover, among all the A-type cases, it was argued in
[8] that apart from the $A_{1}^{24}$ case of Mathieu moonshine, the associated
weight zero Jacobi forms cannot coincide with a partition function or elliptic
genus of a “usual” (chiral) conformal field theory. This is due to the
expectation that the NS-NS ground states lead to a non-vanishing
$q^{0}y^{m-1}$ term in the index $m$ weight zero Jacobi form, which is absent
in all cases other than the $A_{1}^{24}$ case. In this regard, the
$A_{1}^{24}$ case of umbral moonshine is set apart from the other cases that
the mock modular forms can be regarded as arising from the supersymmetric
index of a physical conformal field theory — any $K3$ non-linear sigma model
in fact. This more involved modular property suggests that a novel approach is
likely to be needed in order to understand umbral moonshine, at least in the
cases other than the Mathieu moonshine.

###  2.3 Deformed Drinfel’d Double and its Representations

To set the stage for the generalised umbral moonshine conjecture, we will
quickly review the definition of a deformed Drinfel’d (or quantum) double of a
finite group and its representations. Drinfel’d introduced the quantum double
construction, which associates to a Hopf algebra $\mathcal{A}$ a quasi-
triangular Hopf algebra $D({\cal A})$. This quasi-triangular Hopf algebra
contains ${\cal A}$ as well as its dual algebra ${\cal A}^{\ast}$ as Hopf
algebras and $D(A)={\cal A}^{\ast}\otimes A$ as a vector space [43]. Let $G$
be a finite group and ${\cal A}=kG$ be the group algebra over field $k$. It
was argued that the corresponding quantum double for the case
$k={\mathbb{C}}$, often denoted simply by $D(G)$, is relevant for 3d TQFTs
[44] and relatedly 2d orbifold CFTs [25, 45]. In the former case, $G$ plays
the role of the discrete gauge group in the former case and that of the
orbifold group in the latter case. Here and in the rest of the paper we will
work with $k={\mathbb{C}}$ exclusively.

In [45], Dijkgraaf, Pasquier and Roche introduced an interesting
generalisation of this construction. For $G$ a finite group and $M$ a
$G$-module, one can define the group cohomology
$H^{n}(G,M):=\text{Ker}\,\partial^{(n)}/\text{Im}\,\partial^{(n-1)}$ where the
coboundary operator $\partial^{(n)}$ on the space of $n$-cochains is given by

$(\partial^{(n)}\lambda)(h_{1},\dots,x_{n},x_{n+1})=\lambda(x_{2},\dots,x_{n+1})+{(-1)^{n+1}}\lambda(h_{1},\dots,x_{n})\\\
+\sum_{i=1}^{n}{(-1)^{i}}\lambda(h_{1},\dots,x_{i}x_{i+1},\dots,x_{n+1}).$
(2.19)

From now on we take $M=U(1)$, and to avoid an overload of notation we will
often conflate $x\in{\mathbb{R}}/{\mathbb{Z}}\cong U(1)$ with $e^{2\pi
ix}\in{\mathbb{C}}^{\times}$ when the meaning is clear from the context. Given
$G$ and a 3-cocycle $\omega:G\times G\times G\to{\mathbb{C}}^{\times}$, the
deformed quantum double $D^{\omega}(G)$ is a quasi-triangular quasi-Hopf
algebra generated by the elements444Here we use the notation in [46]. Some
other common choices of notation for $P(x)Q(y)$ include $\delta_{x}y$,
$\mbox{$x$}\,\underset{y}{\leavevmode\hbox to11.87pt{\vbox
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$\langle\xleftarrow[]{y}x\rangle$. We will work only with the so-called
normalised cochains, namely functions $u:G^{n}\to{\mathbb{C}}^{\times}$
satisfying $u(x_{1},\dots,x_{n})=1$ whenever at least one of the $n$ group
elements $x_{1},\dots,x_{n}$ coincides with the identity element $e$. This
corresponds to requiring $\rho(e)={\mathbf{1}}_{V}\in{\rm End}(V)$ in the
$G$-representation $(\rho,V)$.  $P(x)$ and $Q(y)$, with $x,y\in G$[45]. The
multiplications are given by

$\displaystyle P(x)P(y)$ $\displaystyle=\begin{cases}P(x)&{\rm if}~{}x=y\\\
0&\text{otherwise}\end{cases}$ (2.20) $\displaystyle Q(x)Q(y)$
$\displaystyle=\theta(x,y)Q(xy)~{},$ (2.21)

while the co-multilpications are given by

$\displaystyle\Delta P(x)$ $\displaystyle=\sum_{z\in G}P(z)\otimes P(z^{-1}x)$
(2.22) $\displaystyle\Delta Q(x)$ $\displaystyle=\eta(z)Q(x)\otimes Q(x).$
(2.23)

Moreover we have

$\displaystyle Q(x)^{-1}\,P(y)\,Q(x)$ $\displaystyle=P(y\lvert^{x})~{},$
(2.24)

where we use the notation $y\lvert^{x}:=x^{-1}yx$ to denote the conjugate of
$y$ by $x$. The identity element of $D^{\omega}(G)$ is $Q(1)=\sum_{z\in
G}P(z)$.

The multiplication and co-multiplication rules are determined by the 3-cocycle
$\omega$ as

$\displaystyle\theta(x,y)$ $\displaystyle=\sum_{z\in G}\theta_{z}(x,y)P(z)$
(2.25) $\displaystyle\eta(z)$ $\displaystyle=\sum_{x,y\in
G}\eta_{z}(x,y)P(x)\otimes P(y)~{},$ (2.26)

where

$\displaystyle\theta_{g}(x,y):=\frac{\omega(x,g\lvert^{x},y)}{\omega(g,x,y)\omega(x,y,g\lvert^{xy})}$
(2.27)
$\displaystyle\eta_{g}(x,y):=\frac{\omega(x,g,y\lvert^{x})}{\omega(x,y,g)\omega(g,x\lvert^{g},y\lvert^{g})}.$
(2.28)

Note that $\theta_{g}(x,y)=\eta_{g}(x,y)$ when $x,y\in C_{G}(g)$, the
centralizer of $g$ in $G$. It is easy to check that the restriction
$c_{g}:=\theta_{g}\lvert_{C_{G}(g)\times
C_{G}(g)}=\eta_{g}\lvert_{C_{G}(g)\times C_{G}(g)}$ is a 2-cocycle and
moreover, that the corresponding map $Z^{3}(G,U(1))\to Z^{2}(C_{G}(g),U(1))$
induces a map $H^{3}(G,U(1))\to H^{2}(C_{G}(g),U(1))$.

The fact that the above defines a quasi-Hopf algebra is guaranteed by the
3-cocycle condition $\partial^{(3)}\omega=0$. In particular, the co-
multiplication, instead of satisfying
$({\mathbf{1}}\otimes\Delta)\cdot\Delta=(\Delta\otimes{\mathbf{1}})\cdot\Delta$
as in the case of a Hopf algebra, satisfies

$({\mathbf{1}}\otimes\Delta)\cdot\Delta(a)=\varphi(\Delta\otimes{\mathbf{1}})\cdot\Delta(a)\,\varphi^{-1}\qquad\forall
a\in D^{\omega}(G)$ (2.29)

in a quasi-Hopf algebra. In this case the intertwiner is given by

$\varphi=\sum_{x,y,z\in G}\omega(x,y,z)\,P(x)\otimes P(y)\otimes P(z).$ (2.30)

It is easy to see that changing $\omega$ without changing the cohomology class
leads to an isomorphic quasi-Hopf algebra [45]. In the language of holomorphic
orbifold this signals the failure of the fusion between operators belonging to
different twisted sectors to be associative. See [45] and references therein
for more details on $D^{\omega}(G)$ including the $R$-matrices and the
antipode.

Next we will summarize the basic properties of the representations of
$D^{\omega}(G)$. See, for instance, [45, 46, 47] for more information.
Firstly, we will establish that given an $\omega$-compatible projective
representation of $C_{G}(g)$ for some $g\in G$ we can build a corresponding
$D^{\omega}(G)$-representation via the so-called DPR induction.

Recall that a projective representation $(\rho,V)$, where $\rho:H\to{\rm
End}(V)$, satisfies

$\rho(h_{1})\rho(h_{2})=c^{\rho}(h_{1},h_{2})\,\rho(h_{1}h_{2})$ (2.31)

for any $h_{1},h_{2}\in H$, where $c^{\rho}\in Z^{2}(H,U(1))$. Given a
${g_{A}}\in G$ and a projective representation $(\rho_{g_{A}},V_{g_{A}})$ of
$C_{G}(g_{A})$ corresponding to the 2-cocycle $\theta_{g_{A}}\in
Z^{2}(C_{G}(g_{A}),U(1))$, we can construct a DPR-induced representation of
$D^{\omega}(G)$ in the following way [45]. Let ${\cal B}_{g_{A}}$ be the
subalgebra of $D^{\omega}(G)$ spanned by elements of the form $P(g)Q(x)$ with
$g\in G$ and $x\in C_{G}(g_{A})$. We define the action of ${\cal B}_{g_{A}}$
on $V_{g_{A}}$ by $\pi(P(g)Q(x))=\delta_{g,g_{A}}\,\rho_{g_{A}}(x)$. The DPR-
induced $D^{\omega}(G)$ representation, denoted
$\big{(}\pi_{(\rho_{g_{A}},V_{g_{A}})},{\rm Ind}^{{\rm
DPR}}(V_{g_{A}})\big{)}$, is given by ${\rm
Ind}^{{\rm{DPR}}}(V_{g_{A}}):={\mathbb{C}}[G]\otimes_{{\cal
B}_{g_{A}}}V_{g_{A}}$, where we identify ${\mathbb{C}}[G]$ as the subalgebra
spanned by $\sum_{z\in G}P(z)Q(x)$ with $x\in G$. Explicitly, to describe the
action of $D^{\omega}(G)$ we choose a set of representatives
$\\{x_{1},\dots,x_{n}\\}$ of $G/C_{G}(g_{A})$, and it can be checked that, for
$v\in V_{g_{A}}$,

$\pi_{(\rho_{g_{A}},V_{g_{A}})}(P(g)Q(x))(x_{j}\otimes
v)={\theta_{g_{A}}(x,x_{j})\over\theta_{g_{A}}(x_{k},h)}\,\delta_{g\lvert^{x},g_{j}}\,(x_{k}\otimes\rho_{g_{A}}(h)v),$
(2.32)

where $h\in C_{G}(g_{A})$ is determined by $xx_{j}=x_{k}h$.

Secondly, any irreducible representation of $D^{\omega}(G)$ is labelled by a
conjugacy class $A$ of $G$ and an irreducible projective representation of
$C_{G}(g_{A})$ with $g_{A}$ in conjugacy class $A$ with the 2-cocycle
$\theta_{g_{A}}$. Moreover, such a irreducible representation of
$D^{\omega}(G)$ is equivalent to the corresponding DPR-induced representation
described above. As a result, studying the representations of $D^{\omega}(G)$
is equivalent to studying the projective representations of all the
centraliser subgroups with the 2-cocyles given by $\omega$ via (2.27), a fact
that we will exploit to in order to make explicit conjectures in the next
subsection.

The representations of a deformed quantum double share many features with
those of finite groups. For instance there is an orthogonality relation among
irreducible representations. For a given representation $V$ of
$D^{\omega}(G)$, we write its character as ${\rm Ch}_{(x,y)}^{V}:={\rm
Tr}_{V}(P(x)Q(y)).$ These characters satisfy the following properties,

$\displaystyle{\rm Ch}_{(x,y)}^{V}$ $\displaystyle=0~{}~{}{\rm
unless}~{}~{}xy=yx$ (2.33) $\displaystyle{\rm
Ch}_{{(x\lvert^{z},y\lvert^{z})}}^{V}$
$\displaystyle=\frac{\theta_{x}(z,y\lvert^{z})}{\theta_{x}(y,z)}{\rm
Ch}_{(x,y)}^{V},$ (2.34)

which can also be checked explicitly for the DPR-induced representations.
Moreover, $\rm{Rep}(D^{\omega}(G))$ is a modular category. In terms of the
characters, the modular group acts as

$\displaystyle T{\rm Ch}_{(x,y)}^{V}$ $\displaystyle=\theta_{x}(x,y)\,{\rm
Ch}_{(x,xy)}^{V}$ (2.35) $\displaystyle S{\rm Ch}_{(x,y)}^{V}$
$\displaystyle={1\over\theta_{y}(x,x^{-1})}\,{\rm Ch}_{(y,x^{-1})}^{V},$
(2.36)

satisfying $S^{2}=(ST)^{3}$, $S^{4}=\rm{id}$. In particular, the fusion rules
between different representations satisfy the Verlinde formula given by the
above $S$-matrix.

##  3 Generalised Umbral Moonshine

The Generalised Umbral Moonshine Conjecture states that for each of the 23
lambencies of umbral moonshine with umbral group $G$ there is a 3-cocycle
$\omega:G\times G\times G\to{\mathbb{C}}^{\times}$ such that there exists an
infinite-dimensional module for $D^{\omega}(G)$ underlying a set of
distinguished meromorphic Jacobi forms that we will specify in §5. In the
following we will re-formulate the conjecture in explicit terms, using the
relation to the projective representations of the centraliser subgroups
discussed in §2.3. This leads to the first five of the six conditions we will
discuss in this section. The sixth condition is the counterpart of the
Hauptmodul condition and demonstrates the special property of the generalised
umbral moonshine functions.

Given the lambency $\ell$ with index $m$ and the corresponding umbral group
$G$, we propose in §5 the twisted-twined functions $\psi_{(g,h)}$ (and the
corresponding twisted-twined mock modular forms $H_{(g,h)}$) for each
commuting pair $(g,h)$. We conjecture that they satisfy the following six
conditions. The relation between $\psi_{(g,h)}$ and $H_{(g,h)}$ is as
described in (2.4). In the case $\ell=2$, the first five conditions are
equivalent to the ones discussed in [1].

  1. I

Modularity
The modularity condition first states that the function $\psi_{(g,h)}$, which
is meromorphic as a function $\zeta\mapsto\psi_{(g,h)}(\tau,\zeta)$ and can
have poles at most at $2m$-torsion points, $\zeta\in{1\over
2m}{\mathbb{Z}}+{\tau\over 2m}{\mathbb{Z}}$, transforms in the same way as an
index $m$ Jacobi form under the elliptic transformation (cf. equation 2.5).
Namely for all $(\lambda,\mu)\in{\mathbb{Z}}^{2}$,

$\psi_{(g,h)}\big{\lvert}_{m}(\lambda,\mu)=\psi_{(g,h)}~{}.$ (3.1)

Second, under the weight one modular transformation they satisfy for any
${g,h\in G}$ and any $\gamma\in{\rm SL}_{2}({\mathbb{Z}})$

$\varsigma_{(g,h)}(\gamma)\,\psi_{(g,h)}\big{\lvert}_{1,m}\gamma=\psi_{\gamma(g,h)}\;$
(3.2)

for some ${\varsigma_{(g,h)}:{\rm
SL}_{2}({\mathbb{Z}})\to{\mathbb{C}}^{\times}}$. In the above we have defined
$\gamma(g,h):=(g^{a}h^{c},g^{b}h^{d})$ for two commuting elements $g,h\in G$.

This condition reflects the modular property of $D^{\omega}(G)$
representations, embodied by the meromorphic Jacobi forms $\psi_{(g,h)}$, and
can be regarded as the counterpart of condition I of generalised monstrous
moonshine.

  2. II

3-Cocycle

This condition states that there exists a 3-cocycle $\omega:G\times G\times
G\to{\mathbb{C}}^{\times}$ compatible with the multiplier system
$\varsigma_{(g,h)}$ in (3.2) in the following way. For any $g\in G$ let
$\theta_{g}$ be defined as in (2.27). Then the compatibility conditions read

$\displaystyle\varsigma_{(g,h)}\left(\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)\right)=\theta_{g}(g,h)^{-1}$ (3.3a)
$\displaystyle\varsigma_{(g,h)}\left(\left(\begin{smallmatrix}0&-1\\\
1&0\end{smallmatrix}\right)\right)=\theta_{h}(g,g^{-1}).$ (3.3b)

which determine $\varsigma_{(g,h)}$ in terms of $\omega$ together with (3.2).

This condition captures the modular properties (2.35-2.36) of representations
of the quantum deformed double $D^{\omega}(G)$. It is believed that a
3-cocycle underlies the generalised monstrous moonshine in a similar way [1],
although it was not mentioned in any of the five stated conditions of
generalised monstrous moonshine

  3. III

Projective Class Function

This condition states that $\psi_{(g,h)}$ is a projective class function,
satisfying for all $k\in G$

$\psi_{(g\lvert^{k},h\lvert^{k})}=\xi_{(g,h)}(k)\,\psi_{(g,h)}$ (3.4)

with the phases $\xi_{(g,h)}(k)$ determined by the 3-cocycle $\omega$ as

$\xi_{(g,h)}(k)=\frac{\theta_{g}(k,h\lvert^{k})}{\theta_{g}(h,k)}.$ (3.5)

This property together with the modularity property (3.2) implies that the
${\rm SL}_{2}({\mathbb{Z}})$ subgroup for which $H_{(g,h)}$ is a vector-valued
mock modular form is given by

$\Gamma_{g,h}:=\\{\gamma\in{\rm
SL}_{2}({\mathbb{Z}})\big{\lvert}\gamma(g,h)=(g\lvert^{k},h\lvert^{k})\text{
for some }k\in G\\}.$ (3.6)

This condition captures the property (2.34) of representations of the quantum
deformed double $D^{\omega}(G)$ and is the counterpart of condition III of
generalised monstrous moonshine.

  4. IV

Finite Group

The vector-valued mock modular forms $H_{(g,h)}=(H_{(g,h),r})$ have the
following moonshine relation to the finite group $G=G^{(\ell)}$. Namely, there
is a way to assign an infinite-dimensional projective $C_{G}(g)$-module

$K^{g}=\bigoplus_{\begin{subarray}{c}r\in{\mathbb{Z}}/2m{\mathbb{Z}}\\\
\alpha\in{\mathbb{Q}}_{>0}\end{subarray}}K^{g}_{r,\alpha}$ (3.7)

with the 2-cocycle given by $c^{\rho}=\theta_{g}$, such that for all $g\in G$,
$h\in C_{G}(g)$ and $r\in{\mathbb{Z}}/2m{\mathbb{Z}}$,

$\varepsilon_{m}(r)H_{(g,h),r}(\tau)=c_{(g,h),r}q^{-1/4m}+\sum_{\alpha\in{\mathbb{Q}}_{>0}}q^{\alpha}\text{Tr}_{K^{g}_{r,\alpha}}h,$
(3.8)

where $\varepsilon_{m}$ is as in (2.18). The coefficient $c_{(g,h),r}$ equals
$\mp 2$ whenever $H_{(g,h),r}(\tau)=\pm H_{(e,h),1}(\tau)$ and vanishes
otherwise.

The above condition describes the $D^{\omega}(G)$-module as DPR-induced
modules as explained in §2.3 and can be regarded as the counterpart of
condition IV of generalised monstrous moonshine.

  5. V

Consistency
This condition states that the generalised umbral moonshine is compatible with
umbral moonshine in the sense that $\psi_{(e,h)}^{(\ell)}$ coincides with the
weight one meromorphic Jacobi form $\psi^{(\ell)}_{h}$ given in [8, 9]. In
particular, the resulting mock modular form $H_{(e,h)}$ coincides with the
McKay–Thompson series $H^{(\ell)}_{h}=(H^{(\ell)}_{h,r})$ proposed in [8, 9].

This condition establishes the relation between generalised umbral moonshine
and the original umbral moonshine, and can be regarded as the counterpart of
condition V of generalised monstrous moonshine.

  6. VI

Pole Structure
This condition states that the mock modular form $H_{(g,h)}$ is bounded at all
but at most one orbit of the cusps ${\mathbb{Q}}\cup\\{i\infty\\}$ under
$\Gamma_{(g,h)}$. It serves as the counterpart of condition II (the Hauptmodul
condition) of generalised monstrous moonshine.

An immediate consequence of the generalised umbral moonshine conjecture is
certain non-trivial constraints on their twisted-twined functions. To see
this, consider a pair $(g,h)$ of commuting elements in $G$ that is conjugate
to $(g^{-1},h^{-1})$. The conditions (3.2) and (3.4) lead to the equality

$\varsigma_{(g,h)}\left(\left(\begin{smallmatrix}-1&0\\\
0&-1\end{smallmatrix}\right)\right)\psi_{(g,h)}=\,\xi_{(g,h)}(k)\psi_{(g,h)},$
(3.9)

where $k$ a group element satisfying
$(g^{-1},h^{-1})=({g}\lvert^{k},h\lvert^{k})$. It follows that $\psi_{(g,h)}$
must vanish unless

$\varsigma_{(g,h)}\left(\left(\begin{smallmatrix}-1&0\\\
0&-1\end{smallmatrix}\right)\right)=\xi_{g^{-1},h^{-1}}(k).$ (3.10)

We will employ this obstruction555In principle there is another independent
possible obstruction coming from the projective class function property of
$\psi_{(g,h)}$. Namely $\psi_{(g,h)}$ must vanish unless $\xi_{(g,h)}(k)=1$
for all $k$ commuting with both $g$ and $h$. However, in all cases apart from
the $\ell=2$ case treated in [1], this does not lead to new constraints.  in
our computation and list the obstructed twisted-twined functions in Table 5.2.

##  4 Groups, Cohomologies, and Representations

In this section we present the group-theoretic underpinnings of the theory. We
introduce the umbral groups and discuss their cohomology, their rank-2
subgroups, and the projective representations of their centraliser subgroups.

###  4.1 Group Cohomology

The computation of the twisted-twined functions for generalised umbral
moonshine depends crucially on the computation of the third group cohomology
$H^{3}(G^{(\ell)},U(1))$. With the aid of GAP and the module HAP we have been
able to compute the cohomology groups for all of the umbral groups. The
results are listed in Table 4.1, where we include the root system $X$, the
lambency $\ell$, the umbral group $G^{(\ell)}$, of the corresponding cases of
umbral moonshine. Throughout this paper we write ${\mathbb{Z}}/N{\mathbb{Z}}$
as $N$. Also listed in the table is $n$, given by the group automorphism
$G^{(\ell)}\cong n.{\overline{G}}^{(\ell)}$, where ${\overline{G}}^{(\ell)}$
is the group of permutations of the irreducible components of the root systems
of the corresponding Niemeier lattice induced by the lattice automorphism
group. See [9] for more details. Note that $n=2$ or $n=1$ for all lambencies
except for $\ell=6+3$ which has $n=3$.

The subgroups $n\subset G^{(\ell)}$ play a special role in umbral moonshine:
the corresponding twined functions are proportional to the untwined functions,
$H_{(e,z),r}=c_{r}\,H_{(e,e),r}$ for a $c_{r}\in{\mathbb{C}}$ for all $z\in n$
and $r\in{\mathbb{Z}}/2m{\mathbb{Z}}$. More generally, we have
$H_{(e,zg),r}=c_{r}\,H_{(e,g),r}$ for all $g\in G^{(\ell)}$. Together with the
modularity condition (3.2), it suggests that the generalised umbral moonshine
function $H_{(g,h)}$ is really “new” if $(g,h)$ cannot be obtained from the
pair $(z,g^{\prime})$ for some $z\in n$ via an ${\rm SL}_{2}({\mathbb{Z}})$
transformation. This inspires the following terminology: we say that a
subgroup $\langle g,h\rangle$ is old if it is isomorphic to $\langle
z,g^{\prime}\rangle$ for some $g^{\prime}\in G^{(\ell)},z\in n$, and is new
otherwise. We will also call the corresponding twisted-twined function an old
resp. new function. In Table 4.1 we also list $\lvert{\rm new}\rvert$, the
number of conjugacy classes of rank-2 Abelian subgroups of $G^{(\ell)}$ that
are new.

The group theoretic computation for the case $\ell=2$ has been performed in
[1]. For $\ell=3$, direct evaluation of cochains is computationally
challenging but restriction of cochains to the $p$-Sylow subgroups $S_{p}(G)$
provides an isomorphism (cf. [52], Chapter III-10)

$H^{3}(G^{(\ell)},U(1))\cong\bigoplus_{p|\exp(G)}H^{3}(S_{p}(G^{(\ell)}),U(1)),$
(4.1)

where the exponent $\exp(G)$ is the exponent of the group, and
$\exp(G^{(3)})=24$. This allows us to simplify our calculations for $\ell=3$.

Table 1: Umbral Groups and Group Cohomology

${X}$ |  $A_{1}^{24}$ |  $A_{2}^{12}$ |  $A_{3}^{8}$ |  $A_{4}^{6}$ |  $A_{5}^{4}D_{4}$ |  $A_{6}^{4}$ |  $A_{7}^{2}D_{5}^{2}$ |  $A_{8}^{3}$
---|---|---|---|---|---|---|---|---
$\ell$ |  2 |  3 |  4 |  5 |  6 |  7 |  8 | 9
$G^{{X}}$ |  $M_{24}$ |  $2.M_{12}$ |  $2.{\textsl{AGL}}_{3}(2)$ |  ${\textsl{GL}}_{2}(5)/2$ |  ${\textsl{GL}}_{2}(3)$ |  $\operatorname{\textsl{SL}}_{2}(3)$ |  ${\textsl{Dih}}_{4}$ | ${\textsl{Dih}}_{6}$
$\bar{G}^{{X}}$ |  $M_{24}$ |  $M_{12}$ |  ${\textsl{AGL}}_{3}(2)$ |  $\operatorname{\textsl{PGL}}_{2}(5)$ |  $\operatorname{\textsl{PGL}}_{2}(3)$ |  $\operatorname{\textsl{PSL}}_{2}(3)$ |  $2^{2}$ |  ${\textsl{Sym}}_{3}$
$H^{3}(G^{(\ell)},\mathbb{C}^{\times})$ |  12 |  $8\oplus 24$ |  $4\oplus 24$ |  $4\oplus 12$ |  $2\oplus 24$ |  24 |  $2\oplus 2\oplus 4$ |  $2\oplus 2\oplus 6$
$n$ |  1 |  2 |  2 |  2 |  2 |  2 |  2 |  2
$\lvert$new$\rvert$ |  34 |  12 |  8 |  3 |  0 |  0 |  0 |  0
${X}$ |  $A_{9}^{2}D_{6}$ |  $A_{11}D_{7}E_{6}$ |  $A_{12}^{2}$ |  $A_{15}D_{9}$ |  $A_{17}E_{7}$ |  $A_{24}$ |  $D_{4}^{6}$ |  $D_{6}^{4}$
$\ell$ |  10 |  12 |  13 |  16 |  18 |  25 |  6+3 | 10+5
$G^{{X}}$ |  $4$ |  $2$ |  $4$ |  $2$ |  $2$ |  $2$ |  $3.{\textsl{Sym}}_{6}$ | ${\textsl{Sym}}_{4}$
$\bar{G}^{{X}}$ |  $2$ |  $1$ |  $2$ |  $1$ |  $1$ |  $1$ |  ${\textsl{Sym}}_{6}$ |  ${\textsl{Sym}}_{4}$
$H^{3}(G^{(\ell)},\mathbb{C}^{\times})$ |  4 |  1 |  4 |  1 |  1 |  1 |  $2\oplus 6\oplus 12$ |  $2\oplus 12$
$n$ |  2 |  2 |  2 |  2 |  2 |  2 |  3 |  2
$\lvert$new$\rvert$ |  0 |  0 |  0 |  0 |  0 |  0 |  8 |  0
${X}$ |  $D_{8}^{3}$ |  $D_{10}E_{7}^{2}$ |  $D_{12}^{2}$ |  $D_{16}E_{8}$ |  $D_{24}$ |  $E_{6}^{4}$ |  $E_{8}^{3}$ |
$\ell$ |  14+7 |  18+9 |  22+11 |  30+15 |  46+23 |  12+4 |  30+6, 10, 15
$G^{{X}}$ |  ${\textsl{Sym}}_{3}$ |  $2$ |  $2$ |  $1$ |  $1$ |  ${\textsl{GL}}_{2}(3)$ |  ${\textsl{Sym}}_{3}$
$\bar{G}^{{X}}$ |  ${\textsl{Sym}}_{3}$ |  $2$ |  $2$ |  $1$ |  $1$ |  $\operatorname{\textsl{PGL}}_{2}(3)$ |  ${\textsl{Sym}}_{3}$ |
$H^{3}(G^{(\ell)},\mathbb{C}^{\times})$ |  6 |  1 |  1 |  1 |  1 |  $2\oplus 24$ |  6 |
$n$ |  1 |  1 |  1 |  1 |  1 |  2 |  1 |
$\lvert$new$\rvert$ |  0 |  0 |  0 |  0 |  0 |  0 |  0 |

###  4.2 The 3-Cocycles

For most of the lambencies $\ell$ of umbral moonshine, the cohomology class of
the 3-cocycle $\omega$ can be uniquely determined by the multiplier systems of
the old functions $\psi_{(e,g)}$ in the following way. Fix a lambency $\ell$.
For any $g\in G$, $\varsigma_{(e,g)}$ (cf. equation 3.2) can be restricted to
$\Gamma_{0}(\operatorname{ord}g)$ to obtain a multiplier system
$\varsigma_{(e,g)}:\Gamma_{0}(\operatorname{ord}g)\to{\mathbb{C}}^{\times}$,
which we assume to arise from a 3-cocycle $\omega:G\times G\times
G\to{\mathbb{C}}^{\times}$ as discussed in §3. At the same time, these
multipliers are required to coincide with those of the known functions
$\psi_{(e,g)}$[8, 9] (cf. §3, condition V). In this way, we obtain a
consistency condition on $[\omega]$. This consistency condition uniquely fixes
$[\omega]\in H^{3}(G,U(1))$ for all umbral moonshine cases except for
$\ell=3,4,7$. In the latter cases, there are exactly two cohomology classes
that are consistent with the multiplier phases of the known functions
$\psi_{(e,g)}$ for all $g\in G$.

For the $\ell=4$ case, the two cohomology classes provide different multiplier
systems on a weight 1 Jacobi form associated to one of the new functions. One
of the corresponding spaces of weight 1 weak Jacobi forms has dimension 0 ,
but the vanishing of that specific new function would be incompatible with a
decomposition of the characters into projective representations (cf. §3,
condition IV). We can therefore assume that the new function must be non-zero
and this allows us to uniquely select a 3-cohomology class.

For $\ell=3$ and $\ell=7$, both of the 3-cohomology classes that are
consistent with the known functions $\psi_{(e,g)}$ are also consistent with
the remaining twisted-twined functions we propose. As a result, in this sense
generalised umbral moonshine for these two cases is compatible with both
choices.

###  4.3 The Rank Two Subgroups

Given a 3-cocycle $\omega$, the projective class function property (cf.
equation 3.4) guarantees that the generalised umbral moonshine function
$\psi_{(g,h)}$ is determined by $\psi_{(g^{\prime},h^{\prime})}$ whenever the
groups $\langle g,h\rangle$ and $\langle g^{\prime},h^{\prime}\rangle$ are
simultaneously conjugate to each other. It therefore suffices to specify the
twisted-twined functions $\psi_{(g,h)}$ for a set of representatives in the
coset $\\{g,h\in G\lvert\,gh=hg\\}/\\!\\!\sim$, where the equivalence relation
is generated by ${\rm SL}_{2}({\mathbb{Z}})$ transformations and by
conjugation: $(g,h)\sim\gamma(g\lvert^{k},h\lvert^{k})$ for all $k\in G$ and
$\gamma\in{\rm SL}_{2}({\mathbb{Z}})$.

We tabulate these rank-2 subgroups in Table 5.2 and indicate the new rank-2
subgroups whose twisted-twined functions are obstructed by the condition
discussed at the end of §3. In this table we also list the congruence
subgroups $\Gamma_{(g,h)}\subseteq\text{SL}_{2}(\mathbb{Z})$ stabilising the
pair $(g,h)$ up to simultaneous conjugation (cf. equation 3.6). In this list
we employed the following notation for the congruence subgroups

${\rm
H}(p,q,r)=\left\\{\gamma\in\text{SL}_{2}(\mathbb{Z})\,\lvert\gamma=\left(\begin{smallmatrix}1&0\\\
0&1\end{smallmatrix}\right){\rm\;mod\;}\left(\begin{smallmatrix}p&q\\\
r&p\end{smallmatrix}\right)\right\\},$ (4.2)

which is a group if and only if $p|qr$. We also use the familiar convention
for specific congruence subgroups:

${\Gamma(N):={\rm H}(N,N,N)},~{}{\Gamma_{1}(N):={\rm H}(N,1,N)},~{}{\rm
and}~{}{\Gamma_{0}(N):={\rm H}(1,1,N)}.$ (4.3)

Finally, we have

$\Gamma_{(4A,4c)}=\left\\{\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma(2)\,\lvert\,a+b+c\equiv
1\,\text{mod}\,4\right\\}.$ (4.4)

Here and everywhere else we denote the conjugacy class names of group elements
$g\in G$ with upper case letters, while the conjugacy class names of elements
$h\in C_{G}(g)$ in centralizer subgroups are denoted with lower case letters.
See the attached file on the ArXiv for an explicit construction of the umbral
groups.

###  4.4 Projective Representations

In this subsection we briefly explain how we construct the relevant projective
representations. Our discussion follows closely the appendices C and D of [1].

Recall the definition of a projective representation in §2.3. We say that two
projective representations $\rho$ and $\rho^{\prime}$ of a group $H$ are
equivalent if they differ by a 1-cochain $\xi\in Z^{1}(H,U(1))$, namely when
$\rho(x)=\xi(x)\rho^{\prime}(x)$ for all $x\in H$. As a result, inequivalent
projective representations are labelled by the second group cohomology
$H^{2}(H,U(1))$. A convenient way to study the projective representations of a
group $H$ is to study the representations of a Schur cover $X$ of $H$, defined
as a central extension

$1\to M(H)\to X\xrightarrow{\phi}H\to 1,$ (4.5)

where the normal subgroup is the Schur multiplier $M(H)\cong
H_{2}(H,{\mathbb{Z}})$. As we will now explain, every representation of the
Schur cover corresponds to a projective representation of $H$ and vice versa.

Let $\phi^{-1}:H\to X$ be a lift of $H$. Let $\bar{\rho}:X\to{\rm End}(V)$ be
a representation of $X$. Then $\rho:=\bar{\rho}\circ\phi^{-1}$ is a projective
representation of $H$ with the associated cocycle $c^{\rho}$ satisfying

$c^{\rho}(h_{1},h_{2}){{\bf
1}_{V}}=\overline{\rho}\left(\phi^{-1}(h_{1})\,\phi^{-1}(h_{2})\,\phi^{-1}((h_{1}h_{2})^{-1})\right).$
(4.6)

As a result, the 2-cocycle $c^{\rho}$ is fixed by $\bar{\rho}$ restricted to
${M(H)}$. Every projective representation of $H$ is equivalent to a
representation arising from its Schur cover in this way, and different
(necessarily isoclinic) Schur covers of $H$ lead to different but equivalent
projective $H$-representations.

Given a two-cohomology class $[c]\in H^{2}(H,U(1))$, we would like to select
the compatible representations from all the representations of $X$. In other
words, we would like to pick out the $X$-representations that give rise to
projective representations of $H$ whose corresponding 2-cocycle is consistent
with $[c]$. Now we describe how this can be achieved. For a representation
$\bar{\rho}$ of $X$ and $k\in M(H)$, if $h_{1},h_{2}\in H$ satisfy
$k=\phi^{-1}(h_{1})\phi^{-1}(h_{2})\phi^{-1}(h_{1}^{-1})\phi^{-1}(h_{2}^{-1})$,
then $h_{1}$ and $h_{2}$ necessarily commute, and so
$c^{\rho}(h_{1},h_{2})/c^{\rho}(h_{2},h_{1})$ depends only on the cohomology
$[c^{\rho}]$. Moreover, the quotient can be evaluated as

$\displaystyle c^{\rho}(h_{1},h_{2})/c^{\rho}(h_{2},h_{1}){\bf 1}_{V}$
$\displaystyle=\bar{\rho}\left(\phi^{-1}(h_{1})\phi^{-1}(h_{2})\phi^{-1}(h_{1}h_{2})^{-1}\right)\bar{\rho}\left(\phi^{-1}(h_{2})\phi^{-1}(h_{1})\phi^{-1}(h_{2}h_{1})^{-1}\right)^{-1}$
$\displaystyle=\bar{\rho}(k),$ (4.7)

which enables us to identify exactly the projective representations that are
consistent with the given cohomology $[c^{\rho}]$. Finally, irreducible
representations of $X$ give rise to irreducible projective representations of
$H$.

For our purpose, the finite groups whose projective representations we are
interested in are the centralizer groups $C_{G}(g)$ where $G$ is one of the
the umbral groups. The relevant 2-cocycle is determined by the 3-cocycle
$\omega\in Z^{3}(G,U(1))$ as in (2.27). By considering the Schur cover of
$C_{G}(g)$ we obtain all irreducible projective representations $C_{G}(g)$,
and using (4.7) these projective representations can then be filtered to those
consistent with $[\omega]\in H^{3}(G,U(1))$.

##  5 The Generalised Umbral Mock Modular Forms

In this section, we employ the mock modularity property discussed in §3 to
compute the mock modular forms of generalised umbral moonshine, which
conjecturally arise from the generalised umbral moonshine module for the
underlying deformed quantum double ${D^{\omega}(G^{(\ell)})}$. In §5.1 we
compute the $g$-twisted $h$-twined functions when $\langle g,h\rangle$ is an
old group. In §5.2 we compute the remaining functions, the $g$-twisted
$h$-twined functions when $\langle g,h\rangle$ is a new group.

###  5.1 Old Groups

The untwisted twining functions of umbral moonshine were given in [4, 5, 6, 7,
8, 9], and explicit expressions for them can be found in [40]. For a given
$\ell$ with index $m$, these functions, which are denoted
$H_{(e,g)}=(H_{(e,g),r})$, are $2m$-dimensional vector-valued mock modular
forms of weight-1/2 which are related to the meromorphic Jacobi forms
$\psi_{(e,g)}$ as discussed in §2.2 and §3. In this subsection we will discuss
how to obtain the meromorphic Jacobi form $\psi_{(g,h)}$ and the vector-valued
mock modular form $H_{(g,h)}$ from the untwisted function
$\psi_{(e,g^{\prime})}$ for these cases.

In the case there exists an ${\rm SL}_{2}({\mathbb{Z}})$ element $\gamma$ such
that $\gamma(e,g^{\prime})=(g,h)$, by the modularity condition (3.2), we have

$\psi_{(g,h)}=\varsigma_{(e,g^{\prime})}(\gamma)\psi_{(e,g^{\prime})}|_{1,m}\gamma,$
(5.1)

which can be evaluated explicitly from the relation between
$\psi_{(e,g^{\prime})}$ and $\psi_{(e,e)}$ and the explicit expression for
$\psi_{(e,g^{\prime})}$ given in [9, 40].

When $G^{(\ell)}\cong{\bar{G}}^{(\ell)}$ has $n=1$, all old groups are
isomorphic to a cyclic group generated by an element $g^{\prime}$, and the
mock modular forms are found as described above. First consider umbral groups
with $n=2$. Let $z\in n$ be the nontrivial element and note that $n$ is in the
center of $G^{(\ell)}$. For the rank two old groups, namely those groups
$\langle g,h\rangle$ that are isomorphic to $\langle z,g^{\prime}\rangle$ for
some $g^{\prime}\in G^{(\ell)}$, we provide a conjecture and a justification
for the corresponding twisted-twined functions $\psi_{(g,h)}$. From the
modularity property (3.2), we see that it suffices to calculate
$\psi_{(z,g)}$. Fix the lambency $\ell$ with index $m$. We conjecture that

$\psi_{(z,g)}=\varsigma\psi_{(e,g)}|_{1,m}(\tfrac{1}{2},0).$ (5.2)

for all $g\in G$ for some $\varsigma\in{\mathbb{C}}$.

To justify the conjecture, we start by verifying (5.2) whenever
$\operatorname{ord}(g)$ is odd. Let $\gamma$ be an element of ${\rm
SL}_{2}({\mathbb{Z}})$ such that $\gamma(e,zg)=(z,g)$. For instance, we can
take $\gamma=\left(\begin{smallmatrix}1&1\\\
{\operatorname{ord}(g)}&{\operatorname{ord}(g)}+1\end{smallmatrix}\right)$.
Since $g$ is an element of the umbral group with odd order, the function
$\psi_{(e,g)}$ transforms as a weight one Jacobi form for
$\Gamma_{0}({\operatorname{ord}(g)})$ with a multiplier system [9]. Hence we
have

$\displaystyle\psi_{(z,g)}$
$\displaystyle=\varsigma_{(e,zg)}(\gamma)\psi_{(e,zg)}\big{\lvert}_{1,m}\gamma=-\varsigma_{(e,zg)}(\gamma)\psi_{(e,g)}\big{\lvert}_{m}(0,\tfrac{1}{2})\big{\lvert}_{1,m}\gamma$
$\displaystyle=-{\varsigma_{(e,zg)}(\gamma)\over\varsigma_{(e,g)}(\gamma)}\psi_{(e,g)}\big{\lvert}_{m}(\tfrac{1}{2},0)~{},$
(5.3)

where we have used the operator relation (2.7) and the fact that

$\psi_{(e,zg)}=-\psi_{(e,g)}\big{\lvert}_{m}(0,\tfrac{1}{2})$ (5.4)

for all $g\in G^{(\ell)}$.

Using $\varsigma_{(z,g)}\left(\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)\right)\psi_{(z,g)}\big{\lvert}_{1,m}\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)=\psi_{(z,zg)}$, it is easy to also verify the
validity of (5.2) for elements $zg$ that are “paired” with the odd order
elements $g$. Finally, we conjecture that (5.2) is also true when $g$ and $zg$
both have even order.

This conjecture is strongly motivated by the finite group property discussed
in §3, which states that there is an infinite dimensional module for
$C_{G}(z)\cong G$ that underlies $H_{(z,g)}$. Since replacing $\theta_{m}$ by
$\theta_{m}\big{\lvert}_{m}(\tfrac{1}{2},0)$ simply reorders the components of
the vector-valued mock modular forms $H_{(e,g)}=(H_{(e,g),r})$ by a shift
$r\mapsto r+m$, the group theoretic consideration strongly suggests that this
reordering has to happen for all $g\in G,$ given that it is true for all odd
order elements $g$ as well as the elements $zg$ that they are paired with.

For $\ell=6+3$, the unique instance of umbral moonshine with $n=3$, the
conjecture is slightly different. First, instead of (5.4) we have now

$\psi_{(e,zg)}=\psi_{(e,g)}-\frac{1}{2}\sum_{A\in{\mathbb{Z}}/3{\mathbb{Z}}}\psi_{(e,g)}|_{1,m}(0,A/3).$
(5.5)

for $g\in G^{(\ell)}$ with order $n_{g}$ not divisible by 3 that commutes with
$z$. Choose any $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)$ satisfying c= 0 mod n_g = 1 mod 3, d= 1 mod n_g =
0 mod 3, we have

$\psi_{(z,g)}={\varsigma_{(e,zg)}(\gamma)\over\varsigma_{(e,g)}(\gamma)}\left(\psi_{(e,g)}-\frac{1}{2}\sum_{A\in{\mathbb{Z}}/3{\mathbb{Z}}}\psi_{(e,g)}|_{1,m}(A/3,0)\right).$
(5.6)

Repeating the same argument as before by evoking the projective representation
property of moonshine, we conjecture that the above equality also holds for
all $g$ commuting with $z$, and hence for all rank two Abelian old groups
$\langle z,g\rangle$.

###  5.2 New Groups

Let $\langle g,h\rangle$ be a new group and consider the problem of evaluating
$H_{(g,h)}=(H_{(g,h),r})$. First we discuss their analytic properties. We wish
to show that for $\langle g,h\rangle$ a new group, $H_{(g,h)}$ is bounded at
all cusps.

Recall that the umbral moonshine conjecture [8, 9] states that the untwisted
twined functions $H_{(e,g)}$ are bounded at all cusps other than those
equivalent to the cusp at $\tau\to i\infty$ under the group
$\Gamma_{0}(n_{g})$, for all $g\in G$. The number $n_{g}$ is given by the
order of the group element $g$ as permutations of the irreducible components
of the root system of the corresponding Niemeier lattice. See the appendices
of [9] for the list of $n_{g}$. In particular, this means that they have no
poles at the cusp $\tau\to 0$ for all $g\not\in n$ and the modular property of
generalised umbral moonshine (3.2) implies that the corresponding twisted
untwined functions $H_{(g,e)}$ are bounded at the cusp $i\infty$. This fact
together with the finite group property of generalised umbral moonshine (3.8)
then implies that all twisted twined functions $H_{(g,h)}$ are bounded at the
cups $i\infty$ for all $g\in G$, $g\not\in n$ and for all $h\in C_{G}(g)$.
Finally, evoking again the modular property (3.2) and considering
$\gamma\in{\rm SL}_{2}({\mathbb{Z}})$ with
$(g,h)=\gamma(g^{\prime},h^{\prime})$ to transform between cusps shows that
the twisted twined functions $H_{(g,h)}$ are bounded at all cusps whenever
$\langle g,h\rangle$ is a new group.

Next we discuss the modular properties of the new functions. From the above
comment that $H_{(g,h)}$ is bounded at all cusps and the absence of the mock
Jacobi form in the corresponding space with non-trivial shadow (see the proof
of Prop. 4.1.1 of [34]), we expect that the shadow of $H_{(g,h)}$ vanishes. As
a result, we conclude that $H_{(g,h)}$ is a weight $1/2$ vector-valued modular
form for $\Gamma_{(g,h)}$ with the multiplier system determined by the
3-cocyle $\omega$ (cf. equation 3.2). Note that any congruence subgroup, for
instance $\Gamma_{(g,h)}$, contains a subgroup of the form $\Gamma(N)$. Hence
each individual component $H_{(g,h),r}$ is a weight $1/2$ modular form (with
trivial multiplier) for some $\Gamma(N)$. As a result, evoking Corollary 3 of
[53], we conclude that each component $H_{(g,h),r}$ lies in the linear span of
the theta functions

$\theta^{0}_{m^{\prime},s}(\tau):=\theta_{m^{\prime},s}(\tau,0)=\sum_{k\in{\mathbb{Z}}}q^{\frac{(2m^{\prime}k+s)^{2}}{4m^{\prime}}}$
(5.7)

for $m^{\prime}\in{\mathbb{Z}}_{+}$,
$s\in{\mathbb{Z}}/2m^{\prime}{\mathbb{Z}}$.

Table 2: New umbral subgroup representatives $\langle g,h\rangle\subseteq G^{(\ell)}$ $\ell$ | $\langle g,h\rangle$ | $\Gamma_{(g,h)}$ | Obs. | $\psi^{(\ell)}_{(g,h)}(\tau,\zeta)$
---|---|---|---|---
3 | $\langle\text{2B},\text{2a}\rangle$ | $\text{SL}_{2}(\mathbb{Z})$ | ✓ | 0
$\langle\text{2B},\text{2c}\rangle$ | $\Gamma_{1}(2)$ | ✓ | 0
$\langle\text{2B},\text{4a}\rangle$ | $\text{H}(2,2,4)$ | ✓ | 0
$\langle\text{2B},\text{4c}\rangle$ | $\text{H}(2,1,4)$ | ✓ | 0
$\langle\text{2C},\text{2b}\rangle$ | $\Gamma(2)$ | ✓ | 0
$\langle\text{2C},\text{2e}\rangle$ | $\text{SL}_{2}(\mathbb{Z})$ | ✓ | 0
$\langle\text{3A},\text{3a}\rangle$ | $\text{SL}_{2}(\mathbb{Z})$ |  | $-2\theta^{0}_{4,2}\cdot\hat{1}+(\theta^{0}_{4,0}+\theta^{0}_{4,4})\cdot\hat{2}$
$\langle\text{3A},\text{3b}\rangle$ | $\text{SL}_{2}(\mathbb{Z})$ |  | $-2\theta^{0}_{4,2}\cdot\hat{1}+(\theta^{0}_{4,0}+\theta^{0}_{4,4})\cdot\hat{2}$
$\langle\text{3A},\text{3c}\rangle$ | $\Gamma_{1}(3)$ |  | $(\theta^{0}_{9,3}-\theta^{0}_{9,9})\cdot\hat{1}+(\theta^{0}_{9,0}-\theta^{0}_{9,6})\cdot\hat{2}$
$\langle\text{3A},\text{6a}\rangle$ | $\Gamma_{1}(6)$ |  | $-(\theta^{0}_{9,3}-\theta^{0}_{9,9})\cdot\hat{1}+(\theta^{0}_{9,0}-\theta^{0}_{9,6})\cdot\hat{2}$
$\langle\text{3A},\text{6b}\rangle$ | $\Gamma_{1}(2)$ |  | $2\theta^{0}_{4,2}\cdot\hat{1}+(\theta^{0}_{4,0}+\theta^{0}_{4,4})\cdot\hat{2}$
$\langle\text{3A},\text{6h}\rangle$ | $\Gamma_{1}(2)$ |  | $2\theta^{0}_{4,2}\cdot\hat{1}+(\theta^{0}_{4,0}+\theta^{0}_{4,4})\cdot\hat{2}$
4 | $\langle\text{2B},\text{2a}\rangle$ | $\Gamma_{1}(2)$ | ✓ | 0
$\langle\text{2B},\text{2b}\rangle$ | $\text{SL}_{2}(\mathbb{Z})$ | ✓ | 0
$\langle\text{2B},\text{4b}\rangle$ | $\Gamma_{1}(2)$ |  | $2(\theta^{0}_{4,0}-\theta^{0}_{4,4})\cdot\hat{2}$
$\langle\text{2C},\text{2a}\rangle$ | $\text{SL}_{2}(\mathbb{Z})$ | ✓ | 0
$\langle\text{2C},\text{2c}\rangle$ | $\text{SL}_{2}(\mathbb{Z})$ |  | 0
$\langle\text{2C},\text{2i}\rangle$ | $\Gamma_{1}(2)$ | ✓ | 0
$\langle\text{2C},\text{4c}\rangle$ | $\Gamma_{1}(2)$ | ✓ | 0
$\langle\text{4A},\text{4c}\rangle$ | $\Gamma_{(4A,4c)}$ |  | $(2-2i)\theta^{0}_{2,1}\cdot\hat{2}$
5 | $\langle\text{2B},\text{2d}\rangle$ | $\text{SL}_{2}(\mathbb{Z})$ | ✓ | 0
$\langle\text{2B},\text{2e}\rangle$ | $\Gamma_{1}(2)$ | ✓ | 0
$\langle\text{2B},\text{4a}\rangle$ | $\Gamma_{1}(4)$ |  | $2i(\theta^{0}_{20,2}-\theta^{0}_{20,18})\cdot\hat{2}+2i(\theta^{0}_{20,6}-\theta^{0}_{20,14})\cdot\hat{4}$
6+3 | $\langle\text{2A},\text{2a}\rangle$ | $\text{SL}_{2}(\mathbb{Z})$ | ✓ | 0
$\langle\text{2A},\text{2b}\rangle$ | $\Gamma_{1}(2)$ | ✓ | 0
$\langle\text{2A},\text{2c}\rangle$ | $\Gamma_{1}(2)$ | ✓ | 0
$\langle\text{2A},\text{2b}\rangle$ | $\Gamma(2)$ | ✓ | 0
$\langle\text{2A},\text{6a}\rangle$ | $\Gamma_{0}(3)$ | ✓ | 0
$\langle\text{2A},\text{6a}\rangle$ | $\Gamma_{0}(3)$ | ✓ | 0
$\langle\text{2A},\text{2a}\rangle$ | $\text{SL}_{2}(\mathbb{Z})$ | ✓ | 0
$\langle\text{2B},\text{4b}\rangle$ | $\text{H}(2,2,4)$ | ✓ | 0
|  |  |  |

Next we would like to further constrain the space of potential new functions
of generalised umbral moonshine. Consider the congruence condition, which
states that the set of $q$-powers of $H_{(g,h),r}$ with non-vanishing
coefficients must be a subset of that of $H_{(g,e),r}$, a property that
follows from the finite group property (3.8). This defines a subspace
$\Theta_{(g,h),r}$ generated by all theta functions
$\theta^{0}_{m^{\prime},s}$ that $H_{(g,h),r}$ must lie in. Note that till now
the space $\Theta_{(g,h),r}$ is independent of the twining element $h$, as
long as it generates a new group with the given twisting element $g$. We can
further restrict this space in two ways. First we require that also
$H_{\gamma(g,h)}$ must have the correct $q$-powers for all $\gamma\in{\rm
SL}_{2}({\mathbb{Z}})$. Second, elements of $\Gamma_{(g,h)}$ act as a linear
transformation on this space, denoted $\sigma:\Gamma_{(g,h)}\to{\rm
End}(\Theta_{(g,h),r})$ , and $H_{(g,h),r}$ must lie in the simultaneous
eigenspace of $\sigma(\gamma)$ for all $\gamma\in\Gamma_{(g,h)}$, with the
eigenvalues given by the multiplier system $\xi_{(g,h)}$ (cf. equation 3.4).
In all of the unobstructed new groups, these simultaneous eigenspaces has
dimension either 0 or 1. Together with the requirement of the existence of a
decomposition of the first homogeneous into projective representations, this
fixes all the new functions.

In Table 5.2 we also record all the new functions up to a phase. To avoid
cluttering we have adopted the shorthand notation

$\hat{i}:=\theta_{m,i}(\tau,\zeta)-\theta_{m,-i}(\tau,\zeta)$ (5.8)

for $i\in{\mathbb{Z}}/2m{\mathbb{Z}}$.

##  6 Summary and Discussion

In this paper we introduced the notion of generalised umbral moonshine.
Generalised umbral moonshine can be regarded as an extension of umbral
moonshine and is analogous to the generalised monstrous moonshine, albeit with
significant conceptual and technical differences. When restricting to the
$A_{1}^{24}$ case it reduces to the generalised Mathieu moonshine analysed in
[1].

We conjecture that the deformed Drinfel’d double of finite groups is the key
structure underlying the (generalised) umbral moonshine phenomenon. More
specifically, the generalised umbral moonshine conjecture states that for each
of the 23 lambencies of umbral moonshine there is a way to assign an infinite-
dimensional module for a deformed Drinfel’d double of the umbral group such
that its graded character is given by the twisted-twined functions which we
specified in §5. More explicitly, we proposed in §3 six conditions of
generalised umbral moonshine conjecture and provided evidence in the form of
proposing all the twisted–twined functions for all cases of generalised umbral
moonshine.

It is interesting that one does not obtain new mock modular forms (with non-
vanishing shadow) in this generalisation of umbral moonshine. As we reviewed
in §2.2, the origin of the mockness of the the umbral moonshine mock modular
forms $H_{g}^{(\ell)}$ is the poles of the meromorphic Jacobi forms
$\psi_{g}^{(\ell)}$ (2.11), and the residues of the poles of these functions
are given by characters of the umbral group. From (3.2) and the conjecture in
§5.1, we see that the shadow of the mock modular form $H_{(g,h)}^{(\ell)}$
when $\langle g,h\rangle$ is an old group can be obtained from that of certain
umbral moonshine function $H_{g^{\prime}}^{(\ell)}$. In the case when $\langle
g,h\rangle$ is a new group, the shadow of $H_{(g,h)}^{(\ell)}$ vanishes
following the discussion in §5.2. In fact, all the new functions, which do not
play a role in umbral moonshine itself, are given by sums of theta functions
(see Table 5.2).

The existence of generalised umbral moonshine is particularly interesting in
view of the absence (so far) of a clear relation between umbral moonshine and
conventional CFTs666From this point of view, the fact that there is a
generalised moonshine for each of the 23 cases of umbral moonshine is perhaps
more surprising than the existence of generalised Mathieu moonshine alone.
This is because, unlike the other 22 cases of umbral moonshine, Mathieu
moonshine has an obvious relation to CFTs. Indeed, in [1] the existence of
generalised Mathieu moonshine is argued to suggest that a holomorphic VOA may
be underlying Mathieu moonshine. . As mentioned in §2.1, the generalised
monstrous moonshine is believed to be a manifestation of the fact that one can
define the twisted modules in view of the existence of the moonshine VOA. More
generally, the physical context of the deformed Drinfel’d double of a finite
group is usually assumed to be that of holomorphic orbifold CFTs or closely
related structures such as 3D TQFTs. The important lesson generalised umbral
moonshine has taught us is the following: whatever algebraic structure
underlies umbral moonshine, it has to retain the intimate relation to the
deformed Drinfel’d double as chiral conformal field theories do.

## Acknowledgements

We are grateful to John Duncan, Simon Lentner, Terry Gannon, Erik Verlinde for
helpful discussions. We especially John Duncan for many of the group
descriptions and for helpful comments on an earlier version of the manuscript.
The work of M.C. and D.W. was supported by ERC starting grant H2020 ERC StG
2014.

##  Appendix A Examples

In this appendix we focus on three examples of $g$-twisted generalised umbral
moonshine functions for lambency $\ell=3,4,6+3$ of index $m=3,4,6$
respectively. In particular, we list the independent components of the vector-
valued mock modular forms $H_{(g,h)}=(H_{(g,h),r})$,
$r\in{\mathbb{Z}}/2m{\mathbb{Z}}$, for all $h\in C_{G^{\ell}}(g)$. Our
notation for the conjugacy classes can be found in the attached file. In these
tables, the rational numbers in the first column denotes the $q$-power of the
corresponding terms in the Fourier expansion of $H_{(g,h),r}$. We also list
the projective character table for the projective representations compatible
with the 3-cocycle, as well as the decompositions of the first few homogeneous
components $K^{g}_{r,\alpha}$ of the conjectural generalised umbral moonshine
module for these cases (cf. equation 3.8). To increase readability we use “.”
to denote zero.

###  A.1 $\ell$=3, 4B Twist

Table 3: Projective Character Table For $C_{G^{(3)}}(g)$

| $1a$ | $2a$ | $2b$ | $2c$ | $2d$ | $2e$ | $4a$ | $4b$ | $4c$ | $4d$ | $4e$ | $4f$ | $8a$ | $8b$ | $8c$ | $8d$ | $8e$ | $8f$ | $8g$ | $8h$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\chi_{1}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
$\chi_{2}$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$ | $1$
$\chi_{3}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$
$\chi_{4}$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$
$\chi_{5}$ | $1$ | $-1$ | $1$ | $1$ | $-1$ | $-1$ | $i$ | $-i$ | $-i$ | $i$ | $-i$ | $i$ | $\zeta_{8}^{7}$ | $\zeta_{8}^{3}$ | $\zeta_{8}^{5}$ | $\zeta_{8}$ | $\zeta_{8}^{3}$ | $\zeta_{8}^{7}$ | $\zeta_{8}$ | $\zeta_{8}^{5}$
$\chi_{6}$ | $1$ | $-1$ | $1$ | $1$ | $-1$ | $-1$ | $i$ | $-i$ | $-i$ | $i$ | $-i$ | $i$ | $\zeta_{8}^{3}$ | $\zeta_{8}^{7}$ | $\zeta_{8}$ | $\zeta_{8}^{5}$ | $\zeta_{8}^{7}$ | $\zeta_{8}^{3}$ | $\zeta_{8}^{5}$ | $\zeta_{8}$
$\chi_{7}$ | $1$ | $-1$ | $1$ | $1$ | $-1$ | $-1$ | $-i$ | $i$ | $i$ | $-i$ | $i$ | $-i$ | $\zeta_{8}^{5}$ | $\zeta_{8}$ | $\zeta_{8}^{7}$ | $\zeta_{8}^{3}$ | $\zeta_{8}$ | $\zeta_{8}^{5}$ | $\zeta_{8}^{3}$ | $\zeta_{8}^{7}$
$\chi_{8}$ | $1$ | $-1$ | $1$ | $1$ | $-1$ | $-1$ | $-i$ | $i$ | $i$ | $-i$ | $i$ | $-i$ | $\zeta_{8}$ | $\zeta_{8}^{5}$ | $\zeta_{8}^{3}$ | $\zeta_{8}^{7}$ | $\zeta_{8}^{5}$ | $\zeta_{8}$ | $\zeta_{8}^{7}$ | $\zeta_{8}^{3}$
$\chi_{9}$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $i$ | $i$ | $-i$ | $-i$ | $-i$ | $i$ | $\zeta_{8}^{7}$ | $\zeta_{8}^{7}$ | $\zeta_{8}^{5}$ | $\zeta_{8}^{5}$ | $\zeta_{8}^{3}$ | $\zeta_{8}^{3}$ | $\zeta_{8}$ | $\zeta_{8}$
$\chi_{10}$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $i$ | $i$ | $-i$ | $-i$ | $-i$ | $i$ | $\zeta_{8}^{3}$ | $\zeta_{8}^{3}$ | $\zeta_{8}$ | $\zeta_{8}$ | $\zeta_{8}^{7}$ | $\zeta_{8}^{7}$ | $\zeta_{8}^{5}$ | $\zeta_{8}^{5}$
$\chi_{11}$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $-i$ | $-i$ | $i$ | $i$ | $i$ | $-i$ | $\zeta_{8}^{5}$ | $\zeta_{8}^{5}$ | $\zeta_{8}^{7}$ | $\zeta_{8}^{7}$ | $\zeta_{8}$ | $\zeta_{8}$ | $\zeta_{8}^{3}$ | $\zeta_{8}^{3}$
$\chi_{12}$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $-i$ | $-i$ | $i$ | $i$ | $i$ | $-i$ | $\zeta_{8}$ | $\zeta_{8}$ | $\zeta_{8}^{3}$ | $\zeta_{8}^{3}$ | $\zeta_{8}^{5}$ | $\zeta_{8}^{5}$ | $\zeta_{8}^{7}$ | $\zeta_{8}^{7}$
$\chi_{13}$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$ | $-1$ | $-i$ | $i$ | $i$ | $-i$ | $-i$ | $i$ | $i$ | $-i$
$\chi_{14}$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$ | $-1$ | $i$ | $-i$ | $-i$ | $i$ | $i$ | $-i$ | $-i$ | $i$
$\chi_{15}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-i$ | $-i$ | $i$ | $i$ | $-i$ | $-i$ | $i$ | $i$
$\chi_{16}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $i$ | $i$ | $-i$ | $-i$ | $i$ | $i$ | $-i$ | $-i$
$\chi_{17}$ | $2$ | . | $-2$ | . | $-2$ | $2$ | $2$ | . | $2$ | . | $-2$ | $-2$ | . | . | . | . | . | . | . | .
$\chi_{18}$ | $2$ | . | $-2$ | . | $-2$ | $2$ | $-2$ | . | $-2$ | . | $2$ | $2$ | . | . | . | . | . | . | . | .
$\chi_{19}$ | $2$ | . | $-2$ | . | $2$ | $-2$ | $2i$ | . | $-2i$ | . | $2i$ | $-2i$ | . | . | . | . | . | . | . | .
$\chi_{20}$ | $2$ | . | $-2$ | . | $2$ | $-2$ | $-2i$ | . | $2i$ | . | $-2i$ | $2i$ | . | . | . | . | . | . | . | .

Table 4: Coefficients of $H_{(g,h),1}$

| $1a$ | $2a$ | $2b$ | $2c$ | $2d$ | $2e$ | $4a$ | $4b$ | $4c$ | $4d$ | $4e$ | $4f$ | $8a$ | $8b$ | $8c$ | $8d$ | $8e$ | $8f$ | $8g$ | $8h$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$2/12$ | $2$ | . | $2$ | . | $2$ | $2$ | $2$ | . | $2$ | . | $2$ | $2$ | . | $2$ | . | $2$ | . | $2$ | . | $2$
$5/12$ | $4$ | . | $-4$ | . | $4$ | $-4$ | $4i$ | . | $-4i$ | . | $4i$ | $-4i$ | . | . | . | . | . | . | . | .
$8/12$ | $6$ | . | $6$ | . | $6$ | $6$ | $-6$ | . | $-6$ | . | $-6$ | $-6$ | . | $2i$ | . | $-2i$ | . | $2i$ | . | $-2i$
$11/12$ | $8$ | . | $-8$ | . | $8$ | $-8$ | $-8i$ | . | $8i$ | . | $-8i$ | $8i$ | . | . | . | . | . | . | . | .
$14/12$ | $12$ | . | $12$ | . | $12$ | $12$ | $12$ | . | $12$ | . | $12$ | $12$ | . | $-4$ | . | $-4$ | . | $-4$ | . | $-4$
$17/12$ | $16$ | . | $-16$ | . | $16$ | $-16$ | $16i$ | . | $-16i$ | . | $16i$ | $-16i$ | . | . | . | . | . | . | . | .
$20/12$ | $20$ | . | $20$ | . | $20$ | $20$ | $-20$ | . | $-20$ | . | $-20$ | $-20$ | . | $-4i$ | . | $4i$ | . | $-4i$ | . | $4i$
$23/12$ | $28$ | . | $-28$ | . | $28$ | $-28$ | $-28i$ | . | $28i$ | . | $-28i$ | $28i$ | . | . | . | . | . | . | . | .
$26/12$ | $36$ | . | $36$ | . | $36$ | $36$ | $36$ | . | $36$ | . | $36$ | $36$ | . | $4$ | . | $4$ | . | $4$ | . | $4$
$29/12$ | $44$ | . | $-44$ | . | $44$ | $-44$ | $44i$ | . | $-44i$ | . | $44i$ | $-44i$ | . | . | . | . | . | . | . | .

Table 5: Coefficients of $H_{(g,h),2}$

| $1a$ | $2a$ | $2b$ | $2c$ | $2d$ | $2e$ | $4a$ | $4b$ | $4c$ | $4d$ | $4e$ | $4f$ | $8a$ | $8b$ | $8c$ | $8d$ | $8e$ | $8f$ | $8g$ | $8h$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$2/12$ | $2$ | . | $2$ | . | $-2$ | $-2$ | $2i$ | . | $-2i$ | . | $-2i$ | $2i$ | $2\zeta_{8}^{7}$ | . | $2\zeta_{8}^{5}$ | . | $2\zeta_{8}^{3}$ | . | $2\zeta_{8}$ | .
$5/12$ | $4$ | . | $-4$ | . | $-4$ | $4$ | $-4$ | . | $-4$ | . | $4$ | $4$ | . | . | . | . | . | . | . | .
$8/12$ | $6$ | . | $6$ | . | $-6$ | $-6$ | $-6i$ | . | $6i$ | . | $6i$ | $-6i$ | $2\zeta_{8}^{5}$ | . | $2\zeta_{8}^{7}$ | . | $2\zeta_{8}$ | . | $2\zeta_{8}^{3}$ | .
$11/12$ | $8$ | . | $-8$ | . | $-8$ | $8$ | $8$ | . | $8$ | . | $-8$ | $-8$ | . | . | . | . | . | . | . | .
$14/12$ | $12$ | . | $12$ | . | $-12$ | $-12$ | $12i$ | . | $-12i$ | . | $-12i$ | $12i$ | $4\zeta_{8}^{3}$ | . | $4\zeta_{8}$ | . | $4\zeta_{8}^{7}$ | . | $4\zeta_{8}^{5}$ | .
$17/12$ | $16$ | . | $-16$ | . | $-16$ | $16$ | $-16$ | . | $-16$ | . | $16$ | $16$ | . | . | . | . | . | . | . | .
$20/12$ | $20$ | . | $20$ | . | $-20$ | $-20$ | $-20i$ | . | $20i$ | . | $20i$ | $-20i$ | $4\zeta_{8}$ | . | $4\zeta_{8}^{3}$ | . | $4\zeta_{8}^{5}$ | . | $4\zeta_{8}^{7}$ | .
$23/12$ | $28$ | . | $-28$ | . | $-28$ | $28$ | $28$ | . | $28$ | . | $-28$ | $-28$ | . | . | . | . | . | . | . | .
$26/12$ | $36$ | . | $36$ | . | $-36$ | $-36$ | $36i$ | . | $-36i$ | . | $-36i$ | $36i$ | $4\zeta_{8}^{7}$ | . | $4\zeta_{8}^{5}$ | . | $4\zeta_{8}^{3}$ | . | $4\zeta_{8}$ | .
$29/12$ | $44$ | . | $-44$ | . | $-44$ | $44$ | $-44$ | . | $-44$ | . | $44$ | $44$ | . | . | . | . | . | . | . | .

Table 6: Decompositions for the First Few $K^{g}_{1,\alpha}$

| $\chi_{1}$ | $\chi_{2}$ | $\chi_{3}$ | $\chi_{4}$ | $\chi_{5}$ | $\chi_{6}$ | $\chi_{7}$ | $\chi_{8}$ | $\chi_{9}$ | $\chi_{10}$ | $\chi_{11}$ | $\chi_{12}$ | $\chi_{13}$ | $\chi_{14}$ | $\chi_{15}$ | $\chi_{16}$ | $\chi_{17}$ | $\chi_{18}$ | $\chi_{19}$ | $\chi_{20}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$2/12$ | $1$ | $1$ | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | .
$5/12$ | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | $2$ | .
$8/12$ | . | . | . | . | . | . | . | . | . | . | . | . | $2$ | $1$ | $1$ | $2$ | . | . | . | .
$11/12$ | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | $4$
$14/12$ | $2$ | $2$ | $4$ | $4$ | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | .
$17/12$ | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | $8$ | .
$20/12$ | . | . | . | . | . | . | . | . | . | . | . | . | $4$ | $6$ | $6$ | $4$ | . | . | . | .
$23/12$ | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | $14$
$26/12$ | $10$ | $10$ | $8$ | $8$ | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | .
$29/12$ | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | $22$ | .

Table 7: Decompositions for the First Few $K^{g}_{2,\alpha}$

| $\chi_{1}$ | $\chi_{2}$ | $\chi_{3}$ | $\chi_{4}$ | $\chi_{5}$ | $\chi_{6}$ | $\chi_{7}$ | $\chi_{8}$ | $\chi_{9}$ | $\chi_{10}$ | $\chi_{11}$ | $\chi_{12}$ | $\chi_{13}$ | $\chi_{14}$ | $\chi_{15}$ | $\chi_{16}$ | $\chi_{17}$ | $\chi_{18}$ | $\chi_{19}$ | $\chi_{20}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$2/12$ | . | . | . | . | $1$ | . | . | . | $1$ | . | . | . | . | . | . | . | . | . | . | .
$5/12$ | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | $2$ | . | .
$8/12$ | . | . | . | . | . | . | $2$ | $1$ | . | . | $2$ | $1$ | . | . | . | . | . | . | . | .
$11/12$ | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | $4$ | . | . | .
$14/12$ | . | . | . | . | $2$ | $4$ | . | . | $2$ | $4$ | . | . | . | . | . | . | . | . | . | .
$17/12$ | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | $8$ | . | .
$20/12$ | . | . | . | . | . | . | $4$ | $6$ | . | . | $4$ | $6$ | . | . | . | . | . | . | . | .
$23/12$ | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | $14$ | . | . | .
$26/12$ | . | . | . | . | $10$ | $8$ | . | . | $10$ | $8$ | . | . | . | . | . | . | . | . | . | .
$29/12$ | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | $22$ | . | .

###  A.2 $\ell$=4, 2B Twist

Table 8: Projective Character Table For $C_{G^{(4)}}(g)$

| $1a$ | $2a$ | $2b$ | $2c$ | $2d$ | $2e$ | $3a$ | $3b$ | $4a$ | $4b$ | $4c$ | $6a$ | $6b$ | $6c$ | $6d$ | $6e$ | $6f$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\chi_{1}$ | $2$ | . | . | $2$ | $2$ | $2$ | $1$ | $1$ | $2$ | . | . | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
$\chi_{2}$ | $2$ | . | . | $-2$ | $2$ | $-2$ | $1$ | $1$ | . | . | $2$ | $1$ | $-1$ | $-1$ | $-1$ | $-1$ | $1$
$\chi_{3}$ | $2$ | . | . | $-2$ | $-2$ | $2$ | $1$ | $1$ | . | $2$ | . | $-1$ | $-1$ | $-1$ | $1$ | $1$ | $-1$
$\chi_{4}$ | $2$ | . | . | $2$ | $2$ | $2$ | $\zeta_{3}^{2}$ | $\zeta_{3}$ | $2$ | . | . | $\zeta_{3}$ | $\zeta_{3}^{2}$ | $\zeta_{3}$ | $\zeta_{3}^{2}$ | $\zeta_{3}$ | $\zeta_{3}^{2}$
$\chi_{5}$ | $2$ | . | . | $2$ | $2$ | $2$ | $\zeta_{3}$ | $\zeta_{3}^{2}$ | $2$ | . | . | $\zeta_{3}^{2}$ | $\zeta_{3}$ | $\zeta_{3}^{2}$ | $\zeta_{3}$ | $\zeta_{3}^{2}$ | $\zeta_{3}$
$\chi_{6}$ | $2$ | . | . | $-2$ | $2$ | $-2$ | $\zeta_{3}^{2}$ | $\zeta_{3}$ | . | . | $2$ | $\zeta_{3}$ | $\zeta_{6}$ | $\zeta_{6}^{5}$ | $\zeta_{6}$ | $\zeta_{6}^{5}$ | $\zeta_{3}^{2}$
$\chi_{7}$ | $2$ | . | . | $-2$ | $2$ | $-2$ | $\zeta_{3}$ | $\zeta_{3}^{2}$ | . | . | $2$ | $\zeta_{3}^{2}$ | $\zeta_{6}^{5}$ | $\zeta_{6}$ | $\zeta_{6}^{5}$ | $\zeta_{6}$ | $\zeta_{3}$
$\chi_{8}$ | $2$ | . | . | $-2$ | $-2$ | $2$ | $\zeta_{3}^{2}$ | $\zeta_{3}$ | . | $2$ | . | $\zeta_{6}^{5}$ | $\zeta_{6}$ | $\zeta_{6}^{5}$ | $\zeta_{3}^{2}$ | $\zeta_{3}$ | $\zeta_{6}$
$\chi_{9}$ | $2$ | . | . | $-2$ | $-2$ | $2$ | $\zeta_{3}$ | $\zeta_{3}^{2}$ | . | $2$ | . | $\zeta_{6}$ | $\zeta_{6}^{5}$ | $\zeta_{6}$ | $\zeta_{3}$ | $\zeta_{3}^{2}$ | $\zeta_{6}^{5}$
$\chi_{10}$ | $4$ | . | . | $4$ | $-4$ | $-4$ | $-1$ | $-1$ | . | . | . | $1$ | $-1$ | $-1$ | $1$ | $1$ | $1$
$\chi_{11}$ | $4$ | . | . | $4$ | $-4$ | $-4$ | $\zeta_{6}$ | $\zeta_{6}^{5}$ | . | . | . | $\zeta_{3}$ | $\zeta_{6}$ | $\zeta_{6}^{5}$ | $\zeta_{3}^{2}$ | $\zeta_{3}$ | $\zeta_{3}^{2}$
$\chi_{12}$ | $4$ | . | . | $4$ | $-4$ | $-4$ | $\zeta_{6}^{5}$ | $\zeta_{6}$ | . | . | . | $\zeta_{3}^{2}$ | $\zeta_{6}^{5}$ | $\zeta_{6}$ | $\zeta_{3}$ | $\zeta_{3}^{2}$ | $\zeta_{3}$
$\chi_{13}$ | $6$ | . | . | $6$ | $6$ | $6$ | . | . | $-2$ | . | . | . | . | . | . | . | .
$\chi_{14}$ | $6$ | . | . | $-6$ | $6$ | $-6$ | . | . | . | . | $-2$ | . | . | . | . | . | .
$\chi_{15}$ | $6$ | . | . | $-6$ | $-6$ | $6$ | . | . | . | $-2$ | . | . | . | . | . | . | .

Table 9: Coefficients of $H_{(g,h),1}$

| $1a$ | $2a$ | $2b$ | $2c$ | $2d$ | $2e$ | $3a$ | $3b$ | $4a$ | $4b$ | $4c$ | $6a$ | $6b$ | $6c$ | $6d$ | $6e$ | $6f$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$3/16$ | $4$ | . | . | $4$ | $4$ | $4$ | $-1$ | $-1$ | $4$ | . | . | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$
$11/16$ | $4$ | . | . | $-4$ | $4$ | $-4$ | $2$ | $2$ | . | . | $4$ | $2$ | $-2$ | $-2$ | $-2$ | $-2$ | $2$
$19/16$ | $12$ | . | . | $12$ | $12$ | $12$ | . | . | $-4$ | . | . | . | . | . | . | . | .
$27/16$ | $16$ | . | . | $-16$ | $16$ | $-16$ | $-1$ | $-1$ | . | . | . | $-1$ | $1$ | $1$ | $1$ | $1$ | $-1$
$35/16$ | $24$ | . | . | $24$ | $24$ | $24$ | . | . | $8$ | . | . | . | . | . | . | . | .
$43/16$ | $36$ | . | . | $-36$ | $36$ | $-36$ | . | . | . | . | $4$ | . | . | . | . | . | .
$51/16$ | $56$ | . | . | $56$ | $56$ | $56$ | $-2$ | $-2$ | $-8$ | . | . | $-2$ | $-2$ | $-2$ | $-2$ | $-2$ | $-2$
$59/16$ | $76$ | . | . | $-76$ | $76$ | $-76$ | $2$ | $2$ | . | . | $-4$ | $2$ | $-2$ | $-2$ | $-2$ | $-2$ | $2$
$67/16$ | $108$ | . | . | $108$ | $108$ | $108$ | . | . | $12$ | . | . | . | . | . | . | . | .
$75/16$ | $148$ | . | . | $-148$ | $148$ | $-148$ | $-1$ | $-1$ | . | . | $4$ | $-1$ | $1$ | $1$ | $1$ | $1$ | $-1$

Table 10: Coefficients of $H_{(g,h),2}$

| $1a$ | $2a$ | $2b$ | $2c$ | $2d$ | $2e$ | $3a$ | $3b$ | $4a$ | $4b$ | $4c$ | $6a$ | $6b$ | $6c$ | $6d$ | $6e$ | $6f$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$0/2$ | $2$ | . | . | $-2$ | $-2$ | $2$ | $1$ | $1$ | . | $2$ | . | $-1$ | $-1$ | $-1$ | $1$ | $1$ | $-1$
$1/2$ | $8$ | . | . | $8$ | $-8$ | $-8$ | $-2$ | $-2$ | . | . | . | $2$ | $-2$ | $-2$ | $2$ | $2$ | $2$
$2/2$ | $12$ | . | . | $-12$ | $-12$ | $12$ | . | . | . | $-4$ | . | . | . | . | . | . | .
$3/2$ | $16$ | . | . | $16$ | $-16$ | $-16$ | $2$ | $2$ | . | . | . | $-2$ | $2$ | $2$ | $-2$ | $-2$ | $-2$
$4/2$ | $32$ | . | . | $-32$ | $-32$ | $32$ | $-2$ | $-2$ | . | . | . | $2$ | $2$ | $2$ | $-2$ | $-2$ | $2$
$5/2$ | $48$ | . | . | $48$ | $-48$ | $-48$ | . | . | . | . | . | . | . | . | . | . | .
$6/2$ | $64$ | . | . | $-64$ | $-64$ | $64$ | $2$ | $2$ | . | . | . | $-2$ | $-2$ | $-2$ | $2$ | $2$ | $-2$
$7/2$ | $96$ | . | . | $96$ | $-96$ | $-96$ | . | . | . | . | . | . | . | . | . | . | .
$8/2$ | $132$ | . | . | $-132$ | $-132$ | $132$ | . | . | . | $4$ | . | . | . | . | . | . | .
$9/2$ | $184$ | . | . | $184$ | $-184$ | $-184$ | $2$ | $2$ | . | . | . | $-2$ | $2$ | $2$ | $-2$ | $-2$ | $-2$

Table 11: Coefficients of $H_{(g,h),3}$

| $1a$ | $2a$ | $2b$ | $2c$ | $2d$ | $2e$ | $3a$ | $3b$ | $4a$ | $4b$ | $4c$ | $6a$ | $6b$ | $6c$ | $6d$ | $6e$ | $6f$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$3/16$ | $4$ | . | . | $-4$ | $4$ | $-4$ | $-1$ | $-1$ | . | . | $4$ | $-1$ | $1$ | $1$ | $1$ | $1$ | $-1$
$11/16$ | $4$ | . | . | $4$ | $4$ | $4$ | $2$ | $2$ | $4$ | . | . | $2$ | $2$ | $2$ | $2$ | $2$ | $2$
$19/16$ | $12$ | . | . | $-12$ | $12$ | $-12$ | . | . | . | . | $-4$ | . | . | . | . | . | .
$27/16$ | $16$ | . | . | $16$ | $16$ | $16$ | $-1$ | $-1$ | . | . | . | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$
$35/16$ | $24$ | . | . | $-24$ | $24$ | $-24$ | . | . | . | . | $8$ | . | . | . | . | . | .
$43/16$ | $36$ | . | . | $36$ | $36$ | $36$ | . | . | $4$ | . | . | . | . | . | . | . | .
$51/16$ | $56$ | . | . | $-56$ | $56$ | $-56$ | $-2$ | $-2$ | . | . | $-8$ | $-2$ | $2$ | $2$ | $2$ | $2$ | $-2$
$59/16$ | $76$ | . | . | $76$ | $76$ | $76$ | $2$ | $2$ | $-4$ | . | . | $2$ | $2$ | $2$ | $2$ | $2$ | $2$
$67/16$ | $108$ | . | . | $-108$ | $108$ | $-108$ | . | . | . | . | $12$ | . | . | . | . | . | .
$75/16$ | $148$ | . | . | $148$ | $148$ | $148$ | $-1$ | $-1$ | $4$ | . | . | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$

Table 12: Decompositions for the First Few $K^{g}_{1,\alpha}$

| $\chi_{1}$ | $\chi_{2}$ | $\chi_{3}$ | $\chi_{4}$ | $\chi_{5}$ | $\chi_{6}$ | $\chi_{7}$ | $\chi_{8}$ | $\chi_{9}$ | $\chi_{10}$ | $\chi_{11}$ | $\chi_{12}$ | $\chi_{13}$ | $\chi_{14}$ | $\chi_{15}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$3/16$ | . | . | . | $1$ | $1$ | . | . | . | . | . | . | . | . | . | .
$11/16$ | . | $2$ | . | . | . | . | . | . | . | . | . | . | . | . | .
$19/16$ | . | . | . | . | . | . | . | . | . | . | . | . | $2$ | . | .
$27/16$ | . | . | . | . | . | $1$ | $1$ | . | . | . | . | . | . | $2$ | .
$35/16$ | $2$ | . | . | $2$ | $2$ | . | . | . | . | . | . | . | $2$ | . | .
$43/16$ | . | $2$ | . | . | . | $2$ | $2$ | . | . | . | . | . | . | $4$ | .
$51/16$ | . | . | . | $2$ | $2$ | . | . | . | . | . | . | . | $8$ | . | .
$59/16$ | . | $4$ | . | . | . | $2$ | $2$ | . | . | . | . | . | . | $10$ | .
$67/16$ | $6$ | . | . | $6$ | $6$ | . | . | . | . | . | . | . | $12$ | . | .
$75/16$ | . | $6$ | . | . | . | $7$ | $7$ | . | . | . | . | . | . | $18$ | .

Table 13: Decompositions for the First Few $K^{g}_{2,\alpha}$

| $\chi_{1}$ | $\chi_{2}$ | $\chi_{3}$ | $\chi_{4}$ | $\chi_{5}$ | $\chi_{6}$ | $\chi_{7}$ | $\chi_{8}$ | $\chi_{9}$ | $\chi_{10}$ | $\chi_{11}$ | $\chi_{12}$ | $\chi_{13}$ | $\chi_{14}$ | $\chi_{15}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$0/2$ | . | . | $1$ | . | . | . | . | . | . | . | . | . | . | . | .
$1/2$ | . | . | . | . | . | . | . | . | . | $2$ | . | . | . | . | .
$2/2$ | . | . | . | . | . | . | . | . | . | . | . | . | . | . | $2$
$3/2$ | . | . | . | . | . | . | . | . | . | . | $2$ | $2$ | . | . | .
$4/2$ | . | . | . | . | . | . | . | $2$ | $2$ | . | . | . | . | . | $4$
$5/2$ | . | . | . | . | . | . | . | . | . | $4$ | $4$ | $4$ | . | . | .
$6/2$ | . | . | $4$ | . | . | . | . | $2$ | $2$ | . | . | . | . | . | $8$
$7/2$ | . | . | . | . | . | . | . | . | . | $8$ | $8$ | $8$ | . | . | .
$8/2$ | . | . | $6$ | . | . | . | . | $6$ | $6$ | . | . | . | . | . | $16$
$9/2$ | . | . | . | . | . | . | . | . | . | $14$ | $16$ | $16$ | . | . | .

Table 14: Decompositions for the First Few $K^{g}_{3,\alpha}$

| $\chi_{1}$ | $\chi_{2}$ | $\chi_{3}$ | $\chi_{4}$ | $\chi_{5}$ | $\chi_{6}$ | $\chi_{7}$ | $\chi_{8}$ | $\chi_{9}$ | $\chi_{10}$ | $\chi_{11}$ | $\chi_{12}$ | $\chi_{13}$ | $\chi_{14}$ | $\chi_{15}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$3/16$ | . | . | . | . | . | $1$ | $1$ | . | . | . | . | . | . | . | .
$11/16$ | $2$ | . | . | . | . | . | . | . | . | . | . | . | . | . | .
$19/16$ | . | . | . | . | . | . | . | . | . | . | . | . | . | $2$ | .
$27/16$ | . | . | . | $1$ | $1$ | . | . | . | . | . | . | . | $2$ | . | .
$35/16$ | . | $2$ | . | . | . | $2$ | $2$ | . | . | . | . | . | . | $2$ | .
$43/16$ | $2$ | . | . | $2$ | $2$ | . | . | . | . | . | . | . | $4$ | . | .
$51/16$ | . | . | . | . | . | $2$ | $2$ | . | . | . | . | . | . | $8$ | .
$59/16$ | $4$ | . | . | $2$ | $2$ | . | . | . | . | . | . | . | $10$ | . | .
$67/16$ | . | $6$ | . | . | . | $6$ | $6$ | . | . | . | . | . | . | $12$ | .
$75/16$ | $6$ | . | . | $7$ | $7$ | . | . | . | . | . | . | . | $18$ | . | .

###  A.3 $\ell$=6+3, 3B Twist

Table 15: Projective Character Table For $C_{G^{(6+3)}}(g)$ | $1a$ | $2a$ | $3a$ | $3b$ | $3c$ | $3d$ | $3e$ | $6a$ | $6b$
---|---|---|---|---|---|---|---|---|---
$\chi_{1}$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$
$\chi_{2}$ | $1$ | $-1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $-1$ | $-1$
$\chi_{3}$ | $1$ | $-1$ | $\zeta_{3}^{2}$ | $\zeta_{3}$ | $\zeta_{3}$ | $1$ | $\zeta_{3}^{2}$ | $\zeta_{6}$ | $\zeta_{6}^{5}$
$\chi_{4}$ | $1$ | $-1$ | $\zeta_{3}$ | $\zeta_{3}^{2}$ | $\zeta_{3}^{2}$ | $1$ | $\zeta_{3}$ | $\zeta_{6}^{5}$ | $\zeta_{6}$
$\chi_{5}$ | $1$ | $1$ | $\zeta_{3}^{2}$ | $\zeta_{3}$ | $\zeta_{3}$ | $1$ | $\zeta_{3}^{2}$ | $\zeta_{3}^{2}$ | $\zeta_{3}$
$\chi_{6}$ | $1$ | $1$ | $\zeta_{3}$ | $\zeta_{3}^{2}$ | $\zeta_{3}^{2}$ | $1$ | $\zeta_{3}$ | $\zeta_{3}$ | $\zeta_{3}^{2}$
$\chi_{7}$ | $2$ | . | $2$ | $2$ | $-1$ | $-1$ | $-1$ | . | .
$\chi_{8}$ | $2$ | . | $2\zeta_{3}$ | $2\zeta_{3}^{2}$ | $\zeta_{6}$ | $-1$ | $\zeta_{6}^{5}$ | . | .
$\chi_{9}$ | $2$ | . | $2\zeta_{3}^{2}$ | $2\zeta_{3}$ | $\zeta_{6}^{5}$ | $-1$ | $\zeta_{6}$ | . | .
Table 16: Coefficients of $H_{(g,h),1}$ | $1a$ | $2a$ | $3a$ | $3b$ | $3c$ | $3d$ | $3e$ | $6a$ | $6b$
---|---|---|---|---|---|---|---|---|---
$5/72$ | $2$ | $2$ | $2$ | $2$ | $2$ | $2$ | $2$ | $2$ | $2$
$29/72$ | $2$ | $-2$ | $2\zeta_{3}$ | $2\zeta_{3}^{2}$ | $2\zeta_{3}^{2}$ | $2$ | $2\zeta_{3}$ | $2\zeta_{6}^{5}$ | $2\zeta_{6}$
$53/72$ | $4$ | . | $4\zeta_{3}^{2}$ | $4\zeta_{3}$ | $4\zeta_{3}$ | $4$ | $4\zeta_{3}^{2}$ | . | .
$77/72$ | $2$ | $-2$ | $2$ | $2$ | $2$ | $2$ | $2$ | $-2$ | $-2$
$101/72$ | $6$ | $2$ | $6\zeta_{3}$ | $6\zeta_{3}^{2}$ | $6\zeta_{3}^{2}$ | $6$ | $6\zeta_{3}$ | $2\zeta_{3}$ | $2\zeta_{3}^{2}$
$125/72$ | $6$ | $-2$ | $6\zeta_{3}^{2}$ | $6\zeta_{3}$ | $6\zeta_{3}$ | $6$ | $6\zeta_{3}^{2}$ | $2\zeta_{6}$ | $2\zeta_{6}^{5}$
$149/72$ | $8$ | . | $8$ | $8$ | $8$ | $8$ | $8$ | . | .
$173/72$ | $8$ | . | $8\zeta_{3}$ | $8\zeta_{3}^{2}$ | $8\zeta_{3}^{2}$ | $8$ | $8\zeta_{3}$ | . | .
$197/72$ | $12$ | $4$ | $12\zeta_{3}^{2}$ | $12\zeta_{3}$ | $12\zeta_{3}$ | $12$ | $12\zeta_{3}^{2}$ | $4\zeta_{3}^{2}$ | $4\zeta_{3}$
$221/72$ | $14$ | $-2$ | $14$ | $14$ | $14$ | $14$ | $14$ | $-2$ | $-2$
Table 17: Coefficients of $H_{(g,h),3}$ | $1a$ | $2a$ | $3a$ | $3b$ | $3c$ | $3d$ | $3e$ | $6a$ | $6b$
---|---|---|---|---|---|---|---|---|---
$5/72$ | $4$ | . | $4\zeta_{3}$ | $4\zeta_{3}^{2}$ | $2\zeta_{6}$ | $-2$ | $2\zeta_{6}^{5}$ | . | .
$29/72$ | $4$ | . | $4\zeta_{3}^{2}$ | $4\zeta_{3}$ | $2\zeta_{6}^{5}$ | $-2$ | $2\zeta_{6}$ | . | .
$53/72$ | $8$ | . | $8$ | $8$ | $-4$ | $-4$ | $-4$ | . | .
$77/72$ | $4$ | . | $4\zeta_{3}$ | $4\zeta_{3}^{2}$ | $2\zeta_{6}$ | $-2$ | $2\zeta_{6}^{5}$ | . | .
$101/72$ | $12$ | . | $12\zeta_{3}^{2}$ | $12\zeta_{3}$ | $6\zeta_{6}^{5}$ | $-6$ | $6\zeta_{6}$ | . | .
$125/72$ | $12$ | . | $12$ | $12$ | $-6$ | $-6$ | $-6$ | . | .
$149/72$ | $16$ | . | $16\zeta_{3}$ | $16\zeta_{3}^{2}$ | $8\zeta_{6}$ | $-8$ | $8\zeta_{6}^{5}$ | . | .
$173/72$ | $16$ | . | $16\zeta_{3}^{2}$ | $16\zeta_{3}$ | $8\zeta_{6}^{5}$ | $-8$ | $8\zeta_{6}$ | . | .
$197/72$ | $24$ | . | $24$ | $24$ | $-12$ | $-12$ | $-12$ | . | .
$221/72$ | $28$ | . | $28\zeta_{3}$ | $28\zeta_{3}^{2}$ | $14\zeta_{6}$ | $-14$ | $14\zeta_{6}^{5}$ | . | .
Table 18: Decompositions for the First Few $K^{g}_{1,\alpha}$ | $\chi_{1}$ | $\chi_{2}$ | $\chi_{3}$ | $\chi_{4}$ | $\chi_{5}$ | $\chi_{6}$ | $\chi_{7}$ | $\chi_{8}$ | $\chi_{9}$
---|---|---|---|---|---|---|---|---|---
$5/72$ | $2$ | . | . | . | . | . | . | . | .
$29/72$ | . | . | . | $2$ | . | . | . | . | .
$53/72$ | . | . | $2$ | . | $2$ | . | . | . | .
$77/72$ | . | $2$ | . | . | . | . | . | . | .
$101/72$ | . | . | . | $2$ | . | $4$ | . | . | .
$125/72$ | . | . | $4$ | . | $2$ | . | . | . | .
$149/72$ | $4$ | $4$ | . | . | . | . | . | . | .
$173/72$ | . | . | . | $4$ | . | $4$ | . | . | .
$197/72$ | . | . | $4$ | . | $8$ | . | . | . | .
$221/72$ | $6$ | $8$ | . | . | . | . | . | . | .
Table 19: Decompositions for the First Few $K^{g}_{3,\alpha}$ | $\chi_{1}$ | $\chi_{2}$ | $\chi_{3}$ | $\chi_{4}$ | $\chi_{5}$ | $\chi_{6}$ | $\chi_{7}$ | $\chi_{8}$ | $\chi_{9}$
---|---|---|---|---|---|---|---|---|---
$5/72$ | . | . | . | . | . | . | . | $2$ | .
$29/72$ | . | . | . | . | . | . | . | . | $2$
$53/72$ | . | . | . | . | . | . | $4$ | . | .
$77/72$ | . | . | . | . | . | . | . | $2$ | .
$101/72$ | . | . | . | . | . | . | . | . | $6$
$125/72$ | . | . | . | . | . | . | $6$ | . | .
$149/72$ | . | . | . | . | . | . | . | $8$ | .
$173/72$ | . | . | . | . | . | . | . | . | $8$
$197/72$ | . | . | . | . | . | . | $12$ | . | .
$221/72$ | . | . | . | . | . | . | . | $14$ | .

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