${\boldsymbol{\cal N}}$ = 2 SUSY & the Hexipentisteriruncicantitruncated
7-Simplex

Aleksander J. Cianciara111acianciara@princeton.edua,b, Zachary
Coleman222zachary.coleman@trinity.ox.ac.uk,c,d,e,
S. James Gates<EMAIL_ADDRESS>Youngik (Tom) Lee444youngik-
lee@brown.edu,g,h, and Ziyang Zhang555ziyang-zhang1@brown.edu,g,h

_ aJoseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA,
bInstitute for Advanced Study, Princeton, NJ 08540, USA,
cPepperdine University, Natural Science Division, Malibu, CA 90263, USA,
dMathematics Institute, University of Oxford, Oxford OX2 6GG, UK,
e Department of Physics, University of Oxford, Oxford OX1 3PU, UK,
fDepartment of Physics, University of Maryland,
College Park, MD 20742-4111, USA,
gBrown University, Department of Physics,
182 Hope Street, Barus & Holley 545, Providence, RI 02912, USA,
hBrown Center for Theoretical Physics,
340 Brook Street, Barus Hall, Providence, RI 02912, USA, _

ABSTRACT

We study algorithms for recursively creating arbitrary $N$-extended
‘supermultiplets’ given minimal matrix representations of off-shell, $\cal N$
= 1 supermultiplet matrices. We observe connections between the color vertex
problems in graph theory and the different supermultiplet sets locations in
the permutahedron by using the concepts of truncation and chromatic number.
The concept of ‘hopping operators’ is introduced, constructed, and then used
to partition the $8!$ vertices of the permutahedron. We explicitly partition
these into 5,040 octets constrained in locations on the permutahedron by a
magic number rule. Boolean factors in this recursive construction are shown to
obey a doubly even binary flip rule. Although these hopping operators do not
generally constitute normal subgroups of the permutation group, we find that
‘ab-normal cosets’ exist where the same left- and right-hoppers appear as
unordered sets. Finally, using computer simulations, we investigate the types
of faces on higher-order permutahedron which may give rise to lower-order
supermultiplets.

PACS: 11.30.Pb, 12.60.Jv
Keywords: supersymmetry, permutahedron, supermultiplet

##  1 Introduction

The question of how to create arbitrary 4D, $N$-extended supermultiplets
(represented by matrices) starting from minimal 4D, $\cal N$ = 1 realizations
began with [?,?] and was one of the motivations for the DFGHILM collaboration.
Later it was realized [?] the matrices utilized could be interpreted as
modified versions of adjacency matrices for a set of graphs given the names of
“adinkras.” More recently [?], it was shown that the permutahedron plays a
role in the selection of supermultiplets, and that some faces of the
permutahedron polytope related to a given value of $\cal N$ correspond to
valid supermultiplets of a lower degree or extension [?]. In other areas, it’s
been shown that faces of polytopes may correspond to properties of effective
field theories [?]. The present connection between representations of
supersymmetry and permutation polytopes hints at an underlying polytopic
representation theory that may be responsible for generating all valid
representations of supersymmetry. Similar work has been carried out examining
the structure of viruses [?], building off the idea of Watson and Crick that
the crystallographic symmetry of viruses could be explained by the repeated
use of just a few protein subunits (in the present paper we call these hopping
operators) [?].

The results in this work give an extension of the questions and results
determined about the recursive construction of higher $N$-extended 1D, adinkra
matrices from lower $N$-extended 1D, adinkra matrices in the works of [?,?].
In this work similar questions and results are determined for the extension of
adinkra matrices associated with 4D, $\cal N$ = 1 models expanded to adinkra
matrices associated with 4D, $\cal N$ = 2 models.

###  1.1 A problem from the representation theory: at the foundation of
4-dimensional supersymmetry

In mathematics, the four color map conjecture dates back to 1852 [?], and
apparently its extension to a major theorem was the first proven by means of
computers and information technology [?,?]. While at the time of the
announcement in 1976 there was some controversy, by now there seems to be a
general acceptance of the proof’s validity. For a number of years a program of
IT use has been under development [?,?,?,?,?] to explore problems at the
mathematical foundations of supersymmetry. In particular, the work of [?]
showed the concept of “off-shell” 4D, $\cal N$ = 1 supersymmetry leading to
the study of Coxeter Groups. This work demonstrated that there exists a direct
sum partitioning of the totality of BC(4) matrix representations into six
disjoint subsets of matrices representing Clifford Algebras. The permutation
matrices are assigned as shown in Fig. 1 where each permutation matrix is
indicated to belong to one of six (color-coded for convenience sake) subsets.
Upon taking the absolute values of these BC(4) matrix representations, one is
led to twenty-four matrices that provide a matrix representation of the
permutation group of order four. The permutahedron is a well-known concept
related to the study of permutation groups and all the members can properly be
assigned using weak right Bruhat ordering [?] to the appropriate location as
shown in Fig. 3.

This naturally leads to the question of whether there is an invariant
associated with the permutation group of order four that determines the
assignments of the members of the subsets to the distinct patterns of
locations in the permutahedron. This answer is yes. In the past 666See the
works in [?] and [?] for explicit definitions and discussions., we have used
the symbols ${\cal G}_{(1)}$, and ${\cal G}_{(2)}$ to denote maps from adinkra
graphs to the real numbers. In this work, we define a new mapping operator
that will be denoted by ${\cal G}_{(3)}$. Unlike the previous two mapping
operations, this new one is defined by the decoration of the nodes of the
permutahedron by sets of colored spheres.

Figure 1: Permutation matrix assignments to subsets

The permutahedron is shown in Fig. 3, and the six subsets are shown in Fig. 3.

Figure 2: Permutahedron containing subset elements

Figure 3: Permutahedron and subsets

The truncated octahedron is a well known Archimedean solid that contains 24
vertices as shown in Fig. 4. Let us consider the case where 24 spheres (as
shown to the right in Fig. 3 are placed at vertices of the truncated
octahedron, one sphere to each vertex to adorn the solid). However, the
spheres are not all identical. Six disjoint sets each containing four spheres
of the same color, are shown in Fig. 4. After placing all of the spheres at
vertices, a particular sphere can be specified, and then the minimum number of
edges traversed to go to the locations of the remaining three spheres of the
same color can be noted and summed.

A problem to solve is how to place the 24 spheres at the vertices so that the
sum described above is the same independent of the choice of color.

As one example of such a solution, it can be noted that the truncated
octahedron possesses six faces that are squares. So a solution is obtained by
choosing four spheres of the same color and placing them at the vertices of
one of the square faces. Upon choosing one of these spheres as the fiducial
one, there are two other spheres adjacent to it and connected by one edge. The
remaining sphere is connected to this chosen sphere by two edges. Thus the sum
of the numbers of edges connecting the chosen sphere to the other three is
${\cal G}{}_{(3)}$ = 1 + 1 + 2 = 4. As all the square faces are equivalent,
the choice of which color of spheres over which to carry out this calculation
is irrelevant.

An interesting question to entertain is whether there are other placements of
the spheres which have the same property with respect to such an invariant and
what is the numerical value for this invariant for these other placements.

Figure 4: Twenty-four spheres with six different colors next to a truncated
octahedron

There is a second such solution: it is the one obtained by using the image in
Fig. 3 as a guide to all the positioning of the colored spheres. Carrying out
the same calculation upon picking a fiducial sphere of any color leads to
${\cal G}{}_{(3)}$ = 2 + 4 + 6 = 12. Furthermore, this result is independent
of the color of the spheres that are chosen. This choice of decoration is the
one that is required to implement the disassociation of the Coxeter Group
BC(4) matrices in six sets of matrices representing Clifford Algebras. We call
${\cal G}{}_{(3)}$ = 12 the “magic number” and will return to this in the
remainder of the discussion.

The problem described above can be generalized to permutations groups of
higher order. For the permutation group of order $8$, there will be $8!$ nodes
and the number of distinct colors will be $8$. In this case, there are 191,520
faces, 141,120 edges, 40,320 nodes and 8 colors.

Moreover, the solution with ${\cal G}{}_{(3)}$ = 12 can be interpreted as a
solution of a graph vertex coloring problem [?]. The graph vertex coloring
problem is defined as the process of coloring the nodes in such a way that no
two nodes sharing a common edge of the graph have the same color. The smallest
number of colors needed to color a graph $G$ is called the chromatic number,
denoted as $\chi(G)$ [?]. It can be seen that our first proposed solution with
${\cal G}{}_{(3)}$ = 4 is not a solution of the graph vertex coloring problem.

Then one question we can ask is whether any geometric method exists to
construct the second coloring solution suggested above. In order to do this,
first let us define the “restricted chromatic number" under the stronger
restrictions on graph coloring, as the smallest number of colors needed to
color for an arbitrary graph $G$ that ${\cal G}{}_{(3)}$ has a constant value
and independent of the choice of color. Then one can show that the restricted
chromatic number of the permutahedron is 2 as in Fig. 5. Secondly, after
coloring the vertices of the permutahedron with two colors (metallic gold and
silver), we can transform the 12 metallic gold vertices of the permutahedron
onto the cuboctahedron as on the right in Fig. 5. (We can repeat the same
process with all of the silver vertices.)

Figure 5: Coloring a truncated octahedron with 2 colors

Fig. 6 shows the relationship between the cuboctahedron and truncated
octahedron. Starting from the octahedron, if we apply the uniform truncation,
we obtain a truncated octahedron, which is also called a permutahedron (of
order 4). If we continue to do deeper truncation, we can obtain the
cuboctahedron as a result of rectified truncation. This truncation process
also can be interpreted as moving metallic gold nodes on the silver nodes as
in Fig. 5, which is a similar process of compressing the truncated octahedron
from the eight cardinal directions to make a compact polytope (which is a
cuboctahedron in this case) so that the metallic gold nodes are placed closest
to each other.

Figure 6: The truncation process of an octahedron

Then, based on the fact that the restricted chromatic number for the
cuboctahedron is 3 (as in Fig. 7), we can re-color the metallic gold nodes
with a different set of 3 colors (metallic gold, dark orchid, milano red) as
on the right in Fig. 7. And if we process the same step on the silver vertices
as Fig. 8, we can get a new set of 6 $(=2\times 3)$ colors for permutahedron
coloring with the appropriate coloring position that satisfies the restricted
coloring condition as Fig. 3 and maximizes the ${\cal G}{}_{(3)}$, which
coincides with the second solution.

Figure 7: Re-coloring metallic gold nodes in Fig. 5 with 3 colors (metallic
gold, dark orchid, milano red) Figure 8: Re-coloring silver nodes in the Fig.
7 with 3 colors (silver, mai tai, persian blue)

This approach is one of the available polytopic approach to find the positions
of six sets of matrices representing Clifford Algebras on the permutahedron,
and it can also be applied to the higher dimensional permutahedra.

Interestingly there are many different research activities based on developing
the properties of the chromatic number $\chi(G)$. For example, Brook’s Theorem
states that the chromatic number of a graph is at most the maximum vertex
degree, except when the graph is complete or an odd cycle. [?], and Grötzsch’s
theorem states that a triangle-free planar graph can be colored with only
three colors. [?]

Thus, the importance of polytopic representation of the $N$-extended
supermultiplets is the availability it provides for a bridge between various
applications of mathematical fields of study such as graph theory,
combinatorics, and supersymmetry. More detailed examples of SUSY
representation will be presented in the remainder of the discussion.

##  2 Recursive Construction of Supermultiplets

Let us begin by discussing our use of the term “supermultiplet” in this work.
Throughout the following, the term supermultiplet refers to a collection of
matrices. The matrices in question are denoted by ${{\boldsymbol{{\rm
L}}}}{}_{{}_{{\rm I}}}$ and ${{\boldsymbol{{\rm R}}}}{}_{{}_{{\rm I}}}$ where
these are d $\times$ d matrices and the subscripted indices ${{{\rm I}}}$ take
on values ${{{\rm I}}}$ = 1,…, $N$ (for some integer $N$)777In this work, we
primarily consider the case where $N$ = d = 8.. These matrices can be extended
to a set of $N$ (2d) $\times$ (2d) matrices (denoted by
${\widehat{{\boldsymbol{\gamma}}}}{}_{\rm I}$) by use of the formula

$\displaystyle{{\widehat{{\boldsymbol{\gamma}}}}{}_{\rm I}~{}}$ | $\displaystyle{{}=~{}\hbox{\large{${\frac{{1}}{{2}}}$}}\,(\,{{\boldsymbol{\sigma}}}^{1}\,+\,i{{\boldsymbol{\sigma}}}^{2}\,)\,\otimes\,{{\boldsymbol{{\rm L}}}}_{\rm I}~{}+~{}\hbox{\large{${\frac{{1}}{{2}}}$}}\,(\,{{\boldsymbol{\sigma}}}^{1}\,-\,i{{\boldsymbol{\sigma}}}^{2}\,)\,\otimes\,{{\boldsymbol{{\rm R}}}}_{\rm I}}$
---|---
|
$\displaystyle{{}=~{}\hbox{\large{${\frac{{1}}{{\sqrt{2}}}}$}}\,(\,{{\boldsymbol{\sigma}}}^{+}\,)\,\otimes\,{{\boldsymbol{{\rm
L}}}}_{\rm
I}~{}+~{}\hbox{\large{${\frac{{1}}{{\sqrt{2}}}}$}}\,(\,{{\boldsymbol{\sigma}}}^{-}\,)\,\otimes\,{{\boldsymbol{{\rm
R}}}}_{\rm I}~{}~{}~{}}$
(2.1)

for each collection ${{\boldsymbol{{\rm L}}}}{}_{{}_{{\rm I}}}$ and
${{\boldsymbol{{\rm R}}}}{}_{{}_{{\rm I}}}$. Here, the symbol $\otimes$
denotes the outer product where ${{\boldsymbol{\sigma}}}^{1}$ and
${{\boldsymbol{\sigma}}}^{2}$ are the usual first and second Pauli-matrices.

For our purposes, the collections ${{\boldsymbol{{\rm L}}}}{}_{{}_{{\rm
I}}}$-${{\boldsymbol{{\rm R}}}}{}_{{}_{{\rm I}}}$ are distinguished into two
types. When the collections satisfy (and thus form representations of
Euclidean Clifford Algebras)

$\left\\{\,{\widehat{{\boldsymbol{\gamma}}}}{}_{\rm
I}~{},~{}{\widehat{{\boldsymbol{\gamma}}}}{}_{\rm
J}\,\right\\}=2\,\delta_{\rm{IJ}}\,{{\boldsymbol{\rm{I}}}}{}_{2d}~{}~{}~{},$
(2.2)

they are called an “off-shell” set. This condition is realized when the
${{\boldsymbol{{\rm L}}}}{}_{{}_{{\rm I}}}$-${{\boldsymbol{{\rm
R}}}}{}_{{}_{{\rm I}}}$ matrices satisfy the equations

$\displaystyle\begin{split}{{\boldsymbol{{\rm L}}}}{}_{{}_{{\rm
I}}}\,{{\boldsymbol{{\rm R}}}}{}_{{}_{{\rm J}}}~{}+~{}{{\boldsymbol{{\rm
L}}}}{}_{{}_{{\rm J}}}\,{{\boldsymbol{{\rm R}}}}{}_{{}_{{\rm
I}}}&~{}=~{}2\,\delta{}_{{}_{\rm{I\,J}}}\,{{\boldsymbol{\rm{I}}}}{}_{\rm
d}~{}~{}~{},~{}~{}~{}{{\boldsymbol{{\rm R}}}}{}_{{}_{{\rm
I}}}\,{{\boldsymbol{{\rm L}}}}{}_{{}_{{\rm J}}}~{}+~{}{{\boldsymbol{{\rm
R}}}}{}_{{}_{{\rm J}}}\,{{\boldsymbol{{\rm L}}}}{}_{{}_{{\rm
I}}}~{}=~{}2\,\delta{}_{{}_{\rm{I\,J}}}\,{{\boldsymbol{\rm{I}}}}{}_{\rm
d}~{}~{}~{}.\end{split}$ (2.3)

In previous works, these have been called the ${\cal{GR}}$(d, $N$) or “garden
algebra” conditions recognizing the dependence on the parameters $\rm{d}$ and
$N$.

However, some of the sets in the following discussion satisfy a different
condition of the form

$\left\\{\,{\widehat{{\boldsymbol{\gamma}}}}{}_{\rm
I}~{},~{}{\widehat{{\boldsymbol{\gamma}}}}{}_{\rm
J}\,\right\\}=2\,\delta_{\rm{IJ}}\,{{\boldsymbol{\rm{I}}}}{}_{2\rm{d}}~{}+~{}{\cal
N}{}_{\rm{IJ}}{}^{\widehat{\alpha}}\,{{\boldsymbol{\kappa}}}{}_{\widehat{\alpha}}~{}~{}~{},$
(2.4)

where the coefficients ${\cal N}{}_{\rm{IJ}}{}^{\widehat{\alpha}\,({\cal R})}$
and the sets of $2\rm{d}$ $\times$ $2\rm{d}$ matrices
${{\boldsymbol{\kappa}}}{}_{\widehat{\alpha}}{}^{({\cal R})}$ are determined
by the explicit forms of each ${{\boldsymbol{{\rm L}}}}{}_{{}_{{\rm
I}}}$-${{\boldsymbol{{\rm R}}}}{}_{{}_{{\rm I}}}$ collection. In this case
above, the collection is called an “on-shell” set. The coefficients ${\cal
N}{}_{\rm{IJ}}{}^{\widehat{\alpha}\,({\cal R})}$ may be interpreted as
obstructions of an off-shell set to forming a Clifford algebra.

It is worth noting that matrix collections of ${{\boldsymbol{{\rm
L}}}}{}_{{}_{{\rm I}}}$ and ${{\boldsymbol{{\rm R}}}}{}_{{}_{{\rm I}}}$ that
are $\rm d$ $\times$ $\rm d$ matrices can be regarded as special cases of more
general matrices. In this more general case, the $\rm d$ $\times$ $\rm d$
${{\boldsymbol{{\rm L}}}}{}_{{}_{{\rm I}}}$ matrices are replaced by ${\rm
d}_{L}\times{\rm d}_{R}$ ${{\boldsymbol{{\rm L}}}}{}_{{}_{{\rm I}}}$ matrices
where the integers ${\rm d}_{L}$ and ${\rm d}_{R}$ satisfy ${\rm d}_{L}$
$\neq$ ${\rm d}_{R}$ . In a similar manner in the more general case, the $\rm
d$ $\times$ $\rm d$ ${{\boldsymbol{{\rm R}}}}{}_{{}_{{\rm I}}}$ matrices are
replaced by ${\rm d}_{L}\times{\rm d}_{R}$ ${{\boldsymbol{{\rm
R}}}}{}_{{}_{{\rm I}}}$ matrices. When this is done, the condition in (2.3)
requires a replacement of the form

$\displaystyle\begin{split}{{\boldsymbol{{\rm L}}}}{}_{{}_{{\rm
I}}}\,{{\boldsymbol{{\rm R}}}}{}_{{}_{{\rm J}}}~{}+~{}{{\boldsymbol{{\rm
L}}}}{}_{{}_{{\rm J}}}\,{{\boldsymbol{{\rm R}}}}{}_{{}_{{\rm
I}}}&~{}=~{}2\,\delta{}_{{}_{\rm{I\,J}}}\,{{\boldsymbol{\rm{I}}}}{}_{{\rm
d}_{L}}~{}~{}~{},~{}~{}~{}{{\boldsymbol{{\rm R}}}}{}_{{}_{{\rm
I}}}\,{{\boldsymbol{{\rm L}}}}{}_{{}_{{\rm J}}}~{}+~{}{{\boldsymbol{{\rm
R}}}}{}_{{}_{{\rm J}}}\,{{\boldsymbol{{\rm L}}}}{}_{{}_{{\rm
I}}}~{}=~{}2\,\delta{}_{{}_{\rm{I\,J}}}\,{{\boldsymbol{\rm{I}}}}{}_{{\rm
d}_{R}}~{}~{}~{}.\end{split}$ (2.5)

but there have never been uncovered realization where it is possible to
simultaneously satisfy both of these conditions. The situation ${\rm d}_{L}$
$\neq$ ${\rm d}_{R}$. occurs often when a collection of fields only realize
the supersymmetry algebra subject to some set of dynamical equations. In the
following, we will only consider the case were ${\rm d}_{L}$ = ${\rm d}_{R}$,
but in distinction with most previous studies this does not insure the
realization of an “off-shell” system.

It has been proposed that the structures above successfully “mirror” a
situation that often occurs in discussions of supersymmetrical field theories
when these are formulated in terms of the component fields of the
representations. Adinkras are thus graphs defined over the component field
space (but only gauge-invariant component fields) of SUSY representations. The
parameter $\rm d$ counts the number of gauge-invariant field components
independent of dynamics while $N$ counts the number of supercharges. The case
of 4D, $\cal N$ = 1 supergravity can be used to illustrate this approach in an
illuminating manner. Let us illustrate this “mirroring” in an explicit example
provided by four dimensional “supergravity theory.”

The original presentations of the four dimensional supergravity theory were
completed by Freedman, van Nieuwenhuizen, and Ferrara [?] and Deser, and
Zumino [?] whose works were expressed in terms of the graviton
$e{}_{\underline{a}}{}^{\underline{m}}(x)$ and the gravitino
$\psi{}_{\underline{a}}{}^{\alpha}(x)$ fields. These works also contained the
supersymmetry variations $\delta_{Q}e{}_{\underline{a}}{}^{\underline{m}}(x)$
and $\delta_{Q}\psi{}_{\underline{a}}{}^{\alpha}(x)$. Upon calculation of the
commutator of two successive SUSY variations specified by fermionic Grassmann
parameters $\epsilon_{1}$ and $\epsilon_{2}$ on either field variable one
finds

$\displaystyle\left[\,\delta_{Q}(\epsilon_{1})~{},~{}\delta_{Q}(\epsilon_{2})\,\right]~{}=~{}i\,2\left(\epsilon_{1}\,\gamma^{\underline{a}}\,\epsilon_{2}\,\right)\,e{}_{\underline{a}}{}^{\underline{m}}\partial{}_{\underline{m}}~{}+~{}\delta{}_{\rm{Gauge}}~{}+~{}{\delta\,{\cal
L}}{}_{{\rm field}}$ (2.6)

where ${\delta{}_{\rm{Gauge}}}$ denotes an appropriate gauge transformation
for each field, and ${\delta\,{\cal L}}{}_{{\rm field}}$ denotes the variation
of a Lagrangian with respect to the fermionic field, but not the bosonic
field, on which the commutators are being evaluated.

Since adinkras are defined only over the gauge-invariant component fields, the
first term on the RHS in (2.6) has been proposed as the hologram of the first
term on the RHS of (2.4). Since adinkras only contain the gauge-invariant
components, the second term on the RHS of (2.6) vanishes identically. Finally,
the final term on the RHS of (2.6) has been proposed as the hologram of the
last term on the RHS of (2.4).

Starting with the discussions [?] and [?], it has been found that in a number
of cases the ${{\boldsymbol{{\rm R}}}}{}_{{}_{{\rm I}}}$ matrices are the
matrix transpose of the ${{\boldsymbol{{\rm L}}}}{}_{{}_{{\rm I}}}$ matrices
and furthermore the latter can be considered signed permutations, i.e.
elements of Coxeter groups. In the case where ${{\rm I}}$ and $\rm d$ take on
values 1,…,4, the elements of the corresponding Coxeter Group can be “broken”
into six distinct sets

$\displaystyle\\{CM\\}=\\{\langle 1\bar{4}2\bar{3}\rangle,\langle
23\bar{1}\bar{4}\rangle,\langle 3\bar{2}\bar{4}1\rangle,\langle
4132\rangle\\}$ (2.7) $\displaystyle\\{TM\\}=\\{\langle
1\bar{3}\bar{4}\bar{2}\rangle,\langle 24\bar{3}1\rangle,\langle
312\bar{4}\rangle,\langle 4\bar{2}13\rangle\\}$ (2.8)
$\displaystyle\\{VM\\}=\\{\langle 13\bar{2}\bar{4}\rangle,\langle
2\bar{4}1\bar{3}\rangle,\langle 3\bar{1}\bar{4}2\rangle,\langle
4231\rangle\\}$ (2.9) $\displaystyle\left\\{VM_{1}\right\\}=\\{\langle
1\bar{4}\bar{3}2\rangle,\langle\bar{2}\bar{3}41\rangle,\langle
3\bar{2}1\bar{4}\rangle,\langle 4123\rangle\\}$ (2.10)
$\displaystyle\left\\{VM_{2}\right\\}=\\{\langle
12\bar{4}\bar{3}\rangle,\langle\bar{2}13\bar{4}\rangle,\langle
3421\rangle,\langle 4\bar{3}1\bar{2}\rangle\\}$ (2.11)
$\displaystyle\left\\{VM_{3}\right\\}=\\{\langle
12\bar{3}\bar{4}\rangle,\langle\bar{2}1\bar{4}3\rangle,\langle
3412\rangle,\langle 4\bar{3}\bar{2}1\rangle\\}$ (2.12)

(the notation and definitions of these can be found in the work of [?]) whose
union is the complete Coxeter group. Taking the absolute values of the Coxeter
group elements in this discussion leads to the permutation group of order
four. The foundational tool used in [?] to discover the dissection of this
Coxeter group, and its permutation subgroup, caused by the condition in (2.3)
was a computer code used to find all of its solutions.

In the end, this dissection process is equivalent to a sorting problem. The
$4!$ elements of the permutation group of order four are sorted into six (i.e.
3!) disjointed subsets each containing four elements. The equivalent problem
related to the work completed in [?] must begin with the $8!=40,320$ elements
of the permutation group of order eight sorted into 5,040 (i.e. 7!) disjointed
subsets each containing eight elements.

We begin noting some of the results for $\cal N$ = 2 supermultiplets888It
should be noted the symbol $\cal N$ denotes the number of supercharges in a
higher dimensional system while the symbol $N$ denotes the number of one
dimensional supercharges realized in one dimensional systems obtained by
reduction. for supersymmetry [?], which for the chiral-chiral (CC), chiral-
vector (CV) and chiral-tensor (CT) supermultiplets are given by

$\begin{aligned} &\mathbf{L}^{CC}_{1}=(170)_{b}\langle 14235867\rangle\\\
&\mathbf{L}^{CC}_{2}=(204)_{b}\langle 23146758\rangle\\\
&\mathbf{L}^{CC}_{3}=(102)_{b}\langle 32417685\rangle\\\
&\mathbf{L}^{CC}_{4}=(0)_{b}\langle 41328576\rangle\\\
&\mathbf{L}^{CC}_{5}=(15)_{b}\langle 58671423\rangle\\\
&\mathbf{L}^{CC}_{6}=(105)_{b}\langle 67582314\rangle\\\
&\mathbf{L}^{CC}_{7}=(195)_{b}\langle 76853241\rangle\\\
&\mathbf{L}^{CC}_{8}=(165)_{b}\langle
85764132\rangle\end{aligned}\qquad\qquad\begin{aligned}
&\mathbf{L}^{CV}_{1}=(170)_{b}\langle 14236857\rangle\\\
&\mathbf{L}^{CV}_{2}=(204)_{b}\langle 23145768\rangle\\\
&\mathbf{L}^{CV}_{3}=(6)_{b}\langle 32418675\rangle\\\
&\mathbf{L}^{CV}_{4}=(96)_{b}\langle 41327586\rangle\\\
&\mathbf{L}^{CV}_{5}=(210)_{b}\langle 58672413\rangle\\\
&\mathbf{L}^{CV}_{6}=(180)_{b}\langle 67581324\rangle\\\
&\mathbf{L}^{CV}_{7}=(126)_{b}\langle 76854231\rangle\\\
&\mathbf{L}^{CV}_{8}=(24)_{b}\langle
85763142\rangle\end{aligned}\qquad\qquad\begin{aligned}
&\mathbf{L}^{CT}_{1}=(234)_{b}\langle 14235786\rangle\\\
&\mathbf{L}^{CT}_{2}=(76)_{b}\langle 23146875\rangle\\\
&\mathbf{L}^{CT}_{3}=(134)_{b}\langle 32417568\rangle\\\
&\mathbf{L}^{CT}_{4}=(32)_{b}\langle 41328657\rangle\\\
&\mathbf{L}^{CT}_{5}=(11)_{b}\langle 58671342\rangle\\\
&\mathbf{L}^{CT}_{6}=(173)_{b}\langle 67582431\rangle\\\
&\mathbf{L}^{CT}_{7}=(103)_{b}\langle 76853124\rangle\\\
&\mathbf{L}^{CT}_{8}=(193)_{b}\langle 85764213\rangle\end{aligned}$ (2.13)

The elements of the garden algebra, being a subgroup of the Coxeter group, are
composed of a signed matrix (the boolean factors in the above equations) times
a permutation matrix, which together form the Coxeter group elements. Our
method of generating higher $N$ supermultiplets will split the process into
two parts. We generate the permutation matrices and the boolean factors
separately, then combine the two. First we will examine the process of
generating the analogous elements of higher $N$ permutation matrices.

Finally, it is of note that the ${{\boldsymbol{{\rm L}}}}{}_{{}_{{\rm
I}}}$-${{\boldsymbol{{\rm R}}}}{}_{{}_{{\rm I}}}$ matrices associated with the
(CC) set satisfy the equations in (2.13) while those associated with both the
(CV) and (CT) sets satisfy the equations in (2.2).

This is in accord with the observation well known in the physics literature
that there is no known off-shell formulation of the so-called 4D, ${\cal N}$ =
2 “hypermultiplet” described solely in terms of eight component fields. On the
other hand, it is well known that there exist off-shell formulations of the
so-called 4D, ${\cal N}$ = 2 vector supermultiplet and the 4D, ${\cal N}$ = 2
tensor supermultiplet in terms of eight component fields.

###  2.1 Generating higher SUSY permutation matrices from lower ones

We begin by demonstrating the process for recursively constructing the chiral-
chiral $\cal N$=2 supermultiplets from the chiral $\cal N$=1 supermultiplet.
Concatenating two copies of the first element of (2.7) (and ignoring the
boolean factor represented by the overlined numbers for now), we have

$\begin{aligned} \langle 1423\rangle\oplus\langle 1423\rangle=\langle
14231423\rangle\end{aligned}\qquad\rightarrow\qquad\begin{aligned} \langle
1423\rangle\oplus\langle 5867\rangle=\langle 14235867\rangle\end{aligned}$
(2.14)

where we relabeled our second string so that our numbering would be from 1-8.
We see that this indeed matches the permutation matrix component of
$\mathbf{L}^{CC}_{1}$ above. For matrices
$\mathbf{L}^{CC}_{5}-\mathbf{L}^{CC}_{8}$ we instead relabel the first string
(adding 4 to the first four numbers instead of the last four). Similar
manipulations can be carried out for all the remaining supermultiplets,
notably even when we are combining different $\cal N$=1 supermultiplets. For
example, to generate $|\mathbf{L}_{1}^{CT}|$, we combine

$\begin{aligned} \langle 1423\rangle\oplus\langle 1342\rangle=\langle
14231342\rangle\end{aligned}\qquad\rightarrow\qquad\begin{aligned} \langle
1423\rangle\oplus\langle 5786\rangle=\langle 14235786\rangle\end{aligned}$
(2.15)

which matches with what was previously found for $\mathbf{L}_{1}^{CT}$ up to
the boolean factor.

This process is also applied to the creation of the $\mathbf{L}_{2}^{CT}$,
$\mathbf{L}_{3}^{CT}$, and $\mathbf{L}_{4}^{CT}$ matrices. By this means we
have gone from four 4 $\times$ 4 matrices to four 8 $\times$ 8 matrices.

The corresponding $|\mathbf{L}_{5}^{CT}|$ is found from relabeling the first
string instead

$\begin{aligned} \langle 1423\rangle\oplus\langle 1342\rangle=\langle
14231342\rangle\end{aligned}\qquad\rightarrow\qquad\begin{aligned} \langle
5867\rangle\oplus\langle 1342\rangle=\langle 58671342\rangle\end{aligned}$
(2.16)

which indeed matches the permutation element of the 4D, $\cal N$ = 2 chiral-
tensor supermultiplet given above. We apply this process to the creation of
the $\mathbf{L}_{6}^{CT}$, $\mathbf{L}_{7}^{CT}$, and $\mathbf{L}_{8}^{CT}$
matrices. Now we have four more 8 $\times$ 8 matrices.

The discussion above was carried out on the basis of considering the matrices
as realizations of permutation operators. We can summarize these results in
the forms of formulae applied to matrices, but without any reference to
permutations. The discussion above directly implies

$\displaystyle|{\widehat{{\boldsymbol{L}}}}^{(multp1,multp2)}_{\rm
I}|~{}=~{}\frac{1}{2}\,(\,{{\boldsymbol{\rm
I}}}\,+\,{{\boldsymbol{\sigma}}}^{3}\,)\,\otimes\,|{{\boldsymbol{{\rm
L}}}}_{\rm I}^{(multp1)}|~{}+~{}\frac{1}{2}\,(\,{{\boldsymbol{\rm
I}}}\,-\,{{\boldsymbol{\sigma}}}^{3}\,)\,\otimes\,|{{\boldsymbol{{\rm
L}}}}_{\rm
I}^{(multp2)}|~{}~{}~{}~{}~{}~{}~{}~{}~{},~{}~{}~{}{\rm{for}}~{}{\rm
I}~{}=~{}1,...,4~{}~{}~{},$ (2.17)
$\displaystyle|{\widehat{{\boldsymbol{L}}}}^{(multp1,multp2)}_{\rm
I}|~{}=~{}\hbox{\large{${\frac{{1}}{{\sqrt{2}}}}$}}\,(\,{{\boldsymbol{\sigma}}}^{-}\,)\,\otimes\,|{{\boldsymbol{{\rm
L}}}}_{\rm
I}^{(multp1)}|~{}+~{}\hbox{\large{${\frac{{1}}{{\sqrt{2}}}}$}}\,(\,{{\boldsymbol{\sigma}}}^{+}\,)\,\otimes\,|{{\boldsymbol{{\rm
L}}}}_{\rm
I}^{(multp2)}|~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{},~{}~{}~{}{\rm{for}}~{}{\rm
I}~{}=~{}5,...,8~{}~{}~{}.$

Here “multp1”, “multp2” denote the lower SUSY permutation matrices we use. At
this point, it is also convenient to introduce a notational device that will
be convenient later. Upon making the definitions

${{\boldsymbol{\rm P}}}{}^{\pm}~{}\equiv~{}\frac{1}{2}\,(\,{{\boldsymbol{\rm
I}}}\,\pm\,{{\boldsymbol{\sigma}}}^{3}\,)$ (2.18)

we can rewrite (2.17) in the form

$\displaystyle|{\widehat{{\boldsymbol{L}}}}^{(multp1,multp2)}_{\rm
I}|~{}=~{}{{\boldsymbol{\rm P}}}{}^{+}\,\otimes\,|{{\boldsymbol{{\rm
L}}}}_{\rm I}^{(multp1)}|~{}+~{}{{\boldsymbol{\rm
P}}}{}^{-}\otimes\,|{{\boldsymbol{{\rm L}}}}_{\rm
I}^{(multp2)}|~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{},~{}~{}~{}{\rm{for}}~{}{\rm
I}~{}=~{}1,...,4~{}~{}~{},$ (2.19)
$\displaystyle|{\widehat{{\boldsymbol{L}}}}^{(multp1,multp2)}_{\rm
I}|~{}=~{}\hbox{\large{${\frac{{1}}{{\sqrt{2}}}}$}}\,(\,{{\boldsymbol{\sigma}}}^{-}\,)\,\otimes\,|{{\boldsymbol{{\rm
L}}}}_{\rm
I}^{(multp1)}|~{}+~{}\hbox{\large{${\frac{{1}}{{\sqrt{2}}}}$}}\,(\,{{\boldsymbol{\sigma}}}^{+}\,)\otimes\,|{{\boldsymbol{{\rm
L}}}}_{\rm
I}^{(multp2)}|.~{}~{}~{}~{}~{}~{}~{}~{}~{},~{}~{}~{}{\rm{for}}~{}{\rm
I}~{}=~{}5,...,8~{}~{}~{}.$

where we have used the ${{\boldsymbol{\rm P}}}$ notation to distinguish these
matrices from elements in the permutation group such as appearing in (2.20)
below. The reader is reminded that these results [?] follow from two steps
starting with the linear addition of pairs of actions for 4D, $\cal N$ = 1
off-shell supermultiplets. First one obtains fermionic transformation laws
that leave the pairs of actions invariant up to surface terms and then one
reduces these transformation laws to one dimension to obtain the corresponding
L-matrices. Starting from the 4D, $\cal N$ = 1 off-shell chiral, vector, and
tensor supermultiplets, this leads to six sets of L-matrices. We denote these
matrices according to the pairings of the supermultiplets, hence (CC), (CV),
(CT), (VV), (VT), (TT). We may think of these as labeling different
representations. The work in [?] (that is based solely on considerations from
purely one-dimensional SUSY) was the source of a seventh set of matrices, and
we can call this the (O) (for original) representation. The absolute values of
all seven sets lie within the permutation group of order eight. However, only
the (O), (CV), (CT) sets satisfy the garden algebra condition that marks them
as being ‘off-shell’ representations.

###  2.2 Generating the boolean factor for higher dimensional supermultiplets

While in Sec. 2.1 we have recursively generated the unsigned L-matrices, the
boolean factors are slightly more complicated to generate. We illustrate the
general recursive construction for the Chiral-Tensor $\cal N$=2
supermultiplet. We first note that signed L-matrices can be decomposed into a
sign factor matrix (boolean factor) times a permutation group matrix [?], i.e,

$\left(\mathbf{L}_{\mathrm{I}}\right)_{i}{}^{\hat{k}}=\left(\mathcal{S}^{(\mathrm{I})}\right)_{i}{}^{{\ell}}\left(\mathcal{P}_{(\mathrm{I})}\right)_{{\ell}}{}^{\hat{k}}$
(2.20)

We generate the boolean factors from the sign factor matrices as follows:

$\left(\mathcal{S}^{(\mathrm{I})}\right)_{i}{}^{{\ell}}=\left[\begin{array}[]{cccc}(-1)^{b_{1}}&0&0&\cdots\\\
0&(-1)^{b_{2}}&0&\cdots\\\ 0&0&(-1)^{b_{2}}&\cdots\\\
\vdots&\vdots&\vdots&\ddots\end{array}\right]\quad\leftrightarrow\quad\left(\mathcal{R}_{\mathrm{I}}=\sum_{i=1}^{\mathrm{d}}b_{i}2^{i-1}\right)_{b}=\begin{array}[]{c}{\left[b_{1}b_{2}\cdots
b_{\mathrm{d}}\right]_{2}},\end{array}$ (2.21)

Here in (2.21) $\mathcal{R}_{\mathrm{I}}$ represent the arbitrary real number,
where we read binary words from left to right by convention (in contrast to
the typical right to left construction). For example we can consider the
decomposition of $\mathbf{L}_{1}$ for the $\cal N$=1 chiral multiplet, for
which we have

$\left(\mathbf{L}_{1}\right)_{i}{}^{\hat{k}}=\left[\begin{array}[]{cccc}1&0&0&0\\\
0&0&0&-1\\\ 0&1&0&0\\\
0&0&-1&0\end{array}\right]=\left[\begin{array}[]{cccc}1&0&0&0\\\ 0&-1&0&0\\\
0&0&1&0\\\ 0&0&0&-1\end{array}\right]\left[\begin{array}[]{cccc}1&0&0&0\\\
0&0&0&1\\\ 0&1&0&0\\\ 0&0&1&0\end{array}\right]=(10)_{b}\langle
1423\rangle=\langle 1\overline{4}2\overline{3}\rangle.$ (2.22)

Turning now to the construction of $\mathbf{L}_{1}$ for the $\cal N$=2 chiral-
tensor,

$\displaystyle\begin{split}\mathbf{L}_{1}^{(CT)}&=\langle
1\bar{4}2\bar{3}\rangle\oplus\langle 1\bar{3}\bar{4}\bar{2}\rangle\\\
&=[01010111]_{b}\langle 14235786\rangle\end{split}$ (2.23)

where $[01010111]_{b}$ means the boolean factor [01010111]. Going forward we
will bracket these factors and give them a subscript of b.

In the following, ${{\boldsymbol{\cal B}}}_{{{\boldsymbol{\cal(R)}}}}$
corresponds to a bitflip matrix (for given representation $\cal R$) obtained
in the following way. To take the Chiral-Vector multiplet as an example, for
elements $\mathbf{L}^{(CT)}_{5}-\mathbf{L}^{(CT)}_{8}$ we use the following
rule: For the $i$-th element, take the $(i-4)$-th element’s boolean factor and
flip 4 neighboring bits (for example the fifth through the eighth bits. The
choice of where to start is arbitrary). Do this to generate the boolean
factors for $\mathbf{L}^{(CT)}_{5}-\mathbf{L}^{(CT)}_{8}$ and check if the
garden algebra closes. If it does not close, cycle the choice of neighboring
bits by one. For our example of choosing the fifth through the eighth bits, we
would get a factor that wouldn’t satisfy the garden algebra. Therefore we’d
cycle to the next 4 neighboring bits, {6, 7, 8, 1}. In principle, there are 8
choices to cycle through (i.e {1, 2, 3, 4},{2, 3, 4, 5},{3, 4, 5, 6},{4, 5, 6,
7},{5, 6, 7, 8},{6, 7, 8, 1},{7, 8, 1, 2},{8, 1, 2, 3}) for $\cal N$=2. If
none of these combinations work, then the supermultiplet does not have boolean
factors that make it a garden algebra. The example calculation for
$\mathbf{L}^{(CT)}_{5}$ ($\cal N$=2 chiral-tensor) is performed below, where
we apriori know that the correct bits to flip are (6, 7, 8, 1).

$\displaystyle\begin{split}\mathbf{L}_{5}^{(CT)}&=\langle
1\bar{4}2\bar{3}\rangle\oplus\langle 1\bar{3}\bar{4}\bar{2}\rangle\\\
&=\text{Bitflip}_{(6,7,8,1)}([01010111]_{b})\langle 58671342\rangle\\\
&=[11010000]_{b}\langle 58671342\rangle\end{split}$ (2.24)

The corresponding bitflip matrices for the (CV) and (CT) sets take the
respective forms

${{\boldsymbol{\cal B}}}_{(CV)}=\begin{bmatrix}1&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0\\\ 0&0&0&-1&0&0&0&0\\\ 0&0&0&0&-1&0&0&0\\\
0&0&0&0&0&-1&0&0\\\ 0&0&0&0&0&0&-1&0\\\
0&0&0&0&0&0&0&1\end{bmatrix}~{}~{}~{},~{}~{}~{}{{\boldsymbol{\cal
B}}}_{(CT)}=\begin{bmatrix}-1&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0\\\
0&0&1&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0\\\ 0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&-1&0&0\\\
0&0&0&0&0&0&-1&0\\\ 0&0&0&0&0&0&0&-1\end{bmatrix}.$ (2.25)

The boolean factors of $\mathbf{L}^{(CT)}_{5}-\mathbf{L}^{(CT)}_{8}$ are dual
to the boolean factors of $\mathbf{L}^{(CT)}_{1}-\mathbf{L}^{(CT)}_{4}$ by
doubly even (4) bit flips on neighboring bits.

It is obvious that these matrices square to the identity element and as well
are traceless. Let us rewrite these in the outer product notation in the forms

${{\boldsymbol{\cal B}}}_{(CV)}~{}=~{}(\,{{\boldsymbol{\rm
P}}}{}^{+}\,)\,\otimes\,{{\boldsymbol{{\cal
M}}}}_{{{\boldsymbol{\cal(CV)}}}}~{}+~{}(\,{{\boldsymbol{\rm
P}}}{}^{-}\,)\,\otimes\,{{\boldsymbol{\widetilde{\cal
M}}}}_{{{\boldsymbol{\cal(CV)}}}}$ (2.26) ${{\boldsymbol{\cal
B}}}_{(CT)}~{}=~{}(\,{{\boldsymbol{\rm
P}}}{}^{+}\,)\,\otimes\,{{\boldsymbol{\cal
M}}}_{{{\boldsymbol{\cal(CT)}}}}~{}+~{}(\,{{\boldsymbol{\rm
P}}}{}^{-}\,)\,\otimes\,{{\boldsymbol{\widetilde{\cal
M}}}}_{{{\boldsymbol{\cal(CT)}}}}$ (2.27)

The quantities ${{\boldsymbol{\widetilde{\cal
M}}}}_{{{\boldsymbol{\cal(CV)}}}}$ and ${{\boldsymbol{\widetilde{\cal
M}}}}_{{{\boldsymbol{\cal(CT)}}}}$ are the bitflip duals of
${{\boldsymbol{{\cal M}}}}_{{{\boldsymbol{\cal(CV)}}}}$ and
${{\boldsymbol{{\cal M}}}}_{{{\boldsymbol{\cal(CT)}}}}$ and in these equations
${{\boldsymbol{{\cal M}}}}_{{{\boldsymbol{\cal(CV)}}}}$ and
${{\boldsymbol{{\cal M}}}}_{{{\boldsymbol{\cal(CT)}}}}$ are the 4 $\times$ 4
matrices given by

${{\boldsymbol{\cal
M}}}_{{{\boldsymbol{\cal(CV)}}}}~{}=~{}\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&0&0&-1\end{bmatrix}~{}~{}~{},~{}~{}~{}{{\boldsymbol{\cal
M}}}_{{{\boldsymbol{\cal(CT)}}}}~{}=~{}\begin{bmatrix}-1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&0&0&1\end{bmatrix}$ (2.28)

These matrices make it clear that the form of the ${{\boldsymbol{\cal
M}}}_{({{\boldsymbol{{\cal R}}}})}$ matrices determine the forms of the
${{\boldsymbol{\widetilde{\cal M}}}}_{({{\boldsymbol{\cal R}}})}$ matrices
(and thus the forms of the ${{\boldsymbol{\cal B}}}_{({\cal R})}$ matrices)
for the representations $({\cal R})$ = $({CV})$ and $({CT})$.

In terms of matrix notation, we have

$\displaystyle{\widehat{{\boldsymbol{\cal S}}}}{}_{\rm
I}~{}=~{}(\,{{\boldsymbol{\rm P}}}{}^{+}\,)\,\otimes\,{{\boldsymbol{\cal
S}}}_{\rm I}^{(multp1)}~{}+~{}(\,{{\boldsymbol{\rm
P}}}{}^{-}\,)\,\otimes\,{{\boldsymbol{\cal S}}}_{\rm
I}^{(multp2)}~{}~{}~{}~{}~{}~{}~{}~{}~{},~{}~{}~{}{\rm{for}}~{}{\rm
I}~{}=~{}1,...,4$ (2.29)

${\widehat{{\boldsymbol{\cal S}}}}{}_{\rm I}~{}=~{}(\,{{\boldsymbol{\rm
P}}}{}^{+}\,)\,\otimes\,{{\boldsymbol{\cal
M}}}_{{{\boldsymbol{\cal(R)}}}}\cdot{{\boldsymbol{\cal S}}}_{\rm
I}^{(multp1)}~{}+~{}(\,{{\boldsymbol{\rm
P}}}{}^{-}\,)\,\otimes\,{{\boldsymbol{\cal
M}}}_{{{\boldsymbol{\cal(R)}}}}\cdot{{\boldsymbol{\cal S}}}_{\rm
I}^{(multp2)}~{}~{}~{}~{}~{}~{}~{}~{}~{},~{}~{}~{}\\\ {\rm{for}}~{}{\rm
I}~{}=~{}5,...,8$ (2.30)

According to equations (3.3) and (3.6) in [?], we obtain the HYMN value for
each supermultiplet representation. This allows the following dissection:

$\displaystyle{{\boldsymbol{\cal\hat{C}}}}^{{\boldsymbol{(}}CT)}={{\boldsymbol{\cal\hat{C}}}}^{(CV)}={{\boldsymbol{\cal\hat{C}}}}^{(O)}=\sigma_{3}\otimes{\bf{I}}_{8},$
(2.31)
$\displaystyle{{\boldsymbol{\cal\hat{C}}}}^{(CC)}={{\boldsymbol{\cal\hat{C}}}}^{(TT)}={{\boldsymbol{\cal\hat{C}}}}^{(TV)}={{\boldsymbol{\cal\hat{C}}}}^{(VV)}={\bf{I}}_{16}$
(2.32)

Here, the matrix denoted by ${\boldsymbol{\cal\hat{C}}}^{(\cal R)}$ is
associated with each supermultiplet $\cal R$ and has eigenvalues known as HYMN
values. [?]

The first class includes (CT), (CV), and (O), for which the HYMN matrix takes
the form $\sigma_{3}\otimes{\bf{I}}_{8}$. The second class includes (CC),
(TT), (TV), and (VV), for which the HYMN matrix takes the form
${\bf{I}}_{16}$. Based on the results presented in (2.2), the available
bitflip for (CT) was {6, 7, 8, 1}. Although this is not a mathematical proof,
it is an observation that works for (CT), (CV), and (O), but not for any other
combinations.

As part of our investigation, it is necessary to classify other
supermultiplets such as CMN, O($2n$), and Harmonic $N$=2 systems using the
Hexipentisteriruncicantitruncated 7-Simplex. We anticipate that this
classification will align with the HYMN value classification and demonstrate a
clear correlation between the two methods.

###  2.3 A curious property maintained by the recursive construction

There is a curious mathematical property maintained by the six supermultiplets
mentioned from (2.7) to (2.12). If we focus on the permutation matrices within
a single representation, we can always pair the $n$-th matrix to $n$-th to
last matrix so that the sum of the order of their “weight” (see Sec. 3.1) is
$d!+1$, where $d$ is the dimension of the permutation. Take
$\left\\{VM_{1}\right\\}=\\{\langle 1234\rangle,\langle 2143\rangle,\langle
3412\rangle,\langle 4321\rangle\\}$ for example. $\langle 1234\rangle$ is the
smallest possible bracket notation for permutation of order $4$ and $\langle
4321\rangle$ is the largest. So after we arrange all permutations from
smallest to largest, the orders of $\langle 1234\rangle$ and $\langle
4321\rangle$ are $1$ and $4!$ respectively. Their sum is $4!+1$. On the other
hand, the orders of $\langle 2143\rangle$, $\langle 3412\rangle$ are $7$ and
$18$ respectively. Still their sum is $4!+1$.

Another way of stating this property is that we can always pair the bracket
notation so that the sum of digits on any position is $d+1$. For example, one
pair within the VV supermultiplet [?] is $|\mathbf{L}_{3}^{(VV)}|=\langle
42318675\rangle$ and $|\mathbf{L}_{6}^{(VV)}|=\langle 57681324\rangle$
(ignoring the boolean factor). The first digit of 42318675 is 4 and the first
digit of 57681324 is 5. The sum of their first digits is the dimension plus 1:
$4+5=8+1$, as well as the sum of 6-th digit $6+3=8+1$.

These two statements are equivalent. If we send every digit $i$ in braket
notation to $d+1-i$, for instance $45728613$ maps to $54271386$, by symmetry
the order of the preimage counting from small to large is exactly the order of
the image counting from large to small.

This rule is preserved by our construction as well. Suppose we combine two
$\cal N$=1 to obtain a $\cal N$=2 supermultiplet, and the two “ingredients”
$\cal N$=1 satisfy the rule mentioned above. We can pair the $n$-th (for some
$n\leq 4$) L-matrix of the newly obtained supermultiplet with the $(8-i+1)$-th
L-matrix (note that $d$=8 here). Since the ‘ingredients’ satisfy the rule, by
our construction method we know that the sum of the original digits at the
same position would be $4+1$. On the other hand, the last half digits of the
$n$-th L-matrix would be added $4$ from the original $\cal N$=1 matrix,
whereas the first half digits of the $(8-i+1)$-th are added $4$. So the sum of
digits at the same position of this pair would be $(4+1)+4=8+1$, maintaining
the rule.

##  3 Hopping Operators and a General Construction

###  3.1 Introducing hopping operators

Within the domain of compact Lie algebras, the concepts of root and weight
spaces, the maximal torus, etc. have been recognized for decades as valuable
tools to construct the representation theories of these algebras. The Jordan-
Chevelley Decomposition of the generators of these Lie algebras into elements
of the maximal torus accompanied by an associated set of “raising” and
“lowering” operators continually show their value as the foundations of
Cartan’s classification of all compact Lie algebras.

Starting in a lecture entitled “Genomics, Networks, And Computational Concepts
For Polytopic SUSY Representation Theory,” delivered in 2021 [?] it was
proposed that although SUSY algebras are distinctly different from compact Lie
algebras, the concept of a “weight space” does likely exist, even though (to
the authors’ best knowledge) the analog of the Jordan-Chevelley decomposition
apparently does not exist for SUSY algebras. Nevertheless, due to the relation
of adinkras to Coxeter groups [?], a structure has been identified for
supersymmetry to play the role of root and weight spaces. An argument for such
a structure goes as follows.

The elements of Coxeter groups may be regarded as signed permutations. Thus
the Coxeter group of length-$d$ elements—which we denote here by
BC($d$)—consists of $2^{d}\,d!$ elements. Taking the absolute values of the
elements of BC($d$) yields the of the permutation group, which we denoted by
$S_{d}$. There is a well-known polytope called the “permutahedron” [?]
associated with $S_{d}$. It was proposed in the works of [?,?] that the
permutahedron can act as a weight space/root space for adinkras. In order to
achieve further progress along these lines, it becomes a pressing matter to
define quantities that can play the role of “raising and lowering operators”.
What defines the lowest or highest weight state of a representation? In
particular, is there an intrinsic way to define the weight of a permutation
operator?

There are two natural ways to denote permutation operators: (a) cycle
notation, and (b) “bra-ket” notation. As an example, let us consider the set
of permutations given in cycle notation by

$\displaystyle{\left\\{{~{}~{}}\mathcal{P}_{1}=()~{},~{}~{}\mathcal{P}_{2}=(12)(34)~{},~{}~{}\mathcal{P}_{3}=(13)(24)~{},~{}~{}\mathcal{P}_{4}=(14)(23){~{}~{}}\right\\}~{}~{}~{},}$
---
(3.1)

which in “bra-ket” notation can be rewritten as

$\displaystyle{\left\\{{~{}}\langle 1234\rangle,~{}\langle
2143\rangle,~{}\langle 3412\rangle,~{}\langle
4321\rangle{~{}}\right\\}~{}~{}~{},}$
---
(3.2)

respectively. It should be apparent that the “bra-ket” notation provides a
natural mapping between permutation operators and natural numbers, obtained by
treating the list of digits in the brackets as a number in base-$d$. We will
refer to this map as the BK-map. The BK-map thus induces a ‘weight’ for each
permutation, given by the order of their corresponding natural numbers.
Elements can be ordered according to their weight defined by the BK-map.
Inexorably, one is led to the concept that there is always a element of
lightest weight that is natural and intrinsic in the considerations of
permutations.

For example, application of the BK-map to the set in (3.1) leads to the set of
numbers given by

$\displaystyle{\left\\{{~{}}1234,~{}2143,~{}3412,~{}4321{~{}}\right\\}~{}~{}~{}.}$
---
(3.3)

So it is seen that the set of permutations given by (3.1) is already correctly
ordered according by this weight.

This fact leads one to realize given any construction involving permutations,
the elements can be ordered according to their weight defined by the BK-map.
Inexorably, one is led to the concept that there is always a element of
lightest weight that is natural and intrinsic in the considerations of
permutations.

Now we are in position to define “hopping operators” or “hoppers.” There are
two sets of such operators: left hopping operators and right hopping
operators. We denote the A-th left “hoppers” by the symbol
${\mathcal{H}}{}^{(L)}_{\rm A}$ and the A-th right “hoppers” by the symbol
${\mathcal{H}}{}^{(R)}_{\rm A}$. The hoppers are defined by the equations
(where $\mathcal{P}_{1}$ is the permutation operator of lightest weight in a
given set),

${\mathcal{P}_{\rm A}~{}=~{}{\mathcal{H}}{}^{(L)}_{\rm
A}\,\mathcal{P}_{1}~{}~{}~{}~{},~{}~{}~{}\mathcal{P}_{\rm
A}~{}=~{}\mathcal{P}_{1}\,{\mathcal{H}}{}^{(R)}_{\rm A}~{}~{}~{},}$ (3.4)

so the hopping operators can be calculated from the permutation elements via
the equations

${\mathcal{P}_{\rm
A}\,\left[\mathcal{P}_{1}\right]{}^{-1}~{}=~{}{\mathcal{H}}{}^{(L)}_{\rm
A}\,~{}~{}~{},~{}~{}~{}\left[\mathcal{P}_{1}\right]{}^{-1}\,\mathcal{P}_{\rm
A}~{}=~{}{\mathcal{H}}{}^{(R)}_{\rm A}~{}~{}~{}.}$ (3.5)

Before we leave this introduction to hoppers, it is of note that these
operators have appeared in previous works on adinkras [?,?,?,?,?,?]. In these
older works the concepts of “bosonic holoraumy matrices” (${\mathbf{\cal
B}}{\,}_{{}_{\rm{I\,J}}}$) and “fermionic holoraumy matrices” (${\mathbf{\cal
F}}{\,}_{{}_{\rm{I\,J}}}$) were introduced and defined by

$\displaystyle\begin{split}{{\boldsymbol{{\rm L}}}}{}_{{}_{{\rm
I}}}\,{{\boldsymbol{{\rm R}}}}{}_{{}_{{\rm J}}}~{}-~{}{{\boldsymbol{{\rm
L}}}}{}_{{}_{{\rm J}}}\,{{\boldsymbol{{\rm R}}}}{}_{{}_{{\rm
I}}}&~{}=~{}{\mathbf{\cal
B}}{\,}_{{}_{\rm{I\,J}}}~{}~{}~{},~{}~{}~{}{{\boldsymbol{{\rm
R}}}}{}_{{}_{{\rm I}}}\,{{\boldsymbol{{\rm L}}}}{}_{{}_{{\rm
J}}}~{}-~{}{{\boldsymbol{{\rm R}}}}{}_{{}_{{\rm J}}}\,{{\boldsymbol{{\rm
L}}}}{}_{{}_{{\rm I}}}~{}=~{}{\mathbf{\cal
F}}{\,}_{{}_{\rm{I\,J}}}~{}~{}~{}.\end{split}$ (3.6)

It is clear that ${\mathbf{\cal B}}{\,}_{{}_{\rm{I\,J}}}$ and ${\mathbf{\cal
F}}{\,}_{{}_{\rm{I\,J}}}$ matrices have two parts. To examine these two part
we introduce “left jumpers” (${\mathcal{J}}{}^{(L)}_{\rm{A\,B}}$) and “right
jumpers” (${\mathcal{J}}{}^{(R)}_{\rm{A\,B}}$) via the equations

${\mathcal{P}_{A}~{}=~{}{\mathcal{J}}{}^{(L)}_{A\,B}\,\mathcal{P}_{B}~{}~{}~{}~{},~{}~{}~{}\mathcal{P}_{A}~{}=~{}\mathcal{P}_{B}\,{\mathcal{J}}{}^{(R)}_{B\,A}~{}~{}~{},}$
(3.7)

and thus we have the relations

${\mathcal{P}_{A}\,[\mathcal{P}_{B}]{}^{-1}~{}=~{}{\mathcal{J}}{}^{(L)}_{A\,B}~{}~{}~{}~{},~{}~{}~{}[\mathcal{P}_{B}]{}^{-1}\,\mathcal{P}_{A}~{}=~{}{\mathcal{J}}{}^{(R)}_{B\,A}~{}~{}~{},}$
(3.8)

and hence jumpers take an arbitrary permutation described by $\mathcal{P}_{\rm
A}$ and “jumps” it to the permutation described by $\mathcal{P}_{\rm B}$. The
holoraumy matrices are therefore given by

$\displaystyle\begin{split}{\mathbf{\cal
B}}{\,}_{{}_{\rm{A\,B}}}~{}=~{}{\mathcal{J}}{}^{(L)}_{A\,B}~{}-~{}{\mathcal{J}}{}^{(L)}_{B\,A}~{}~{}~{},~{}~{}~{}{\mathbf{\cal
F}}{\,}_{{}_{\rm{A\,B}}}~{}=~{}{\mathcal{J}}{}^{(R)}_{A\,B}~{}-~{}{\mathcal{J}}{}^{(R)}_{B\,A}~{}~{}~{}.\end{split}$
(3.9)

Thus, the hoppers are only some of the components of the larger ${\mathbf{\cal
B}}{\,}_{{}_{\rm{A\,B}}}$ and ${\mathbf{\cal F}}{\,}_{{}_{\rm{A\,B}}}$
matrices. In particular, we see

${{\mathcal{J}}{}^{(L)}_{A\,1}~{}=~{}{\mathcal{H}}{}^{(L)}_{A}~{}~{}~{}~{},~{}~{}~{}{\mathcal{J}}{}^{(R)}_{1\,A}~{}=~{}{\mathcal{H}}{}^{(R)}_{A}~{}~{}~{}.}$
(3.10)

As noted in the work of [?], the holoraumy matrices themselves satisfy the
equations

$\displaystyle{\big{[}\,{\mathbf{\cal B}}{\,}_{{}_{\rm{I\,J}}}~{},~{}{\mathbf{\cal B}}{\,}_{{}_{\rm{K\,L}}}\,\big{]}~{}}$ | $\displaystyle{{}=~{}2\delta_{IL}\,{\mathbf{\cal B}}_{\rm{JK}}-2\delta_{\rm{IK}}\,{\mathbf{\cal B}}_{JL}+2\delta_{\rm{JK}}\,{\mathbf{\cal B}}_{\rm{IL}}-2\delta_{\rm{JL}}\,{\mathbf{\cal B}}_{\rm{IK}}~{}~{}~{},}$
---|---
(3.11) $\displaystyle{\big{[}\,{\mathbf{\cal F}}{\,}_{{}_{\rm{I\,J}}}~{},~{}{\mathbf{\cal F}}{\,}_{{}_{\rm{K\,L}}}\,\big{]}~{}}$ | $\displaystyle{{}=~{}2\delta_{IL}\,{\mathbf{\cal F}}_{\rm{JK}}-2\delta_{\rm{IK}}\,{\mathbf{\cal F}}_{JL}+2\delta_{\rm{JK}}\,{\mathbf{\cal F}}_{\rm{IL}}-2\delta_{\rm{JL}}\,{\mathbf{\cal F}}_{\rm{IK}}~{}~{}~{}.}$
---|---
(3.12)

Once the existence of weights and hoppers is accepted, then one can recall
some structures from the theory of compact Lie algebras. In this latter case,
the Jordan-Chevalley decomposition implies that all of the generators of a
compact Lie algebra can be partitioned into elements of either an abelian
“torus” algebra, raising operators, or lowering operators. The eigenvalues of
the generators belonging to the abelian torus algebra define a space of
weights for a set of states. The raising operators act on the state space
which possesses a state of lowest weight. When a raising operator is applied
to this lowest state, this produces a state of higher weight in a given
representation.

In close analogy, for the permutations the application of the hoppers to the
state with the lowest weight yields a higher weight state. So effectively
hoppers are ‘raising operators’ from the lowest weight state in the set of
permutations.

###  3.2 Using Hopping Operators

The sets of permutations presented at the beginning of Sec. 2 can be viewed as
sets of hopping operators. Given one element of a supermultiplet, the hopping
operators allow us to generate the rest of the supermultiplet. In $\cal N$=1,
the Klein group elements are the hopping operators. In $\cal N$=2, the Rana
octet $\mathcal{R}_{8}$ plays the role of the hopping operators. In general,
the hopping operators will be given by a group of permutations isomorphic to
$(Z_{2})^{\oplus\mathcal{N}+1}$. See Sec. 3.3. For example, for $\cal N$=1,
the hopping operators (listed in cycle notation) are given by

$\displaystyle\mathcal{P}_{1}=()$ (3.13)
$\displaystyle\mathcal{P}_{2}=(12)(34)$
$\displaystyle\mathcal{P}_{3}=(13)(24)$
$\displaystyle\mathcal{P}_{4}=(14)(23)$

Figure 9: Elements of the Chiral multiplet. Starting at 1423, the hopping
operators will generate the three other elements.

Similarly, for 4D $\cal N$=2 (dimensionally reduced to D=1, $\cal N$=8), we
can generate all the elements of a supermultiplet given a single element and
the hopping operators. For example, for the chiral-vector supermultiplet,
starting with $\mathbf{L}^{CV}_{1}=(170)_{b}\langle 14236857\rangle$ (and
focusing only on the permutation matrix component), we can act with the
hopping operators

$\begin{array}[]{llcl}\mathcal{P}_{1}=\mathbf{I}_{2}\otimes\mathbf{I}_{2}\otimes\mathbf{I}_{2}=&()&=&\langle
12345678\rangle\\\
\mathcal{P}_{2}=\mathbf{I}_{2}\otimes\mathbf{I}_{2}\otimes\sigma^{1}=&(12)(34)(56)(78)&=&\langle
21436587\rangle\\\
\mathcal{P}_{3}=\mathbf{I}_{2}\otimes\sigma^{1}\otimes\mathbf{I}_{2}=&(13)(24)(57)(68)&=&\langle
34127856\rangle\\\
\mathcal{P}_{4}=\mathbf{I}_{2}\otimes\sigma^{1}\otimes\sigma^{1}=&(14)(23)(58)(67)&=&\langle
43218765\rangle\\\
\mathcal{P}_{5}=\sigma^{1}\otimes\mathbf{I}_{2}\otimes\mathbf{I}_{2}=&(15)(26)(37)(48)&=&\langle
56781234\rangle\\\
\mathcal{P}_{6}=\sigma^{1}\otimes\mathbf{I}_{2}\otimes\sigma^{1}=&(16)(25)(38)(47)&=&\langle
65872143\rangle\\\
\mathcal{P}_{7}=\sigma^{1}\otimes\sigma^{1}\otimes\mathbf{I}_{2}=&(17)(28)(35)(46)&=&\langle
78563412\rangle\\\
\mathcal{P}_{8}=\sigma^{1}\otimes\sigma^{1}\otimes\sigma^{1}=&(18)(27)(36)(45)&=&\langle
87654321\rangle\end{array}$ (3.14)

in which case we can reproduce all elements of the chiral-vector
supermultiplet as follows.

$\displaystyle\mathcal{P}_{1}|\mathbf{L}^{CV}_{1}|=()\langle
14236857\rangle=\langle 14236857\rangle=|\mathbf{L}^{CV}_{1}|$ (3.15)
$\displaystyle\mathcal{P}_{2}|\mathbf{L}^{CV}_{1}|=(12)(34)(56)(78)\langle
14236857\rangle=\langle 23145768\rangle=|\mathbf{L}^{CV}_{2}|$
$\displaystyle\mathcal{P}_{3}|\mathbf{L}^{CV}_{1}|=(13)(24)(57)(68)\langle
14236857\rangle=\langle 32418675\rangle=|\mathbf{L}^{CV}_{3}|$
$\displaystyle\mathcal{P}_{4}|\mathbf{L}^{CV}_{1}|=(14)(23)(58)(67)\langle
14236857\rangle=\langle 41327586\rangle=|\mathbf{L}^{CV}_{4}|$
$\displaystyle\mathcal{P}_{5}|\mathbf{L}^{CV}_{1}|=(15)(26)(37)(48)\langle
14236857\rangle=\langle 58672413\rangle=|\mathbf{L}^{CV}_{5}|$
$\displaystyle\mathcal{P}_{6}|\mathbf{L}^{CV}_{1}|=(16)(25)(38)(47)\langle
14236857\rangle=\langle 67581324\rangle=|\mathbf{L}^{CV}_{6}|$
$\displaystyle\mathcal{P}_{7}|\mathbf{L}^{CV}_{1}|=(17)(28)(35)(46)\langle
14236857\rangle=\langle 76854231\rangle=|\mathbf{L}^{CV}_{7}|$
$\displaystyle\mathcal{P}_{8}|\mathbf{L}^{CV}_{1}|=(18)(27)(36)(45)\langle
14236857\rangle=\langle 85763142\rangle=|\mathbf{L}^{CV}_{8}|$

###  3.3 Partitioning the Permutahedron via Cosets

It was found in [?] that the partitioning of $S_{4}$ can be understood in
terms of cosets of Klein’s Vierergruppe, i.e. the set

$V=\\{\langle 1234\rangle,\langle 2143\rangle,\langle 3412\rangle,\langle
4321\rangle\\}.$ (3.16)

The Vierergruppe is a normal subgroup of $S_{4}$, and it is isomorphic to the
group $(Z_{2})^{\oplus 2}$.

Here, we define cosets and normal subgroups as follows:

###### Definition 1

Let $\mathcal{H}=\\{h_{i}\\}$ be a subgroup of $S_{n}$ and $g\in S_{n}$ be any
group element. We call the set $g\mathcal{H}=\\{gh_{i}\\}$ the left coset by
$g$, and the set $\mathcal{H}g=\\{h_{i}g\\}$ the right coset by $g$.

###### Definition 2

A subgroup $\mathcal{X}$ of the group $G$ is a normal subgroup of $G$ if and
only if $g^{-1}xg\in\mathcal{X}$ for every $x\in\mathcal{X}$ and $g\in G$.

A consequence of this definition is that, if $\mathcal{X}$ is normal in $G$,
left and right cosets of $\mathcal{X}$ are equivalent. That is,
$\mathcal{X}g=g\mathcal{X},$ $\forall g\in G$.

In particular, each of the supermultiplets of $\cal N$=1 supersymmetry can be
written as right cosets (or equivalently, left cosets, since $V$ is normal in
$S_{4}$) of $V$:

$\displaystyle VM_{3}$ $\displaystyle=V$ (3.17) $\displaystyle VM_{2}$
$\displaystyle=V\langle 2134\rangle$ (3.18) $\displaystyle VM_{1}$
$\displaystyle=V\langle 3214\rangle$ (3.19) $\displaystyle VM$
$\displaystyle=V\langle 1324\rangle$ (3.20) $\displaystyle CM$
$\displaystyle=V\langle 2314\rangle$ (3.21) $\displaystyle TM$
$\displaystyle=V\langle 3124\rangle.$ (3.22)

When applied to the permutahedron, we recall findings regarding the magic
number rules within and between these sets [?]. With a suitable choice of sign
prefactors, the Vierergruppe and its cosets form valid garden algebras.

Returning to [?], the authors note that using the analogue of the Vierergruppe
in $S_{8}$, the same partitioning can be accomplished. We can construct this
analogue, which we call the Rana subgroup $\mathcal{R}_{8}$, by following the
method outlined in Sec. 2, starting with the Vierergruppe. The result is

$\displaystyle\mathcal{R}_{8}=\\{$ $\displaystyle\langle
12345678\rangle,\langle 21436587\rangle,\langle 34127856\rangle,\langle
43218765\rangle,$ (3.23) $\displaystyle\langle 56781234\rangle,\langle
65872143\rangle,\langle 78563412\rangle,\langle 87654321\rangle\\}.$

This $\cal N$=2 supermultiplet is also identified in [?] and is isomorphic to
the group $(Z_{2})^{\oplus 3}$. As $\mathcal{R}_{8}$ forms a subgroup of
$S_{8}$, its cosets will partition $S_{8}$ into octets. However, in contrast
with the Vierergruppe in $S_{4}$, $\mathcal{R}_{8}$ is not normal in $S_{8}$,
so taking left and right cosets will not necessarily yield the same
partitions. Indeed, there are 30 conjugate subgroups of $\mathcal{R}_{8}$
that, when cosets are taken, will each yield a separate partition of $S_{8}$
[?]. The chiral-chiral, chiral-vector, and chiral-tensor supermultiplets are
right cosets of $\mathcal{R}_{8}$. See Sec. 3.2 for an explicit construction
of the chiral-vector supermultiplet by this method. Again, with suitable
choices of sign prefactors, $\mathcal{R}_{8}$, its conjugates, and their
cosets for valid garden algebras.

We can further employ this method to generate a partition of $S_{16}$ via a
recursive construction of the Rana subgroup (here, we must delineate terms of
a permutation with commas to avoid ambiguity):

$\displaystyle\mathcal{R}_{16}=\\{$ $\displaystyle\langle
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16\rangle,\langle
2,1,4,3,6,5,8,7,10,9,12,11,14,13,16,15\rangle,$ (3.24) $\displaystyle\langle
3,4,1,2,7,8,5,6,11,12,9,10,15,16,13,14\rangle,\langle
4,3,2,1,8,7,6,5,12,11,10,9,16,15,14,13\rangle,$ $\displaystyle\langle
5,6,7,8,1,2,3,4,13,14,15,16,9,10,11,12\rangle,\langle
6,5,8,7,2,1,4,3,14,13,16,15,10,9,12,11\rangle,$ $\displaystyle\langle
7,8,5,6,3,4,1,2,15,16,13,14,11,12,9,10\rangle,\langle
8,7,6,5,4,3,2,1,16,15,14,13,12,11,10,9\rangle,$ $\displaystyle\langle
9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8\rangle,\langle
10,9,12,11,14,13,16,15,2,1,4,3,6,5,8,7\rangle,$ $\displaystyle\langle
11,12,9,10,15,16,13,14,3,4,1,2,7,8,5,6\rangle,\langle
12,11,10,9,16,15,14,13,4,3,2,1,8,7,6,5\rangle,$ $\displaystyle\langle
13,14,15,16,9,10,11,12,5,6,7,8,1,2,3,4\rangle,\langle
14,13,16,15,10,9,12,11,6,5,8,7,2,1,4,3\rangle,$ $\displaystyle\langle
15,16,13,14,11,12,9,10,7,8,5,6,3,4,1,2\rangle,\langle
16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1\rangle\\}.$

We once again note that $\mathcal{R}_{16}$ is isomorphic to $(Z_{2})^{\oplus
4}$, and it is a subgroup of $S_{16}$ (though not a normal subgroup). Thus,
the set forms two separate partitions of $S_{16}$: that formed by left cosets,
and that formed by right cosets. However, in contrast with previous cases, we
find by direct search that there is no suitable choice of signs to convert
$\mathcal{R}_{16}$ into a garden algebra. The results in [?] indicate that we
may have to go as far as $S_{128}$ to find a valid $N=16$ garden algebra. The
reason for this breach of pattern we leave for future exploration.

###  3.4 The magic number

In [?], the idea of a number (hereafter called “the magic number”) that
constrains the “positions” of elements within a supermultiplet relative to the
rest of the permutahedron was introduced. The magic number intriguingly refers
to two different but equivalent notions: the intra-tet correlator, and the
inter-tet correlator. (These were originally dubbed “intra-quartet” and
“inter-quartet” correlators, as they were studied in the context of $\cal N$=4
supersymmetry.) First, the intra-tet correlator of a given supermultiplet
$\mathcal{A}$ and element $\pi\in\mathcal{A}$, labeled
$\rm{Intra}_{\pi}(\mathcal{A})$, is the sum of the distances between $\pi$ and
each other element of $\mathcal{A}$; that is,

$\rm{Intra}_{\pi}(\mathcal{A})=\sum_{\alpha\in\mathcal{A}}d(\pi,\alpha),$
(3.25)

where $\rm d(\cdot,\cdot)$ is the hamming or Bruhat distance function detailed
in [?]. Second, the inter-tet correlator of a given permutation
$\pi\in\mathcal{A}$ to a supermultiplet $\mathcal{B}\neq\mathcal{A}$, labeled
$\rm{Inter}_{\pi}(\mathcal{A},\mathcal{B})$ is the sum of the distances
between $\pi$ and each element of $\mathcal{B}$; that is,

$\rm{Inter}_{\pi}(\mathcal{A},\mathcal{B})=\sum_{\beta\in\mathcal{B}}d(\pi,\beta).$
(3.26)

What is remarkable about these quantities, and what in fact merits their
collective description as the magic number, is that, in both $S_{4}$ and
$S_{8}$, for any supermultiplets $\mathcal{A},\mathcal{B}$ and any elements
$\pi,\sigma\in\mathcal{A}$,

$\rm{Intra}_{\pi}(\mathcal{A})=\rm{Intra}_{\sigma}(\mathcal{A})=\rm{Inter}_{\pi}(\mathcal{A},\mathcal{B})=\rm{Inter}_{\sigma}(\mathcal{A},\mathcal{B}).$
(3.27)

In $S_{4}$ and $S_{8}$, this has been confirmed by brute force calculation,
the magic number in $S_{4}$ being 12 and in $S_{8}$ being 112. In $S_{16}$,
$\rm{Intra}_{\langle
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16\rangle}(\mathcal{R}_{16})=960$, though
the sheer size of $S_{16}$ has precluded a full validation of the identity
(3.27). We conjecture that this magic number rule will hold for partitions in
arbitrary $S_{2^{n}}$, the magic number being ${\cal
G}_{(3)}=$$2^{2n-2}(2^{n}-1)$.

Notice that for a permutahedron of dimension $D$, the average distance between
two vertices would be $(D-1)D/4$, which can be proved by induction. When
$D=2^{n}$ and there are $D$ vertices in a multiplet, we can then approximate
the magic number by the average distance

$(D-1)D/4\times D=(2^{n}-1)2^{2n}/4=2^{2n-2}(2^{n}-1)$ (3.28)

which exactly matches with our conjectured magic number.

Future work may focus on the theoretical grounding for this magic number rule,
while the rule itself may function as a crucial technique for the validation
of supermultiplets.

##  4 The Appearance of Ab-normal Cosets in BC(8)

We have seen in Sec. 3.3 that, in the 4D $\mathcal{N}=1$ case, left and right
cosets are equivalent for purposes of partitioning the permutahedron, but in
the 4D $\mathcal{N}=2$ case, they are not. Here, we investigate some
consequences of this fact.

###  4.1 4D $\mathcal{N}=1$ ab-normal coset

Let us first explicitly calculate left and right cosets for the 4D
$\mathcal{N}=1$ case using the Vierergruppe $VM_{3}$ as the set of hopping
operators. This set has also been dubbed the diadem set $O$ [?]. Also the
diadem $O$ is the same as the object called Rana subgroup $\mathcal{R}_{n}$ in
Sec. 3.3. We calculate the left- and right-cosets of 4D $\mathcal{N}=1$ case
by $O$ with every element of $S_{4}$. Using Def. 1, we calculate the left and
right cosets of $O$ by elements in $S_{4}$. As expected, we find six distinct
cosets. Table. LABEL:tab01 and LABEL:tab02 shows the left- and right-coset
list for 4D $\mathcal{N}=1$ case.

Table 1: $S_{4}$ left-cosets list No. | Element 1 | Element 2 | Element 3 | Element 4 | Type
---|---|---|---|---|---
1 | 1234 | 2143 | 3412 | 4321 |  $O$/$VM_{3}$
2 | 1243 | 2134 | 3421 | 4312 | $VM_{2}$
3 | 1324 | 2413 | 3142 | 4231 | $VM$
4 | 1342 | 2431 | 3124 | 4213 | $TM$
5 | 1423 | 2314 | 3241 | 4132 | $CM$
6 | 1432 | 2341 | 3214 | 4123 | $\rm VM_{1}$
Table 2: $S_{4}$ right-cosets list No. | Element 1 | Element 2 | Element 3 | Element 4 | Type
---|---|---|---|---|---
1 | 1234 | 2143 | 3412 | 4321 |  $O$/$VM_{3}$
2 | 1243 | 2134 | 4312 | 3421 | $VM_{2}$
3 | 1324 | 3142 | 2413 | 4231 | $VM$
4 | 1342 | 3124 | 4213 | 2431 | $TM$
5 | 1423 | 4132 | 2314 | 3241 | $CM$
6 | 1432 | 4123 | 3214 | 2341 | $VM_{1}$

Here, we write 1234 to mean the permutation $\langle 1234\rangle$ for
convinience. Also, we call each row of Table LABEL:tab01 and Table LABEL:tab02
a Kevin’s Pizza slice. Kevin’s Pizza refers to a partitioning of the
permutation group of order four [?] (as in Fig. 1). Table LABEL:tab01 and
Table LABEL:tab02 show that although the Vierergruppe generates identical left
and right cosets, the order in which these elements occur changes. For
example, the third row of Table LABEL:tab01 and Table LABEL:tab02 contain the
same set elements but have different orders. To describe this particular
mathematical property, we define

###### Definition 3

Let $O$ be a diadem of $S_{n}$, and let ${\rm KP}_{i}^{(n)}$ be the $i$-th
slice of Kevin’s Pizza on $S_{n}$. If $Og=gO$ $\forall g\in{\rm
KP}_{i}^{(n)}$, then we call ${\rm KP}_{i}^{(n)}$ an ab-normal coset.

Here the $i$-th slice of Kevin’s Pizza on $S_{n}$ is defined as

${\rm KP}_{i}^{(n)}=Og_{i},$ (4.1)

where $O$ is a diadem of $S_{n}$.

where the diadem $O$ is Rana subgroup $\mathcal{R}_{n}$ defined in Sec.
3.2-3.3 and $g_{i}$ is the $i$-th element in $S_{n}$ (weighted by the BK-map).

Based on Def. 3, in the $S_{4}$ case, we can say that all of the Kevin’s pizza
slices are ab-normal cosets.

###  4.2 Left versus right hoppers

In $S_{8}$, using the set $\mathcal{R}_{8}$, explicit construction of the
left- and right-hoppers yields

Table 3: Left Hoppers in 4D $\mathcal{N}=2$ Left Hoppers | Element1 | Element2 | Element3 | Element4 | Element5 | Element6 | Element7 | Element8
---|---|---|---|---|---|---|---|---
$CMCM$ | 12345678 | 21436587 | 34127856 | 43218765 | 56781234 | 65872143 | 78563412 | 87654321
$CMTM$ | 12345678 | 21436587 | 34127856 | 43218765 | 56781234 | 65872143 | 78563412 | 87654321
$CMVM$ | 12345678 | 21436587 | 34127856 | 43218765 | 56781234 | 65872143 | 78563412 | 87654321
$TMVM$ | 12345678 | 21436587 | 34127856 | 43218765 | 56781234 | 65872143 | 78563412 | 87654321
$TMTM$ | 12345678 | 21436587 | 34127856 | 43218765 | 56781234 | 65872143 | 78563412 | 87654321
$VMVM$ | 12345678 | 21436587 | 34127856 | 43218765 | 56781234 | 65872143 | 78563412 | 87654321
$O$ | 12345678 | 21436587 | 34127856 | 43218765 | 56781234 | 65872143 | 78563412 | 87654321
Table 4: Right Hoppers in 4D $\mathcal{N}=2$ Right Hoppers | Element1 | Element2 | Element3 | Element4 | Element5 | Element6 | Element7 | Element8
---|---|---|---|---|---|---|---|---
$CMCM$ | 12345678 | 34127856 | 43218765 | 21436587 | 56781234 | 78563412 | 87654321 | 65872143
$CMTM$ | 12345678 | 34128765 | 43216587 | 21437856 | 57861423 | 86573241 | 68754132 | 75682314
$CMVM$ | 12345678 | 34127856 | 43216587 | 21438765 | 76583214 | 58761432 | 85672341 | 67854123
$TMVM$ | 12345678 | 43217856 | 21436587 | 34128765 | 78654312 | 56871243 | 87563421 | 65782134
$TMTM$ | 12345678 | 43218765 | 21436587 | 34127856 | 56781234 | 87654321 | 65872143 | 78563412
$VMVM$ | 12345678 | 34127856 | 21436587 | 43218765 | 56781234 | 78563412 | 65872143 | 87654321
$O$ | 12345678 | 21436587 | 34127856 | 43218765 | 56781234 | 65872143 | 78563412 | 87654321

Here $CM$, $TM$, $VM$ are chiral-, tensor-, vector-multiplets defined from
(2.7), (2.8), (2.9) and $CMCM$ is same as $(CC)$ which is the chiral-chiral
$\mathcal{N}=2$ supermultiplet generated from concatenating two copies of the
$\mathcal{N}=1$ supermultiplet in Sec. 2.1. And one can observe that in
general the left and right hoppers of $CMCM$, $VMVM$, $TMTM$, and $O$ have a
different ordering of elements, however, when taken as a set they are the
same. This shows that there is an ab-normal coset in which the left and right
cosets are the same as a set at 4D $\mathcal{N}=2$ case, which we can do a
0-brane reduction on 8D $\mathcal{N}=1$ (Eight color) case.

###  4.3 Eight color ab-normal coset

In general for the group $S_{n}$, we have [?]:

###### Theorem 4

Let n $\geq$ 5\. Then the only normal subgroups of $S_{n}$ are {()}, $A_{n}$,
and $S_{n}$.

Here $A_{n}$ is the alternating group: the group of even permutations on a set
of length $n$. Therefore, from $n\geq 5$, the diadem $O$ is not a normal
subgroup of $S_{n}$.

In 4D $\mathcal{N}=2$ case, which is equivalent to 8D $\mathcal{N}=1$ after a
0-brane reduction we constructed the 8-th Extended Kevin’s Pizza using a
diadem $O$ (Rana subgroup $\mathcal{R}_{8}$). Thus the total number of 8D
$\mathcal{N}=1$ 8-th Extended Kevin’s Pizza slices are 7!=5040 because one
8-th Extended Kevin’s Pizza slice has 8 elements.

Then for every $g\in{\rm{KP}}_{i}^{(n)}$, $n=8$, we have

$g{\mathcal{H}^{(R)}={\rm KP}_{i}^{(n)}},\quad{\mathcal{H}^{(L)}}g={\rm
KP}_{i}^{(n)},$ (4.2)

${\rm\mathcal{H}^{(R)}=\\{\mathcal{H}_{1}^{(R)},\mathcal{H}_{2}^{(R)},...,\mathcal{H}_{8}^{(R)}}\\},\quad{\rm\mathcal{H}^{(L)}}=\\{\mathcal{H}_{1}^{(L)},\mathcal{H}_{2}^{(L)},...,\mathcal{H}_{8}^{(L)}\\}.$
(4.3)

Here $\rm\mathcal{H}_{A}^{(L)}$, $\rm\mathcal{H}_{A}^{(R)}$ are defined in
(3.4).

And when $\mathcal{H}^{(L)}$ and $\mathcal{H}^{(R)}$ are the same, we find

${\rm\mathcal{H}^{(L)}}g=g{\rm\mathcal{H}^{(L)}}={\rm KP}_{i}^{(n)}.$ (4.4)

By using (4.1) and (4.4),

${\rm\mathcal{H}^{(L)}}={O}$ (4.5)

Therefore, if there is any ${\rm KP}_{i}^{(n)}$ which has the same left- and
right-hoppers simultaneously, then from (4.4), (4.5) and Def. 3 we can call
this kind of $n$-th extended Kevin’s Pizza slice an ab-normal coset.

In the appendix, Table. LABEL:tab5 shows the ab-normal coset list on eight
color case with the lexicographic order (BK-map) from $\langle
12345678\rangle$ to $\langle 18726345\rangle$. As a result, one can see there
are 168 sets in which the set of left- and right-hoppers are the same, which
means each 8-th extended Kevin’s Pizza slices on the list is an ab-normal
coset. And the highlighted rows’ elements of Table. LABEL:tab5 have (4D
$\mathcal{N}=1$ supermultiplet)+(4D $\mathcal{N}=1$ supermultiplet)
decomposable structures, such as $VMVM$. And from row 72, the highlighted row
patterns are based on the repetition of the first 24 rows, but it does not
have particular type naming compared to 4D $\mathcal{N}=1$ supermultiplet.

Thus, in further investigation, one need is to clarify why the highlighted
supermultiplet type consists of the same copies of 4D $\mathcal{N}=1$
supermultiplet such as (CC), (VV), (TT) rather than different combinations,
eg., (CT), (CV), (VT), and why there is particular number (168) of ab-normal
cosets which exists in 4D $\mathcal{N}=2$.

##  5 Towards a Polytopic Representation Theory for Supersymmetry

In our previous work [?], we discussed the potential to view faces of the
permutahedron as being composed of valid supermultiplets of a lower degree. In
fact, we can realize this by examining how $\cal N$=3 permutahedra (hexagons)
conspire to create the $\cal N$=4 permutahedron (truncated octahedron). In
Fig. 10 and Fig. 11, the construction of an $\cal N$=4 permutahedron out of
$\cal N$=3 permutahedra is demonstrated. We start by taking the three element
string “123” and using bruhat weak ordering to create two hexagons. Then, we
add a “4” to the left hand side of all elements of one hexagon, and a “4” to
the right hand side of all elements of the other hexagon. Fig. 10 shows how
this pair of hexagons fits on the completed $\cal N$=4 permutahedron. By
following this construction for three more pairs of hexagons, where the
respective added element is a “1”, “2”, and “3” (as depicted in Fig. 11), we
can complete the construction of the $\cal N$=4 permutahedron. The square
faces (which in our previous work [?] were identified as supermultiplets of a
lower degree or extension) are seen to arise as a consequence of creating the
pairs of hexagonal faces.

Figure 10: Illustration of the hexagon pair and its position on the 3rd order
permutahedron

Figure 11: Illustration of the four hexagon pairs that generate the 3rd order
permutahedron

A similar construction may be carried out for higher $\cal N$ permutahedra.
For example, the permutahedra of order 5, the omnitruncated 5-cell, depicted
in Fig. 12. This permutahedra is composed of 10 truncated octahedra, and 20
hexagonal prisms. If we were to carry out the analogous construction as for
the $\cal N$=4 permutahedra, we would find that the truncated octahedra being
put together would make the hexagonal prisms similarly emergent as the square
faces were in the $\cal N$=4 case. This is strongly suggestive that our
approach of creating higher degree supermultiplets out of lower degree
supermultiplets is mathematically robust. Creating higher order
supermultiplets out of lower order supermultiplets can be reduced to a problem
of identifying vertices on a permutahedron of higher degree.

Figure 12: The omnitruncated 5-cell, constructed from truncated octahedra and
hexagonal prisms

It’s notable that the types of 2-faces remain the same even as $\cal N$
scales. This isn’t true for any of the other types of faces we examined (up to
5-faces).

Types of 2-faces
---
$\cal{N}$ |  {4} |  {6}
3 |  0 |  1
4 |  6 |  8
5 |  90 |  60
6 |  1080 |  480
7 |  12600 |  4200
8 |  151200 |  40320

Types of 3-faces
---
$\cal{N}$ |  {8} |  {12} |  {24}
4 |  0 |  0 |  1
5 |  0 |  20 |  10
6 |  90 |  360 |  90
7 |  2520 |  5040 |  840
8 |  50400 |  67200 |  8400

Types of 4-faces
---
$\cal{N}$ |  {16} |  {24} |  {36} |  {48} |  {120}
5 |  0 |  0 |  0 |  0 |  1
6 |  0 |  0 |  20 |  30 |  12
7 |  0 |  630 |  420 |  630 |  126
8 |  2520 |  20160 |  6720 |  10080 |  1344

Types of 5-faces
---
$\cal{N}$ |  {72} |  {96} |  {144} |  {240} |  {720}
6 |  0 |  0 |  0 |  0 |  1
7 |  0 |  0 |  70 |  42 |  14
8 |  1680 |  1260 |  1680 |  1008 |  168

Types of 6-faces
---
$\cal{N}$ |  {72} |  {96} |  {144} |  {240}
7 |  0 |  0 |  0 |  1
8 |  70 |  112 |  56 |  16

##  6 Conclusions and Summary

In this work, we have begun the exploration of the permutahedron associated
with the omnitruncated 7-simplex. With regard to supersymmetry, this is the
appropriate setting for models where eight independent supercharges occur.

In Sec. 1, we reviewed past works and brought a new perspective to them. We
began our presentation by reviewing some “traditional” color problems of
mathematics such as the four color map problem and the Graph Vertex Coloring
Problem. The latter can be considered in the case where the polytope is the
truncated octahedron. This polytope occurs in the structure of the
permutahedron associated with the permutation group of order four,
$\\{{\mathbb{S}}{}_{4}\\}$. We also noted in Fig. 3 that Clifford Algebra
matrices are associated with the color-coded permutation matrices. While we
did not mention it in Sec. 1.1, that pattern is also a solution to the Graph
Coloring Vertex Problem!

An examination of that image shows that no edge is connected to two vertices
associated with the same color. This is true of the square faces, and it is
also true for the hexagonal faces! So adinkras, which have adjacency matrices
that form Clifford Algebras, in the cases under study can be associated with
solutions of the ‘traditional’ Graph Vertex Coloring Problem. Should this hold
in general, the emergence of SUSY via graph theory, would be intimately
connected with the “traditional” Graph Vertex Coloring Problem. This is one of
the most suggestive, and perhaps important, realization presented in the
current work!

In Sec. 2, the focus was on how previously uncovered adinkras and associated
Clifford Algebras in the case of four colors, can be as a starting to
construct the similar adinkras and associated Clifford Algebras in the case of
eight colors. We demonstrated a set of constructive rules that accomplish this
goal.

In Sec. 3, the challenge of uncovering operators that play the role of raising
operators in compact Lie Algebras (or alternating as “hopping operator” in
systems of lattices in physics problems) for the space of adinkras was
considered. Although the Jordan-Chevelley Decomposition of the generators in
Lie algebras appear not to apply, in any obvious fashion, due to the
appearance of permutation matrices, a concept that induces a weighted ordering
can be obtained from another viewpoint. As noted in equations (3.2) and (3.3),
there is a natural notion of the weight of permutation matrices that is
induced by a map from the action of the permutation matrices on their carrier
spaces into the real numbers. This notion of ordering is actually implicit in
the results that follow from Bruhat ordering. Using these concepts, we
constructed elementary left hopping and right hopping operators which have the
effect of generating translation on the space of the permutahedron. These were
given for both the cases of four-color and eight-color systems.

This same section also formally introduced the concept of the “magic number.”
In the first chapter, a verbal description of it was given and as well the
symbol ${\cal G}{}_{(3)}$ was introduced for this quantity. In equations
(3.25) and (3.26) two formal mathematical definitions have been presented to
precisely define the values of ${\cal G}{}_{(3)}$. For the cases of
$\\{{\mathbb{S}}{}_{4}\\}$, $\\{{\mathbb{S}}{}_{8}\\}$, and
$\\{{\mathbb{S}}{}_{16}\\}$, our studies indicate the results shown in Table.
LABEL:tab06

Table 5: $S_{4}$ left-coset ${\cal G}{}_{(3)}$-Value | Permutation Group
---|---
12 | ${~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}\\{{\mathbb{S}}{}_{4}\\}$
112 | ${~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}\\{{\mathbb{S}}{}_{8}\\}$
960 | ${~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}\\{{\mathbb{S}}{}_{16}\\}$

and we conjecture the formula in equation (3.28) will hold for the special
cases of $\\{{\mathbb{S}}{}_{2^{r}}\\}$.

In Sec. 4 we report on a new phenomenon. For the case of
$\\{{\mathbb{S}}{}_{4}\\}$ the left hoppers and the right hoppers are one and
the same when considered as unordered sets. This is due to the fact that the
quartets that appear are cosets of $\\{{\mathbb{S}}{}_{4}\\}$ with respect to
the Klein Group and the latter is a normal subgroup in forming these cosets.
In the case of $\\{{\mathbb{S}}{}_{8}\\}$ we are no longer dealing with a
normal subgroup. But remarkably enough when one looks at sets instead of
individual elements, there are one-hundred and sixty-eight subsets that act as
if they are normal groups. We have given the name ‘ab-normal’ to these sets.
In further investigations, we can do more analysis on how many ab-normal
cosets appear in more high-order dimensions.

Finally, in Sec. 5 we investigated the $k$-face analysis of higher-order
permutahedron. We constructed a method to build 4-order permutahedron from 2D
hexagon pairs. The $k$-face analysis yields the polytopic way to construct
higher-degree supermultiplets out of lower-degree supermultiplets. By using
the computer simulation, we obtain the partial $k$-face spectrum up to
$\mathcal{N}=8$ and $k=5$. In further investigations, we can do more analysis
on how many numbers of $k$-face type exists on particular $\mathcal{N}$-order
permutahedron.

Let us end by making a comment about a new possible restriction that is
revealed by the current work. We can refer to this as the ‘restriction of
decomposability.’

In the case of the four color case, there were found to be six subsets of the
permutation group $\\{{\mathbb{S}}{}_{4}\\}$, embedded within the Coxeter
Group, that led to 1D, $N$ = 4 supermultiplets, i.e. provide the basis for
Clfford Algebras. Thus, if one begins with these six subsets as a starting
point to construct 1D, $N$ = 8 off-shell supermultiplets in pairs, then the
experience of the work in [?] informs us that each member of the pairs must be
chosen to be distinct. So there must be 6 $\times$ 5 = 30 such pairings as the
basis to construct 1D, $N$ = 8 off-shell supermultiplets that are
decomposable! It is notable that the number 30 has also appeared in one of our
previous investigations [?]. Whether these occurrences are simply the result
of accidents is a topic for future study. This observation has the potential
to imposed drastic reductions in the space of acceptable ${{\boldsymbol{{\rm
L}}}}{}_{{}_{{\rm I}}}$-${{\boldsymbol{{\rm R}}}}{}_{{}_{{\rm I}}}$
collections.

“Behind it all is surely an idea so simple, so beautiful, that when we grasp
it - in a decade, a century, or a millennium - we will all say to each other,
how could it have been otherwise? How could we have been so stupid?”

\- John Archibald Wheeler

Acknowledgements

We wish to acknowledge conversation with Dr. D. E. A. Gates for useful
comments on this work. The research of S. J. G., A. C. and Z. Z. was supported
during a portion of the time it was carried out by the endowment of the Ford
Foundation Professorship of Physics at Brown University. All also gratefully
acknowledge the support of the Brown Theoretical Physics Center. In addition,
the research of S. J. G. is supported by the Clark Leadership Chair in Science
endowment at the University of Maryland - College Park.

##  7 Appendix

###  7.1 Eight color ab-normal cosets

Table 6: Eight color ab-normal cosets list No. | Element1 | Element2 | Element3 | Element4 | Element5 | Element6 | Element7 | Element8 | Type
---|---|---|---|---|---|---|---|---|---
1 | 12345678 | 21436587 | 34127856 | 43218765 | 56781234 | 65872143 | 78563412 | 87654321 | O/$\rm VM_{3}VM_{3}$
2 | 12346587 | 21435678 | 34128765 | 43217856 | 56782143 | 65871234 | 78564321 | 87653412 |
3 | 12347856 | 21438765 | 34125678 | 43216587 | 56783412 | 65874321 | 78561234 | 87652143 |
4 | 12348765 | 21437856 | 34126587 | 43215678 | 56784321 | 65873412 | 78562143 | 87651234 |
5 | 12435687 | 21346578 | 34217865 | 43128756 | 56871243 | 65782134 | 78653421 | 87564312 | $\rm VM_{2}VM_{2}$
6 | 12436578 | 21345687 | 34218756 | 43127865 | 56872134 | 65781243 | 78654312 | 87563421 |
7 | 12437865 | 21348756 | 34215687 | 43126578 | 56873421 | 65784312 | 78651243 | 87562134 |
8 | 12438756 | 21347865 | 34216578 | 43125687 | 56874312 | 65783421 | 78652134 | 87561243 |
9 | 12563478 | 21654387 | 34781256 | 43872165 | 56127834 | 65218743 | 78345612 | 87436521 |
10 | 12564387 | 21653478 | 34782165 | 43871256 | 56128743 | 65217834 | 78346521 | 87435612 |
11 | 12567834 | 21658743 | 34785612 | 43876521 | 56123478 | 65214387 | 78341256 | 87432165 |
12 | 12568743 | 21657834 | 34786521 | 43875612 | 56124387 | 65213478 | 78342165 | 87431256 |
13 | 12653487 | 21564378 | 34871265 | 43782156 | 56217843 | 65128734 | 78435621 | 87346512 |
14 | 12654378 | 21563487 | 34872156 | 43781265 | 56218734 | 65127843 | 78436512 | 87345621 |
15 | 12657843 | 21568734 | 34875621 | 43786512 | 56213487 | 65124378 | 78431265 | 87342156 |
16 | 12658734 | 21567843 | 34876512 | 43785621 | 56214378 | 65123487 | 78432156 | 87341265 |
17 | 12783456 | 21874365 | 34561278 | 43652187 | 56347812 | 65438721 | 78125634 | 87216543 |
18 | 12784365 | 21873456 | 34562187 | 43651278 | 56348721 | 65437812 | 78126543 | 87215634 |
19 | 12785634 | 21876543 | 34567812 | 43658721 | 56341278 | 65432187 | 78123456 | 87214365 |
20 | 12786543 | 21875634 | 34568721 | 43657812 | 56342187 | 65431278 | 78124365 | 87213456 |
21 | 12873465 | 21784356 | 34651287 | 43562178 | 56437821 | 65348712 | 78215643 | 87126534 |
22 | 12874356 | 21783465 | 34652178 | 43561287 | 56438712 | 65347821 | 78216534 | 87125643 |
23 | 12875643 | 21786534 | 34657821 | 43568712 | 56431287 | 65342178 | 78213465 | 87124356 |
24 | 12876534 | 21785643 | 34658712 | 43567821 | 56432178 | 65341287 | 78214356 | 87123465 |
25 | 13245768 | 24136857 | 31427586 | 42318675 | 57681324 | 68572413 | 75863142 | 86754231 | $\rm VMVM$
26 | 13246857 | 24135768 | 31428675 | 42317586 | 57682413 | 68571324 | 75864231 | 86753142 |
27 | 13247586 | 24138675 | 31425768 | 42316857 | 57683142 | 68574231 | 75861324 | 86752413 |
28 | 13248675 | 24137586 | 31426857 | 42315768 | 57684231 | 68573142 | 75862413 | 86751324 |
29 | 13425786 | 24316875 | 31247568 | 42138657 | 57861342 | 68752431 | 75683124 | 86574213 | $\rm TMTM$
30 | 13426875 | 24315786 | 31248657 | 42137568 | 57862431 | 68751342 | 75684213 | 86573124 |
31 | 13427568 | 24318657 | 31245786 | 42136875 | 57863124 | 68754213 | 75681342 | 86572431 |
32 | 13428657 | 24317568 | 31246875 | 42135786 | 57864213 | 68753124 | 75682431 | 86571342 |
33 | 13572468 | 24681357 | 31754286 | 42863175 | 57136824 | 68245713 | 75318642 | 86427531 |
34 | 13574286 | 24683175 | 31752468 | 42861357 | 57138642 | 68247531 | 75316824 | 86425713 |
35 | 13576824 | 24685713 | 31758642 | 42867531 | 57132468 | 68241357 | 75314286 | 86423175 |
36 | 13578642 | 24687531 | 31756824 | 42865713 | 57134286 | 68243175 | 75312468 | 86421357 |
37 | 13682457 | 24571368 | 31864275 | 42753186 | 57246813 | 68135724 | 75428631 | 86317542 |
38 | 13684275 | 24573186 | 31862457 | 42751368 | 57248631 | 68137542 | 75426813 | 86315724 |
39 | 13685724 | 24576813 | 31867542 | 42758631 | 57241368 | 68132457 | 75423186 | 86314275 |
40 | 13687542 | 24578631 | 31865724 | 42756813 | 57243186 | 68134275 | 75421368 | 86312457 |
41 | 13752486 | 24861375 | 31574268 | 42683157 | 57316842 | 68425731 | 75138624 | 86247513 |
42 | 13754268 | 24863157 | 31572486 | 42681375 | 57318624 | 68427513 | 75136842 | 86245731 |
43 | 13756842 | 24865731 | 31578624 | 42687513 | 57312486 | 68421375 | 75134268 | 86243157 |
44 | 13758624 | 24867513 | 31576842 | 42685731 | 57314268 | 68423157 | 75132486 | 86241375 |
45 | 13862475 | 24751386 | 31684257 | 42573168 | 57426831 | 68315742 | 75248613 | 86137524 |
46 | 13864257 | 24753168 | 31682475 | 42571386 | 57428613 | 68317524 | 75246831 | 86135742 |
47 | 13865742 | 24756831 | 31687524 | 42578613 | 57421386 | 68312475 | 75243168 | 86134257 |
48 | 13867524 | 24758613 | 31685742 | 42576831 | 57423168 | 68314257 | 75241386 | 86132475 |
49 | 14235867 | 23146758 | 32417685 | 41328576 | 58671423 | 67582314 | 76853241 | 85764132 | $\rm CMCM$
50 | 14236758 | 23145867 | 32418576 | 41327685 | 58672314 | 67581423 | 76854132 | 85763241 |
51 | 14237685 | 23148576 | 32415867 | 41326758 | 58673241 | 67584132 | 76851423 | 85762314 |
52 | 14238576 | 23147685 | 32416758 | 41325867 | 58674132 | 67583241 | 76852314 | 85761423 |
53 | 14325876 | 23416785 | 32147658 | 41238567 | 58761432 | 67852341 | 76583214 | 85674123 | $\rm VM_{1}VM_{1}$
54 | 14326785 | 23415876 | 32148567 | 41237658 | 58762341 | 67851432 | 76584123 | 85673214 |
55 | 14327658 | 23418567 | 32145876 | 41236785 | 58763214 | 67854123 | 76581432 | 85672341 |
56 | 14328567 | 23417658 | 32146785 | 41235876 | 58764123 | 67853214 | 76582341 | 85671432 |
57 | 14582367 | 23671458 | 32764185 | 41853276 | 58146723 | 67235814 | 76328541 | 85417632 |
58 | 14583276 | 23674185 | 32761458 | 41852367 | 58147632 | 67238541 | 76325814 | 85416723 |
59 | 14586723 | 23675814 | 32768541 | 41857632 | 58142367 | 67231458 | 76324185 | 85413276 |
60 | 14587632 | 23678541 | 32765814 | 41856723 | 58143276 | 67234185 | 76321458 | 85412367 |
61 | 14672358 | 23581467 | 32854176 | 41763285 | 58236714 | 67145823 | 76418532 | 85327641 |
62 | 14673285 | 23584176 | 32851467 | 41762358 | 58237641 | 67148532 | 76415823 | 85326714 |
63 | 14675823 | 23586714 | 32857641 | 41768532 | 58231467 | 67142358 | 76413285 | 85324176 |
64 | 14678532 | 23587641 | 32856714 | 41765823 | 58234176 | 67143285 | 76412358 | 85321467 |
65 | 14762385 | 23851476 | 32584167 | 41673258 | 58326741 | 67415832 | 76148523 | 85237614 |
66 | 14763258 | 23854167 | 32581476 | 41672385 | 58327614 | 67418523 | 76145832 | 85236741 |
67 | 14765832 | 23856741 | 32587614 | 41678523 | 58321476 | 67412385 | 76143258 | 85234167 |
68 | 14768523 | 23857614 | 32586741 | 41675832 | 58324167 | 67413258 | 76142385 | 85231476 |
69 | 14852376 | 23761485 | 32674158 | 41583267 | 58416732 | 67325841 | 76238514 | 85147623 |
70 | 14853267 | 23764158 | 32671485 | 41582376 | 58417623 | 67328514 | 76235841 | 85146732 |
71 | 14856732 | 23765841 | 32678514 | 41587623 | 58412376 | 67321485 | 76234158 | 85143267 |
72 | 14857623 | 23768514 | 32675841 | 41586732 | 58413267 | 67324158 | 76231485 | 85142376 |
73 | 15263748 | 26154837 | 37481526 | 48372615 | 51627384 | 62518473 | 73845162 | 84736251 |
74 | 15264837 | 26153748 | 37482615 | 48371526 | 51628473 | 62517384 | 73846251 | 84735162 |
75 | 15267384 | 26158473 | 37485162 | 48376251 | 51623748 | 62514837 | 73841526 | 84732615 |
76 | 15268473 | 26157384 | 37486251 | 48375162 | 51624837 | 62513748 | 73842615 | 84731526 |
77 | 15372648 | 26481537 | 37154826 | 48263715 | 51736284 | 62845173 | 73518462 | 84627351 |
78 | 15374826 | 26483715 | 37152648 | 48261537 | 51738462 | 62847351 | 73516284 | 84625173 |
79 | 15376284 | 26485173 | 37158462 | 48267351 | 51732648 | 62841537 | 73514826 | 84623715 |
80 | 15378462 | 26487351 | 37156284 | 48265173 | 51734826 | 62843715 | 73512648 | 84621537 |
81 | 15482637 | 26371548 | 37264815 | 48153726 | 51846273 | 62735184 | 73628451 | 84517362 |
82 | 15483726 | 26374815 | 37261548 | 48152637 | 51847362 | 62738451 | 73625184 | 84516273 |
83 | 15486273 | 26375184 | 37268451 | 48157362 | 51842637 | 62731548 | 73624815 | 84513726 |
84 | 15487362 | 26378451 | 37265184 | 48156273 | 51843726 | 62734815 | 73621548 | 84512637 |
85 | 15623784 | 26514873 | 37841562 | 48732651 | 51267348 | 62158437 | 73485126 | 84376215 |
86 | 15624873 | 26513784 | 37842651 | 48731562 | 51268437 | 62157348 | 73486215 | 84375126 |
87 | 15627348 | 26518437 | 37845126 | 48736215 | 51263784 | 62154873 | 73481562 | 84372651 |
88 | 15628437 | 26517348 | 37846215 | 48735126 | 51264873 | 62153784 | 73482651 | 84371562 |
89 | 15732684 | 26841573 | 37514862 | 48623751 | 51376248 | 62485137 | 73158426 | 84267315 |
90 | 15734862 | 26843751 | 37512684 | 48621573 | 51378426 | 62487315 | 73156248 | 84265137 |
91 | 15736248 | 26845137 | 37518426 | 48627315 | 51372684 | 62481573 | 73154862 | 84263751 |
92 | 15738426 | 26847315 | 37516248 | 48625137 | 51374862 | 62483751 | 73152684 | 84261573 |
93 | 15842673 | 26731584 | 37624851 | 48513762 | 51486237 | 62375148 | 73268415 | 84157326 |
94 | 15843762 | 26734851 | 37621584 | 48512673 | 51487326 | 62378415 | 73265148 | 84156237 |
95 | 15846237 | 26735148 | 37628415 | 48517326 | 51482673 | 62371584 | 73264851 | 84153762 |
96 | 15847326 | 26738415 | 37625148 | 48516237 | 51483762 | 62374851 | 73261584 | 84152673 |
97 | 16253847 | 25164738 | 38471625 | 47382516 | 52617483 | 61528374 | 74835261 | 83746152 |
98 | 16254738 | 25163847 | 38472516 | 47381625 | 52618374 | 61527483 | 74836152 | 83745261 |
99 | 16257483 | 25168374 | 38475261 | 47386152 | 52613847 | 61524738 | 74831625 | 83742516 |
100 | 16258374 | 25167483 | 38476152 | 47385261 | 52614738 | 61523847 | 74832516 | 83741625 |
101 | 16382547 | 25471638 | 38164725 | 47253816 | 52746183 | 61835274 | 74528361 | 83617452 |
102 | 16384725 | 25473816 | 38162547 | 47251638 | 52748361 | 61837452 | 74526183 | 83615274 |
103 | 16385274 | 25476183 | 38167452 | 47258361 | 52741638 | 61832547 | 74523816 | 83614725 |
104 | 16387452 | 25478361 | 38165274 | 47256183 | 52743816 | 61834725 | 74521638 | 83612547 |
105 | 16472538 | 25381647 | 38254716 | 47163825 | 52836174 | 61745283 | 74618352 | 83527461 |
106 | 16473825 | 25384716 | 38251647 | 47162538 | 52837461 | 61748352 | 74615283 | 83526174 |
107 | 16475283 | 25386174 | 38257461 | 47168352 | 52831647 | 61742538 | 74613825 | 83524716 |
108 | 16478352 | 25387461 | 38256174 | 47165283 | 52834716 | 61743825 | 74612538 | 83521647 |
109 | 16523874 | 25614783 | 38741652 | 47832561 | 52167438 | 61258347 | 74385216 | 83476125 |
110 | 16524783 | 25613874 | 38742561 | 47831652 | 52168347 | 61257438 | 74386125 | 83475216 |
111 | 16527438 | 25618347 | 38745216 | 47836125 | 52163874 | 61254783 | 74381652 | 83472561 |
112 | 16528347 | 25617438 | 38746125 | 47835216 | 52164783 | 61253874 | 74382561 | 83471652 |
113 | 16742583 | 25831674 | 38524761 | 47613852 | 52386147 | 61475238 | 74168325 | 83257416 |
114 | 16743852 | 25834761 | 38521674 | 47612583 | 52387416 | 61478325 | 74165238 | 83256147 |
115 | 16745238 | 25836147 | 38527416 | 47618325 | 52381674 | 61472583 | 74163852 | 83254761 |
116 | 16748325 | 25837416 | 38526147 | 47615238 | 52384761 | 61473852 | 74162583 | 83251674 |
117 | 16832574 | 25741683 | 38614752 | 47523861 | 52476138 | 61385247 | 74258316 | 83167425 |
118 | 16834752 | 25743861 | 38612574 | 47521683 | 52478316 | 61387425 | 74256138 | 83165247 |
119 | 16835247 | 25746138 | 38617425 | 47528316 | 52471683 | 61382574 | 74253861 | 83164752 |
120 | 16837425 | 25748316 | 38615247 | 47526138 | 52473861 | 61384752 | 74251683 | 83162574 |
121 | 17283546 | 28174635 | 35461728 | 46352817 | 53647182 | 64538271 | 71825364 | 82716453 |
122 | 17284635 | 28173546 | 35462817 | 46351728 | 53648271 | 64537182 | 71826453 | 82715364 |
123 | 17285364 | 28176453 | 35467182 | 46358271 | 53641728 | 64532817 | 71823546 | 82714635 |
124 | 17286453 | 28175364 | 35468271 | 46357182 | 53642817 | 64531728 | 71824635 | 82713546 |
125 | 17352846 | 28461735 | 35174628 | 46283517 | 53716482 | 64825371 | 71538264 | 82647153 |
126 | 17354628 | 28463517 | 35172846 | 46281735 | 53718264 | 64827153 | 71536482 | 82645371 |
127 | 17356482 | 28465371 | 35178264 | 46287153 | 53712846 | 64821735 | 71534628 | 82643517 |
128 | 17358264 | 28467153 | 35176482 | 46285371 | 53714628 | 64823517 | 71532846 | 82641735 |
129 | 17462835 | 28351746 | 35284617 | 46173528 | 53826471 | 64715382 | 71648253 | 82537164 |
130 | 17463528 | 28354617 | 35281746 | 46172835 | 53827164 | 64718253 | 71645382 | 82536471 |
131 | 17465382 | 28356471 | 35287164 | 46178253 | 53821746 | 64712835 | 71643528 | 82534617 |
132 | 17468253 | 28357164 | 35286471 | 46175382 | 53824617 | 64713528 | 71642835 | 82531746 |
133 | 17532864 | 28641753 | 35714682 | 46823571 | 53176428 | 64285317 | 71358246 | 82467135 |
134 | 17534682 | 28643571 | 35712864 | 46821753 | 53178246 | 64287135 | 71356428 | 82465317 |
135 | 17536428 | 28645317 | 35718246 | 46827135 | 53172864 | 64281753 | 71354682 | 82463571 |
136 | 17538246 | 28647135 | 35716428 | 46825317 | 53174682 | 64283571 | 71352864 | 82461753 |
137 | 17642853 | 28531764 | 35824671 | 46713582 | 53286417 | 64175328 | 71468235 | 82357146 |
138 | 17643582 | 28534671 | 35821764 | 46712853 | 53287146 | 64178235 | 71465328 | 82356417 |
139 | 17645328 | 28536417 | 35827146 | 46718235 | 53281764 | 64172853 | 71463582 | 82354671 |
140 | 17648235 | 28537146 | 35826417 | 46715328 | 53284671 | 64173582 | 71462853 | 82351764 |
141 | 17823564 | 28714653 | 35641782 | 46532871 | 53467128 | 64358217 | 71285346 | 82176435 |
142 | 17824653 | 28713564 | 35642871 | 46531782 | 53468217 | 64357128 | 71286435 | 82175346 |
143 | 17825346 | 28716435 | 35647128 | 46538217 | 53461782 | 64352871 | 71283564 | 82174653 |
144 | 17826435 | 28715346 | 35648217 | 46537128 | 53462871 | 64351782 | 71284653 | 82173564 |
145 | 18273645 | 27184536 | 36451827 | 45362718 | 54637281 | 63548172 | 72815463 | 81726354 |
146 | 18274536 | 27183645 | 36452718 | 45361827 | 54638172 | 63547281 | 72816354 | 81725463 |
147 | 18275463 | 27186354 | 36457281 | 45368172 | 54631827 | 63542718 | 72813645 | 81724536 |
148 | 18276354 | 27185463 | 36458172 | 45367281 | 54632718 | 63541827 | 72814536 | 81723645 |
149 | 18362745 | 27451836 | 36184527 | 45273618 | 54726381 | 63815472 | 72548163 | 81637254 |
150 | 18364527 | 27453618 | 36182745 | 45271836 | 54728163 | 63817254 | 72546381 | 81635472 |
151 | 18365472 | 27456381 | 36187254 | 45278163 | 54721836 | 63812745 | 72543618 | 81634527 |
152 | 18367254 | 27458163 | 36185472 | 45276381 | 54723618 | 63814527 | 72541836 | 81632745 |
153 | 18452736 | 27361845 | 36274518 | 45183627 | 54816372 | 63725481 | 72638154 | 81547263 |
154 | 18453627 | 27364518 | 36271845 | 45182736 | 54817263 | 63728154 | 72635481 | 81546372 |
155 | 18456372 | 27365481 | 36278154 | 45187263 | 54812736 | 63721845 | 72634518 | 81543627 |
156 | 18457263 | 27368154 | 36275481 | 45186372 | 54813627 | 63724518 | 72631845 | 81542736 |
157 | 18542763 | 27631854 | 36724581 | 45813672 | 54186327 | 63275418 | 72368145 | 81457236 |
158 | 18543672 | 27634581 | 36721854 | 45812763 | 54187236 | 63278145 | 72365418 | 81456327 |
159 | 18546327 | 27635418 | 36728145 | 45817236 | 54182763 | 63271854 | 72364581 | 81453672 |
160 | 18547236 | 27638145 | 36725418 | 45816327 | 54183672 | 63274581 | 72361854 | 81452763 |
161 | 18632754 | 27541863 | 36814572 | 45723681 | 54276318 | 63185427 | 72458136 | 81367245 |
162 | 18634572 | 27543681 | 36812754 | 45721863 | 54278136 | 63187245 | 72456318 | 81365427 |
163 | 18635427 | 27546318 | 36817245 | 45728136 | 54271863 | 63182754 | 72453681 | 81364572 |
164 | 18637245 | 27548136 | 36815427 | 45726318 | 54273681 | 63184572 | 72451863 | 81362754 |
165 | 18723654 | 27814563 | 36541872 | 45632781 | 54367218 | 63458127 | 72185436 | 81276345 |
166 | 18724563 | 27813654 | 36542781 | 45631872 | 54368127 | 63457218 | 72186345 | 81275436 |
167 | 18725436 | 27816345 | 36547218 | 45638127 | 54361872 | 63452781 | 72183654 | 81274563 |
168 | 18726345 | 27815436 | 36548127 | 45637218 | 54362781 | 63451872 | 72184563 | 81273654 |

Here column ‘type’ means the type of the supermultiplet. For example $\rm
VM_{3}VM_{3}$ means, the 8-string has $\rm VM_{3}$+$\rm VM_{3}$ structure. To
be specific, if we have element 2143, which is an element of $\rm VM_{3}$,
then $\rm VM_{3}VM_{3}$ should contains 21436587, where 6587 also has 2143
order with $1\rightarrow 5,\;2\rightarrow 6,\;3\rightarrow 7,\;4\rightarrow 8$
mapping.

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