# Crystal Structure Generation Based on Polyhedra using Dual Periodic Graphs

Tomoyasu Yokoyama  Technology Division, Panasonic Holdings Corporation, Osaka
571-8501, Japan<EMAIL_ADDRESS>Kazuhide Ichikawa
Technology Division, Panasonic Holdings Corporation, Osaka 571-8501, Japan
Hisashi Naito  [

###### Abstract

Crystal structure design is important for the discovery of new highly
functional materials because crystal structure strongly influences material
properties. Crystal structures are composed of space-filling polyhedra, which
affect material properties such as ionic conductivity and dielectric constant.
However, most conventional methods of crystal structure prediction use random
structure generation methods that do not take space-filling polyhedra into
account, contributing to the inefficiency of materials development. In this
work, we propose a crystal structure generation method based on discrete
geometric analysis of polyhedra information. In our method, the shape and
connectivity of a space-filling polyhedron are represented as a dual periodic
graph, and the crystal structure is generated by the standard realization of
this graph. We demonstrate that this method can correctly generate face-
centered cubic, hexagonal close-packed, and body-centered cubic structures
from dual periodic graphs. This work is a first step toward generating
undiscovered crystal structures based on the target polyhedra, leading to
major advances in materials design in areas including electronics and energy
storage.

###### keywords:

crystal structure generation, crystal structure prediction, polyhedron, dual
graph, graph theory, standard realization

Nagoya University] Graduate School of Mathematics, Nagoya University, Nagoya
464-8602, Japan

##  1 Introduction

The fundamental parameters that determine material properties are composition
and structure. The smallest unit of composition is an atom, and the
composition can be designed by changing the type and ratio of atoms. The
smallest unit of structure can be regarded as a polyhedron, especially in
crystal structures. This is because a crystal structure is an assembly of
repeating units periodically in which each unit consists of atoms arranged at
the vertices of a space-filling polyhedron.

The shape and connectivity of such polyhedra strongly affect material
properties such as ionic conductivity and dielectric constant. For example, it
has been shown theoretically that ionic crystals with a body-centered cubic
(BCC) framework structure exhibit higher ionic conductivity than those with a
face-centered cubic (FCC) or hexagonal close-packed (HCP) framework structure
1. Ionic conduction is caused by transitions of mobile ions through the
central sites of the polyhedra in a crystal. The FCC structure is composed of
tetrahedral and octahedral tiles. The coordination polyhedra affect the
energies of ions at their central sites. These different site energies lead to
large energy gaps in the transition state for ion diffusion, which leads to
low ionic conductivity. The BCC structure is composed of tetrahedra only, and
mobile ions can diffuse from one tetrahedral site to another without passing
through octahedral sites. Because the site energy remains constant, the energy
barrier for mobile ion diffusion is lower in the BCC framework compared with
that in the FCC framework. Thus, structural design based on polyhedra is
crucial for developing efficient materials.

Methods for designing crystal structures based on polyhedra have not yet been
established. Most crystal structure prediction methods rely on random
structure generation, in which atoms are randomly arranged. We previously
reported a method for predicting crystal structure by inputting the local
structure as clusters and randomly arranging these clusters 2; however, it is
difficult to generate a structure with a targeted polyhedron. Recently,
methods using generative models such as generative adversarial network (GAN),
variational autoencoder (VAE), and diffusion models have been proposed to
reverse engineer crystal structures from material properties 3, 4, 5, 6.
However, this approach requires a huge amount of data to accurately reproduce
a structure. Furthermore, it is difficult to predict extrapolation regions
using such methods.

In this work, we use discrete geometric analysis called the theory of standard
realization to construct a new method of crystal structure generation. The
theory of standard realization of topological crystals proposed by Kotani and
Sunada 7 is based on the principle of least energy and graph theory, which
gives the most symmetric crystal structure among all placements of given
periodic graphs. The standard realization satisfies the equilibrium condition
and symmetry of each vertex. The equilibrium condition is that the sum of the
vectors of the edges starting from an edge is zero. Symmetry is that each
symmetry of the graph structure of a crystal lattice generates a Euclidean
motion that preserves the crystal structure. For example, graphene (periodic
regular hexagonal lattices) and diamond structures are standard realizations
of hexagonal and diamond lattices, respectively 8. This method has yielded a
new metallic carbon structure 9, 10 and a new carbon material structure with
negative curvature 11. Although a wide variety of coordination numbers have
been reported for crystalline materials, the application of this theory has so
far been limited to three-coordinate carbon materials.

Figure 1: Conceptual diagram of our method for crystal structure generation.
The dual periodic graph generated from the input polyhedra is transformed into
a standard realization of a dual crystal structure. The obtained dual crystal
structure is transformed into a crystal structure by centroidal Voronoi
tessellation.

This work aims to establish a new approach to generate crystal structures from
polyhedron-based graphs using standard realization theory. We use dual
periodic graphs instead of periodic graphs to generate structures based on
space-filling polyhedra. A dual periodic graph, as we define it here, is a
periodic graph obtained from a dual crystal structure. A dual crystal
structure is a structure formed by connecting the central points of space-
filling polyhedra in a crystal structure. The dual periodic graph directly
represents the space-filling polyhedra in a crystal structure, which means
that the dual periodic graph can be used to generate a crystal structure based
on polyhedra. A conceptual diagram of our proposed method is shown in Figure
1. Given a dual periodic graph based on a space-filling polyhedron, we
generate a dual crystal structure from the graph using standard realization
theory and then transform the obtained dual crystal structure into a crystal
structure. Using our method, we demonstrate the generation of FCC, HCP, and
BCC structures, which are typical structures of metallic compounds. A new
finding in this work is the demonstration of the crystal structure generation
from space-filling polyhedra by combining dual periodic graphs and the theory
of standard realization. We also discuss the challenges and potential of our
method to discover new crystal structures based on polyhedra.

##  2 Experimental Section

###  2.1 Standard Realizations

A standard realization of a topological crystal, as proposed by Kotani and
Sunada 7, has the least energy obtained by the harmonic oscillator model, and
is the most symmetric among all placements in the space for a given
combinatorial structure. Construction of a standard realization is a purely
mathematical method. In the following, the method to construct a standard
realization is briefly described. More detailed mathematical descriptions can
be found elsewhere 7, 12, 13, 8.

Let $X_{0}=(V_{0},E_{0})$ be the dual graph of the target structure, and
$X_{T}=(V_{0},E_{T})$ be a spanning tree of $X_{0}$, where $V_{0}$ is the set
of vertices and $E_{0}$ are the set of edges. The Betti number $b=b(X_{0})$ is
defined by $b=|E_{0}|-|E_{T}|$, and expresses the number of essential closed
paths in $X_{0}$. First, we set directions for all edges
$\\{e_{j}\\}_{j=1}^{N}$ of $X_{0}$ ($N=|E_{0}|$), and consider a space

$C_{1}(X_{0},{\mathbb{Z}})=\\{a_{1}e_{1}+\cdots+a_{N}e_{N}\mid\\{e_{j}\\}_{j=1}^{N}=E_{0},a_{n}\text{
are integers}\\}.$

Then we may establish the notion of ${\mathbb{Z}}$-linear independence for
elements in $C_{1}(X_{0},{\mathbb{Z}})$. A set of closed paths
$\\{\alpha_{j}\\}_{j=1}^{b}$ in $X_{0}$, which are mutually
${\mathbb{Z}}$-linearly independent is selected, and then we can calculate the
$N\times b$ matrix
$B=\begin{bmatrix}{\left\langle{e_{j}},{\alpha_{k}}\right\rangle}\end{bmatrix}$,
the $b\times b$ matrix
$G_{0}=\begin{bmatrix}{\left\langle{\alpha_{j}},{\alpha_{k}}\right\rangle}\end{bmatrix}$,
and $A=G_{0}^{-1}B$, where ${\left\langle{\cdot},{\cdot}\right\rangle}$ is the
inner product on $C_{1}(X_{0},{\mathbb{Z}})$, which is defined by
${\left\langle{e_{j}},{e_{k}}\right\rangle}=\delta_{jk}$ and
${\left\langle{e_{j}},{\overline{e_{k}}}\right\rangle}=-\delta_{jk}$. Then,
the matrix $A$ gives us the edges of a standard realization described by the
basis $\\{\alpha_{j}\\}_{j=1}^{b}$. However, in general, a standard
realization is constructed in the $b$-dimensional space
$H_{1}(X_{0},{\mathbb{R}})$, which consists of linear combinations of all
essential closed paths in $X_{0}$. Therefore, we should construct a projection
onto the three-dimensional (3D) space.

By selecting $\alpha_{4},\cdots,\alpha_{b}$ as the basis of
$(b-3)$-dimensional subspace $H$, defining the $3\times 3$ matrix as

$G=G_{11}-G_{12}G_{22}^{-1}G_{21},\quad G_{0}=\begin{bmatrix}G_{11}&G_{12}\\\
G_{21}&G_{22}\end{bmatrix},$

and projecting $A$ onto $H$, we obtain the edges of a standard realization in
$3$D space described by the basis $p_{x}$, $p_{y}$, $p_{z}$, where
$G=\begin{bmatrix}{\left\langle{\beta_{j}},{\beta_{k}}\right\rangle}\end{bmatrix}$.
Next, vectors of parallel transformation ${\bm{p}}_{x}$, ${\bm{p}}_{y}$, and
${\bm{p}}_{z}$ are calculated in $3$D Euclidean space; these vectors satisfy
$G=\begin{bmatrix}{\left\langle{{\bm{p}}_{j}},{{\bm{p}}_{k}}\right\rangle}\end{bmatrix}$.
This provides $3$D vectors that describe the edges of a standard realization.
By fixing a vertex ${\bm{v}}_{0}=(0,0,0)$, and, for each vertex $v_{k}$,
($k=1,\ldots,|V|-1$), taking the shortest path $c_{k}$ from $v_{0}$ to $v_{k}$
in the spanning tree $X_{T}$, we may calculate the coordinate ${\bm{v}}_{k}$
along the path $c_{k}$. Finally, we calculate coordinates of extra vertices
using vectors of edges in $X_{0}\setminus X_{T}$.

A $b$-dimensional crystal lattice can be realized in 3D space by choosing a
$(b-3)$-dimensional subspace $H$ from $H_{1}(X_{0},{\mathbb{R}})$ and by using
the above method. However, in order to construct a good (symmetric) 3D crystal
lattice, $H$ must be chosen appropriately, and for this purpose, it is
important to choose the basis of $H_{1}(X_{0},{\mathbb{R}})$ appropriately.

###  2.2 Dual Crystal Structure and Dual Periodic Graph

In general, a dual relationship is between two objects that are paired with
each other; the dual relationship of a dual structure is the original
structure. An octahedron and a cube or a Delaunay diagram and Voronoi diagram
are in dual relationships. An example of a two-dimensional ($2$D) crystal
structure and its dual crystal structure is shown in Figure 2a. This structure
is the kagome lattice. Note that a $b$-dimensional crystal structure here
means a group of given points that repeat periodically in $b$-dimensional
space. By connecting the given points with edges, we can see that this $2$D
crystal structure is a tessellation with two triangles and one hexagon. The
periodic graph obtained from this crystal structure does not reveal that the
$2$D crystal structure corresponding to this graph is composed of triangles
and hexagons.

Figure 2: Relationship between crystal structures and dual crystal structures,
periodic graphs, and dual periodic graphs in (a) two-dimensional planes and
(b) three-dimensional space. The points in blue correspond to atomic sites,
and the dual vertices shown in red and green correspond to interlattice sites.
The crystal structure and dual crystal structure are obtained by connecting
the given points and dual vertices, respectively. The periodic graph and dual
periodic graph are obtained from the crystal structure and dual periodic
graph, respectively.

To explicitly consider these tiles in the graph, we introduce a dual crystal
structure. A dual crystal structure is a structure formed by connecting the
central points of tiles in a crystal structure. For example, the dual crystal
structure of the kagome lattice is obtained by connecting the central points
of the triangles and hexagons. The dual periodic graph is the periodic graph
obtained from the dual crystal structure. The edges of the dual periodic graph
correspond to the number of polygonal edges in the crystal structure. The dual
periodic graph of the kagome lattice has three edges connecting to the green
vertex and six edges connecting to the red vertex in Figure 2a, which
correspond to the number of edges in the triangle and hexagon, respectively.
The symmetry of the original crystal structure and the dual crystal structure
obtained by polyhedral packing are same. Therefore, the group of periodicity
$X/X_{0}$ is also same for the structures constructed from each graph.

As in the case of the $2$D plane, dual crystal structures and dual periodic
graphs can also be defined in $3$D space. As an example, the $3$D crystal
structure of the FCC lattice and its dual crystal structure are shown in
Figure 2b. Connecting the given vertices by edges, we can see that the
primitive cell of this $3$D crystal structure consists of two tetrahedra and
one octahedron. The dual crystal structure of the FCC lattice is obtained by
connecting the central points of the tetrahedra and octahedra. Although the
periodic graph of the FCC structure does not reveal that it is composed of two
tetrahedra and an octahedron, the dual periodic graph has four edges
connecting to the green vertex and eight edges connecting to the red vertex,
which correspond to the face number of tetrahedra and octahedra, respectively.
Thus, a dual periodic graph represents a space-filling polyhedron, and a
standard realization generates a dual crystal structure that satisfies the
crystal structure of an appropriately selected space-filling polyhedra.

In general, the dual graph cannot be uniquely defined for a given graph,
however, it can be uniquely defined by determining the placement of graph
vertices in space. Our method defines bonds between atoms in a crystal. It
then forms a dual crystal structure using these bonds and vertices. Finally, a
dual periodic graph is obtained from that structure. In other words, the dual
periodic graph is defined based on the polyhedral packing of the crystal
structure. Our method uses centroidal Voronoi tessellation (CVT) as this
polyhedral packing. CVT is a method of defining a boundary between two
adjacent points for a given point and decomposing the region by the boundary.
2D planes are decomposed into polygons, and 3D space is decomposed into
polyhedra. If the intersection of such boundaries is defined as a dual vertex,
CVT can transform from a crystal structure to a dual structure or from a dual
structure to a crystal structure. If we can uniquely decompose a crystal
structure into polyhedra as in the CVT, it is possible to uniquely define a
dual graph. By using CVT as a polyhedral packing, a dual crystal structure
with the same symmetry as the original crystal structure can be obtained for
FCC, HCP, and BCC structures.

###  2.3 Workflow

Figure 3 shows a flowchart of our method. In this method, a dual periodic
graph is used as an input. The crystal structure corresponding to the dual
periodic graph is generated in eight steps. Steps 1 to 7 describe the process
of generating the dual structure from the dual periodic graph based on the
theory of standard realization of topological crystals. Step 8 is the process
of obtaining the crystal structure from the dual crystal structure. These
steps are conducted as follows.

Figure 3: Flowchart of our method for crystal structure generation.

  1. 1.

Calculate the Betti number $b$ from the given dual periodic graph $X_{0}$.
Here, $b$ is equal to the number of edges in the graph minus the number of
maximal spanning trees. In the case of the FCC dual graph, the number of edges
is eight and the number of maximal spanning trees is two, so $b$ is six.

  2. 2.

Define basis set $\\{\alpha_{j}\\}_{j=1}^{b}$ based on the number of selected
closed paths corresponding to $b$ obtained in step 1.

  3. 3.

Calculate the matrices $G_{0}$, $G$, and $A$ from the basis set
$\\{\alpha_{j}\\}_{j=1}^{b}$ obtained in step 2 to project from the
$b$-dimensional vector space to the $3$D vector subspace. The matrices $G_{0}$
and $G$ represent the conditions satisfied by the $b$\- and $3$D lattice
vectors, respectively. The matrix $A$ represents the basis of the edges in the
$b$-dimensional vector space.

  4. 4.

Define lattice vectors ${\bm{p}}_{x}$, ${\bm{p}}_{y}$, and ${\bm{p}}_{z}$ such
that the matrix $G$ obtained in step 3 is satisfied. The lattice vectors can
also be obtained by Cholesky decomposition of the matrix $G$.

  5. 5.

Calculate the edge vectors $\\{e_{j}\\}_{j=1}^{N}$ from the matrix $A$
obtained in step 3 and the lattice vectors ${\bm{p}}_{x}$, ${\bm{p}}_{y}$, and
${\bm{p}}_{z}$ obtained in step 4.

  6. 6.

Calculate the vertex vectors $\\{v_{k}\\}_{k=0}^{|V|-1}$ from the lattice
vectors ${\bm{p}}_{x}$, ${\bm{p}}_{y}$, and ${\bm{p}}_{z}$ obtained in step 4
and the edge vectors $\\{e_{j}\\}_{j=1}^{N}$ obtained in step 5. The vertex
vectors correspond to the primitive coordinates.

  7. 7.

Generate a dual crystal structure from the lattice vectors ${\bm{p}}_{x}$,
${\bm{p}}_{y}$, ${\bm{p}}_{z}$ obtained in step 4 and the vertex vectors
$\\{v_{k}\\}_{k=0}^{|V|-1}$ obtained in step 6.

  8. 8.

Transform the dual structure obtained in step 7 into a crystal structure by
CVT.

The steps were performed using Python with NetworkX for graph operations 14,
Numpy for operations of linear algebra 15, Pymatgen for generating the dual
structure 16, and SciPy for computing CVT 17. An example of Python code that
generates a dual crystal structure from a dual periodic graph is shown in
Supporting Information. Crystal structures were visualized by VESTA 18.

##  3 Results and Discussion

###  3.1 Crystal Structure Generation for FCC Structures

We demonstrated crystal structure generation from dual periodic graphs of FCC
structures using our method. The FCC structure and dual periodic graph are
shown in Figure 4a and b, respectively. The primitive FCC structure has two
interstitial tetrahedral sites and one interstitial octahedral site. In other
words, the primitive FCC structure is tessellated by two tetrahedra and one
octahedron. All the faces of each tetrahedron are shared with octahedra, and
each octahedron shares all its faces with tetrahedra . The dual periodic graph
has four edges connecting to the green vertex (${v_{1}}$ and ${v_{2}}$) and
eight edges connecting to the red vertex (${v_{0}}$), which correspond to the
face number of tetrahedra and octahedra, respectively.

Figure 4: (a) The primitive cell of the FCC structure consists of one
octahedron (red) and two tetrahedra (green). (b) Dual periodic graph of the
FCC structure, where ${v_{i}}$ and ${e_{i}}$ are the vertex and edge numbers,
respectively. (c) Dual crystal structure and (d) crystal structure generated
from the dual periodic graph using our method.

The specific process of generating the FCC structure from its dual periodic
graph using standard realization theory is described below. For the FCC
structure, the dual periodic graph $X_{0}$ has $b=6$. Taking a $Z$-basis of
$H_{1}(X_{0},{\mathbb{Z}})$ as

$\displaystyle{\alpha}_{1}$ $\displaystyle={e}_{1}-{e}_{2},$
$\displaystyle\quad{\alpha}_{2}$ $\displaystyle={e}_{1}-{e}_{3},$
$\displaystyle\quad{\alpha}_{3}$ $\displaystyle={e}_{1}-{e}_{4},$
$\displaystyle{\alpha}_{4}$ $\displaystyle={e}_{1}-{e}_{2}+{e}_{5}-{e}_{6},$
$\displaystyle\quad{\alpha}_{5}$
$\displaystyle={e}_{1}-{e}_{3}+{e}_{5}-{e}_{7},$
$\displaystyle\quad{\alpha}_{6}$
$\displaystyle={e}_{1}-{e}_{4}+{e}_{5}-{e}_{8},$

The closed path corresponding to each basis is shown in Figure S1. Then, by
projecting onto the subspace spanning $\alpha_{1}$, $\alpha_{2}$, and
$\alpha_{3}$, we obtain a standard realization in $3$D space as

$\displaystyle e_{1}$
$\displaystyle=\frac{1}{4}p_{x}+\frac{1}{4}p_{y}+\frac{1}{4}p_{z},$
$\displaystyle\quad e_{5}$
$\displaystyle=-\frac{1}{4}p_{x}-\frac{1}{4}p_{y}-\frac{1}{4}p_{z}=-e_{1},$
(1) $\displaystyle e_{2}$
$\displaystyle=-\frac{3}{4}p_{x}+\frac{1}{4}p_{y}+\frac{1}{4}p_{z},$
$\displaystyle\quad e_{6}$
$\displaystyle=\frac{3}{4}p_{x}-\frac{1}{4}p_{y}-\frac{1}{4}p_{z}=-e_{2},$
$\displaystyle e_{3}$
$\displaystyle=\frac{1}{4}p_{x}-\frac{3}{4}p_{y}+\frac{1}{4}p_{z},$
$\displaystyle\quad e_{7}$
$\displaystyle=-\frac{1}{4}p_{x}+\frac{3}{4}p_{y}-\frac{1}{4}p_{z}=-e_{3},$
$\displaystyle e_{4}$
$\displaystyle=\frac{1}{4}p_{x}+\frac{1}{4}p_{y}-\frac{3}{4}p_{z},$
$\displaystyle\quad e_{8}$
$\displaystyle=-\frac{1}{4}p_{x}-\frac{1}{4}p_{y}+\frac{3}{4}p_{z}=-e_{4}.$

The period vectors (a basis of Bravais lattice)
$\\{{\bm{p}}_{x},{\bm{p}}_{y},{\bm{p}}_{z}\\}$ satisfies

$G=\begin{bmatrix}|p_{x}|^{2}&{\left\langle{p_{x}},{p_{y}}\right\rangle}&{\left\langle{p_{x}},{p_{z}}\right\rangle}\\\
{\left\langle{p_{y}},{p_{z}}\right\rangle}&|p_{y}|^{2}&{\left\langle{p_{y}},{p_{z}}\right\rangle}\\\
{\left\langle{p_{z}},{p_{x}}\right\rangle}&{\left\langle{p_{z}},{p_{y}}\right\rangle}&|p_{z}|^{2}\\\
\end{bmatrix}=\frac{1}{2}\begin{bmatrix}2&1&1\\\ 1&2&1\\\ 1&1&2\\\
\end{bmatrix}.$ (2)

Take ${\bm{p}}_{x}$, ${\bm{p}}_{y}$, and ${\bm{p}}_{z}$

${\bm{p}}_{x}=(1,0,1),\quad{\bm{p}}_{y}=(1,1,0),\quad{\bm{p}}_{z}=(0,1,1),$
(3)

which satisfies Equation (2). Substituting Equation (3) into Equation (1), we
obtain positions of vertices as

$\displaystyle{\bm{v}}_{0}$ $\displaystyle=\left(0,0,0\right),$
$\displaystyle{\bm{v}}_{1}$
$\displaystyle={\bm{v}}_{0}+{\bm{e}}_{1}=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right),$
$\displaystyle\quad{\bm{v}}_{2}$
$\displaystyle={\bm{v}}_{0}+{\bm{e}}_{2}=\left(-\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right),$
$\displaystyle{\bm{v}}_{3}$
$\displaystyle={\bm{v}}_{0}+{\bm{e}}_{3}=\left(\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right),$
$\displaystyle\quad{\bm{v}}_{4}$
$\displaystyle={\bm{v}}_{0}+{\bm{e}}_{4}=\left(-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right),$
$\displaystyle{\bm{v}}_{5}$
$\displaystyle={\bm{v}}_{0}+{\bm{e}}_{5}=\left(-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right),$
$\displaystyle\quad{\bm{v}}_{6}$
$\displaystyle={\bm{v}}_{0}+{\bm{e}}_{6}=\left(\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right),$
$\displaystyle{\bm{v}}_{7}$
$\displaystyle={\bm{v}}_{0}+{\bm{e}}_{7}=\left(-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right),$
$\displaystyle\quad{\bm{v}}_{8}$
$\displaystyle={\bm{v}}_{0}+{\bm{e}}_{8}=\left(\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right).$

The dual crystal structure and crystal structure obtained from this graph are
shown in Figure4c and d, respectively. The obtained crystal structure is
consistent with the FCC structure. Therefore, the FCC structure can be
reproduced from the dual periodic graph using our method.

###  3.2 Crystal Structure Generation for HCP Structures

We also demonstrated crystal structure generation from dual periodic graphs of
HCP structures using our method. The HCP structure and dual periodic graph are
shown in Figure 5a and b, respectively. The HCP structure has four
interstitial tetrahedral sites and two interstitial octahedral sites. In other
words, the primitive HCP structure is tessellated by four tetrahedra and two
octahedra. Unlike the FCC structure, tetrahedra share faces with each other,
and octahedra share faces with each other in the HCP structure. The dual
periodic graph has four edges connecting to the green vertex (${v_{2}}$,
${v_{3}}$, ${v_{4}}$, and ${v_{5}}$) and eight edges connecting to the red
vertex (${v_{0}}$ and ${v_{1}}$), which correspond to the face number of
tetrahedra and octahedra, respectively.

Figure 5: (a) The primitive cell of the HCP structure consisting of two
octahedra (red) and four tetrahedra (green). (b) Dual periodic graph of the
HCP structure, where ${v_{i}}$ and ${e_{i}}$ are the vertex and edge numbers,
respectively. (c) Dual crystal structure and (d) crystal structure generated
from the dual periodic graph using our method.

The specific process of generating the FCC structure from its dual periodic
graph using standard realization theory is described below. The dual periodic
graph $X_{0}$ of HCP structure had $b=11$. A ${\mathbb{Z}}$-basis of
$H_{1}(X_{0},{\mathbb{Z}})$ was taken as

$\displaystyle\alpha_{1}$ $\displaystyle=e_{1}-e_{2},$
$\displaystyle\quad\alpha_{2}$ $\displaystyle=e_{7}-e_{8},$
$\displaystyle\quad\alpha_{3}$ $\displaystyle=e_{4}-e_{5},$
$\displaystyle\alpha_{4}$ $\displaystyle=e_{7}-e_{8}+e_{15}-e_{16},$
$\displaystyle\quad\alpha_{5}$ $\displaystyle=e_{4}-e_{5}+e_{10}-e_{12},$
$\displaystyle\alpha_{6}$ $\displaystyle=e_{1}+e_{6}+e_{3}+e_{7},$
$\displaystyle\quad\alpha_{7}$ $\displaystyle=e_{1}+e_{6}+e_{4}+e_{9},$
$\displaystyle\alpha_{8}$ $\displaystyle=e_{1}+e_{6}+e_{5}+e_{8},$
$\displaystyle\quad\alpha_{9}$ $\displaystyle=e_{2}+e_{13}+e_{10}+e_{14},$
$\displaystyle\alpha_{10}$ $\displaystyle=e_{2}+e_{13}+e_{11}+e_{15},$
$\displaystyle\quad\alpha_{11}$ $\displaystyle=e_{2}+e_{13}+e_{12}+e_{16},$

The closed path corresponding to each basis is shown in Figure S2. Then,
projecting onto the subspace spanning $\alpha_{1}$, $\alpha_{2}$, and
$\alpha_{3}$, we obtain a standard realization in the $3$D space as

$\displaystyle e_{1}$ $\displaystyle=\frac{1}{2}p_{x},$ $\displaystyle\quad
e_{2}$ $\displaystyle=-\frac{1}{2}p_{x},$ (4) $\displaystyle e_{3}$
$\displaystyle=-\frac{1}{10}p_{x}-\frac{2}{3}p_{y}-\frac{1}{3}p_{z},$
$\displaystyle\quad e_{4}$
$\displaystyle=-\frac{1}{10}p_{x}+\frac{1}{3}p_{y}+\frac{2}{3}p_{z},$
$\displaystyle e_{5}$
$\displaystyle=-\frac{1}{10}p_{x}-\frac{1}{3}p_{y}-\frac{1}{3}p_{z},$
$\displaystyle\quad e_{6}$ $\displaystyle=-\frac{3}{10}p_{x},$ $\displaystyle
e_{7}$ $\displaystyle=-\frac{1}{10}p_{x}+\frac{2}{3}p_{y}+\frac{1}{3}p_{z},$
$\displaystyle\quad e_{8}$
$\displaystyle=-\frac{1}{10}p_{x}+\frac{2}{3}p_{y}-\frac{1}{3}p_{z},$
$\displaystyle e_{9}$
$\displaystyle=-\frac{1}{10}p_{x}-\frac{1}{3}p_{y}-\frac{2}{3}p_{z},$
$\displaystyle\quad e_{10}$
$\displaystyle=\frac{1}{10}p_{x}-\frac{1}{3}p_{y}-\frac{2}{3}p_{z},$
$\displaystyle e_{11}$
$\displaystyle=\frac{1}{10}p_{x}+\frac{2}{3}p_{y}+\frac{1}{3}p_{z},$
$\displaystyle\quad e_{12}$
$\displaystyle=\frac{1}{10}p_{x}-\frac{1}{3}p_{y}-\frac{1}{3}p_{z},$
$\displaystyle e_{13}$ $\displaystyle=\frac{3}{10}p_{x},$ $\displaystyle\quad
e_{14}$ $\displaystyle=\frac{1}{10}p_{x}+\frac{1}{3}p_{y}+\frac{2}{3}p_{z},$
$\displaystyle e_{15}$
$\displaystyle=\frac{1}{10}p_{x}-\frac{2}{3}p_{y}-\frac{1}{3}p_{z},$
$\displaystyle\quad e_{16}$
$\displaystyle=\frac{1}{10}p_{x}+\frac{1}{3}p_{y}-\frac{1}{3}p_{z},$

The period vectors (a basis of Bravais lattice)
$\\{{\bm{p}}_{x},{\bm{p}}_{y},{\bm{p}}_{z}\\}$ satisfies

$G=\begin{bmatrix}|p_{x}|^{2}&{\left\langle{p_{x}},{p_{y}}\right\rangle}&{\left\langle{p_{x}},{p_{z}}\right\rangle}\\\
{\left\langle{p_{y}},{p_{z}}\right\rangle}&|p_{y}|^{2}&{\left\langle{p_{y}},{p_{z}}\right\rangle}\\\
{\left\langle{p_{z}},{p_{x}}\right\rangle}&{\left\langle{p_{z}},{p_{y}}\right\rangle}&|p_{z}|^{2}\\\
\end{bmatrix}=\frac{1}{4}\begin{bmatrix}5&0&0\\\ 0&2&-1\\\ 0&-1&2\\\
\end{bmatrix},$ (5)

Take ${\bm{p}}_{x}$, ${\bm{p}}_{y}$, and ${\bm{p}}_{z}$

${\bm{p}}_{x}=\left(\sqrt{5},0,0\right),\quad{\bm{p}}_{y}=\left(1,\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}}\right),\quad{\bm{p}}_{z}=\left(0,\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}}\right),$
(6)

which satisfy Equation (5), and substituting Equation (6) into Equation (4),
we obtain vectors of edges as

$\displaystyle{\bm{e}}_{1}$
$\displaystyle=\left(\frac{\sqrt{5}}{2},0,0\right),$
$\displaystyle\quad{\bm{e}}_{2}$
$\displaystyle=\left(-\frac{\sqrt{5}}{2},0,0\right),$
$\displaystyle{\bm{e}}_{3}$
$\displaystyle=\left(-\frac{1}{2\sqrt{5}},-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{6}}\right),$
$\displaystyle\quad{\bm{e}}_{4}$
$\displaystyle=\left(-\frac{1}{2\sqrt{5}},\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{6}}\right),$
$\displaystyle{\bm{e}}_{5}$
$\displaystyle=\left(-\frac{1}{2\sqrt{5}},0,\sqrt{\frac{2}{3}}\right),$
$\displaystyle\quad{\bm{e}}_{6}$
$\displaystyle=\left(-\frac{3}{2\sqrt{5}},0,0\right),$
$\displaystyle{\bm{e}}_{7}$
$\displaystyle=\left(-\frac{1}{2\sqrt{5}},\frac{1}{\sqrt{2}},\frac{1}{\sqrt{6}}\right),$
$\displaystyle\quad{\bm{e}}_{8}$
$\displaystyle=\left(-\frac{1}{2\sqrt{5}},0,-\sqrt{\frac{2}{3}}\right),$
$\displaystyle{\bm{e}}_{9}$
$\displaystyle=\left(-\frac{1}{2\sqrt{5}},-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{6}}\right),$
$\displaystyle\quad{\bm{e}}_{10}$
$\displaystyle=\left(\frac{1}{2\sqrt{5}},-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{6}}\right),$
$\displaystyle{\bm{e}}_{11}$
$\displaystyle=\left(\frac{1}{2\sqrt{5}},\frac{1}{\sqrt{2}},\frac{1}{\sqrt{6}}\right),$
$\displaystyle\quad{\bm{e}}_{12}$
$\displaystyle=\left(\frac{1}{2\sqrt{5}},0,-\sqrt{\frac{2}{3}}\right),$
$\displaystyle{\bm{e}}_{13}$
$\displaystyle=\left(\frac{3}{2\sqrt{5}},0,0\right),$
$\displaystyle\quad{\bm{e}}_{14}$
$\displaystyle=\left(\frac{1}{2\sqrt{5}},\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{6}}\right),$
$\displaystyle{\bm{e}}_{15}$
$\displaystyle=\left(\frac{1}{2\sqrt{5}},-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{6}}\right),$
$\displaystyle\quad{\bm{e}}_{16}$
$\displaystyle=\left(\frac{1}{2\sqrt{5}},0,\sqrt{\frac{2}{3}}\right)$

and positions of vertices as

$\displaystyle{\bm{v}}_{0}$ $\displaystyle=(0,0,0),$
$\displaystyle{\bm{v}}_{1}$
$\displaystyle={\bm{v}}_{0}+{\bm{e}}_{1}=\left(\frac{\sqrt{5}}{2},0,0\right)$
$\displaystyle\quad{\bm{v}}_{2}$
$\displaystyle={\bm{v}}_{0}-{\bm{e}}_{7}=\left(\frac{1}{2\sqrt{5}},-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{6}}\right),$
$\displaystyle{\bm{v}}_{3}$
$\displaystyle={\bm{v}}_{1}+{\bm{e}}_{3}=\left(-\frac{1}{2\sqrt{5}},-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{6}}\right)$
$\displaystyle\quad{\bm{v}}_{4}$
$\displaystyle={\bm{v}}_{0}-{\bm{e}}_{14}=\left(-\frac{1}{2\sqrt{5}},-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{6}}\right)$
$\displaystyle{\bm{v}}_{5}$
$\displaystyle={\bm{v}}_{1}+{\bm{e}}_{10}=\left(\frac{1}{2\sqrt{5}},-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{6}}\right)$
$\displaystyle\quad{\bm{v}}_{0a}$
$\displaystyle={\bm{v}}_{4}+{\bm{e}}_{15}=\left(0,-\sqrt{2},0\right),$
$\displaystyle{\bm{v}}_{0b}$
$\displaystyle={\bm{v}}_{4}+{\bm{e}}_{16}=\left(0,-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}}\right)$
$\displaystyle\quad{\bm{v}}_{0c}$
$\displaystyle={\bm{v}}_{2}+{\bm{e}}_{8}=\left(0,-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}}\right)$
$\displaystyle{\bm{v}}_{0d}$
$\displaystyle={\bm{v}}_{2}+{\bm{e}}_{9}=\left(0,-\sqrt{2},0\right)$
$\displaystyle\quad{\bm{v}}_{1a}$
$\displaystyle={\bm{v}}_{0}+{\bm{e}}_{2}=\left(-\frac{\sqrt{5}}{2},0,0\right)$
$\displaystyle{\bm{v}}_{2a}$
$\displaystyle={\bm{v}}_{3}+{\bm{e}}_{6}=\left(-\frac{2}{\sqrt{5}},-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{6}}\right)$
$\displaystyle\quad{\bm{v}}_{3a}$
$\displaystyle={\bm{v}}_{1}+{\bm{e}}_{4}=\left(\frac{2}{\sqrt{5}},\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{6}}\right)$
$\displaystyle{\bm{v}}_{3b}$
$\displaystyle={\bm{v}}_{1}+{\bm{e}}_{5}=\left(\frac{2}{\sqrt{5}},0,\sqrt{\frac{2}{3}}\right)$
$\displaystyle\quad{\bm{v}}_{4a}$
$\displaystyle={\bm{v}}_{5}+{\bm{e}}_{13}=\left(\frac{2}{\sqrt{5}},-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{6}}\right)$
$\displaystyle{\bm{v}}_{5a}$
$\displaystyle={\bm{v}}_{1}+{\bm{e}}_{12}=\left(\frac{3}{\sqrt{5}},0,-\sqrt{\frac{2}{3}}\right)$
$\displaystyle\quad{\bm{v}}_{5b}$
$\displaystyle={\bm{v}}_{1}+{\bm{e}}_{11}=\left(\frac{3}{\sqrt{5}},\frac{1}{\sqrt{2}},\frac{1}{\sqrt{6}}\right)$

The dual crystal structure and crystal structure obtained from this graph are
shown in Figure 5c and d. The obtained crystal structure is consistent with
the HCP structure. Therefore, the HCP structure can be reproduced from the
graph using our method.

###  3.3 Crystal Structure Generation for BCC Structures

Next, we demonstrate crystal structure generation from dual periodic graphs of
BCC structures using our method. The BCC structure and dual periodic graph are
shown in Figure 6a and b, respectively. The primitive BCC structure has six
interstitial tetrahedral sites. Unlike the FCC and HCP structures, the BCC
structure is tessellated from only tetrahedra. The dual periodic graph has
four edges connecting to the red vertex, corresponding to the face number of
tetrahedra.

Figure 6: (a) Primitive cell of the BCC structure consisting of six
tetrahedra. (b) Dual periodic graph of the BCC structure, where ${v_{i}}$ and
${e_{i}}$ are the vertex and edge numbers, respectively. (c) Dual crystal
structure and (d) crystal structure generated from the dual periodic graph
using our method.

The specific process of generating the BCC structure from its dual periodic
graph using standard realization theory is described below. The dual periodic
graph $X_{0}$ of BCC structure had $b=7$. A ${\mathbb{Z}}$-basis of
$H_{1}(X_{0},{\mathbb{Z}})$ was taken as

$\displaystyle\alpha_{1}$ $\displaystyle=e_{1}+e_{11}+e_{12},$
$\displaystyle\quad\alpha_{2}$ $\displaystyle=e_{2}+e_{9}+e_{10},$
$\displaystyle\quad\alpha_{3}$ $\displaystyle=e_{3}+e_{7}+e_{8},$
$\displaystyle\alpha_{4}$ $\displaystyle=e_{5}-e_{8}+e_{2}-e_{11},$
$\displaystyle\quad\alpha_{5}$ $\displaystyle=e_{6}-e_{9}+e_{3}-e_{12},$
$\displaystyle\quad\alpha_{6}$ $\displaystyle=e_{7}-e_{4}-e_{1}+e_{10},$
$\displaystyle\alpha_{7}$
$\displaystyle=e_{7}+e_{8}+e_{9}+e_{10}+e_{11}+e_{12},$

The closed path corresponding to each basis is shown in Figure S3. Then,
projecting onto the subspace spanning $\alpha_{1}$, $\alpha_{2}$, and
$\alpha_{3}$, we obtain a standard realization in $3$D space as

$\displaystyle e_{1}$
$\displaystyle=\frac{1}{2}p_{x}+\frac{1}{4}p_{y}+\frac{1}{4}p_{z},$
$\displaystyle\quad e_{2}$
$\displaystyle=\frac{1}{4}p_{x}+\frac{1}{2}p_{y}+\frac{1}{4}p_{z},$ (7)
$\displaystyle e_{3}$
$\displaystyle=\frac{1}{4}p_{x}+\frac{1}{4}p_{y}+\frac{1}{2}p_{z},$
$\displaystyle\quad e_{4}$
$\displaystyle=-\frac{1}{2}p_{x}-\frac{1}{4}p_{y}-\frac{1}{4}p_{z},$
$\displaystyle e_{5}$
$\displaystyle=-\frac{1}{4}p_{x}-\frac{1}{2}p_{y}-\frac{1}{4}p_{z},$
$\displaystyle\quad e_{6}$
$\displaystyle=-\frac{1}{4}p_{x}-\frac{1}{4}p_{y}-\frac{1}{2}p_{z},$
$\displaystyle e_{7}$ $\displaystyle=-\frac{1}{4}p_{y}+\frac{1}{4}p_{z},$
$\displaystyle\quad e_{8}$ $\displaystyle=-\frac{1}{4}p_{x}+\frac{1}{4}p_{z},$
$\displaystyle e_{9}$ $\displaystyle=\frac{1}{4}p_{x}+\frac{1}{4}p_{y},$
$\displaystyle\quad e_{10}$ $\displaystyle=\frac{1}{4}p_{y}-\frac{1}{4}p_{z},$
$\displaystyle e_{11}$ $\displaystyle=\frac{1}{4}p_{x}-\frac{1}{4}p_{z},$
$\displaystyle\quad e_{12}$ $\displaystyle=\frac{1}{4}p_{x}-\frac{1}{4}p_{y}$

The period vectors (a basis of a Bravais lattice)
$\\{{\bm{p}}_{x},{\bm{p}}_{y},{\bm{p}}_{z}\\}$ satisfy

$G=\begin{bmatrix}|p_{x}|^{2}&{\left\langle{p_{x}},{p_{y}}\right\rangle}&{\left\langle{p_{x}},{p_{z}}\right\rangle}\\\
{\left\langle{p_{y}},{p_{z}}\right\rangle}&|p_{y}|^{2}&{\left\langle{p_{y}},{p_{z}}\right\rangle}\\\
{\left\langle{p_{z}},{p_{x}}\right\rangle}&{\left\langle{p_{z}},{p_{y}}\right\rangle}&|p_{z}|^{2}\\\
\end{bmatrix}=\begin{bmatrix}3&-1&-1\\\ -1&3&-1\\\ -1&-1&3\\\ \end{bmatrix}.$
(8)

Take ${\bm{p}}_{x}$, ${\bm{p}}_{y}$, and ${\bm{p}}_{z}$

${\bm{p}}_{x}=\left(-1,1,1\right),\quad{\bm{p}}_{y}=\left(1,-1,1\right),\quad{\bm{p}}_{z}=\left(1,1,-1\right),$
(9)

which satisfy Equation (8), and substituting Equation (9) into Equation (7),
we obtain vectors of edges as

$\displaystyle{\bm{e}}_{1}$
$\displaystyle=\left(0,\frac{1}{2},\frac{1}{2}\right),$
$\displaystyle\quad{\bm{e}}_{2}$
$\displaystyle=\left(\frac{1}{2},0,\frac{1}{2}\right),$
$\displaystyle{\bm{e}}_{3}$
$\displaystyle=\left(\frac{1}{2},\frac{1}{2},0\right),$
$\displaystyle\quad{\bm{e}}_{4}$
$\displaystyle=\left(0,-\frac{1}{2},-\frac{1}{2}\right),$
$\displaystyle{\bm{e}}_{5}$
$\displaystyle=\left(-\frac{1}{2},0,-\frac{1}{2}\right),$
$\displaystyle\quad{\bm{e}}_{6}$
$\displaystyle=\left(-\frac{1}{2},-\frac{1}{2},0\right),$
$\displaystyle{\bm{e}}_{7}$
$\displaystyle=\left(0,\frac{1}{2},-\frac{1}{2}\right),$
$\displaystyle\quad{\bm{e}}_{8}$
$\displaystyle=\left(\frac{1}{2},0,-\frac{1}{2}\right),$
$\displaystyle{\bm{e}}_{9}$
$\displaystyle=\left(\frac{1}{2},-\frac{1}{2},0\right),$
$\displaystyle\quad{\bm{e}}_{10}$
$\displaystyle=\left(0,-\frac{1}{2},\frac{1}{2}\right),$
$\displaystyle{\bm{e}}_{11}$
$\displaystyle=\left(-\frac{1}{2},0,\frac{1}{2}\right),$
$\displaystyle\quad{\bm{e}}_{12}$
$\displaystyle=\left(-\frac{1}{2},\frac{1}{2},0\right)$

and positions of vertices as

$\displaystyle{\bm{v}}_{0}$ $\displaystyle=(0,0,0),$
$\displaystyle{\bm{v}}_{1}$
$\displaystyle={\bm{v}}_{0}+{\bm{e}}_{1}=\left(0,\frac{1}{2},\frac{1}{2}\right)$
$\displaystyle\quad{\bm{v}}_{2}$
$\displaystyle={\bm{v}}_{0}-{\bm{e}}_{7}=\left(0,-\frac{1}{2},\frac{1}{2}\right)$
$\displaystyle{\bm{v}}_{3}$
$\displaystyle={\bm{v}}_{1}+{\bm{e}}_{3}=\left(\frac{1}{2},\frac{1}{2},0\right)$
$\displaystyle\quad{\bm{v}}_{4}$
$\displaystyle={\bm{v}}_{0}-{\bm{e}}_{12}=\left(\frac{1}{2},-\frac{1}{2},0\right)$
$\displaystyle{\bm{v}}_{5}$
$\displaystyle={\bm{v}}_{1}+{\bm{e}}_{10}=\left(0,-\frac{1}{2},\frac{1}{2}\right)$
$\displaystyle\quad{\bm{v}}_{1a}$
$\displaystyle={\bm{v}}_{2}+{\bm{e}}_{5}=\left(-\frac{1}{2},-\frac{1}{2},0\right)$
$\displaystyle{\bm{v}}_{1b}$
$\displaystyle={\bm{v}}_{3}+{\bm{e}}_{11}=\left(0,\frac{1}{2},\frac{1}{2}\right)$
$\displaystyle\quad{\bm{v}}_{2a}$
$\displaystyle={\bm{v}}_{5}+{\bm{e}}_{4}=\left(0,-1,0\right)$
$\displaystyle{\bm{v}}_{4a}$
$\displaystyle={\bm{v}}_{3}+{\bm{e}}_{2}=\left(1,\frac{1}{2},\frac{1}{2}\right)$
$\displaystyle\quad{\bm{v}}_{4b}$
$\displaystyle={\bm{v}}_{2}+{\bm{e}}_{8}=\left(\frac{1}{2},-\frac{1}{2},0\right)$
$\displaystyle{\bm{v}}_{5a}$
$\displaystyle={\bm{v}}_{1}+{\bm{e}}_{6}=\left(-\frac{1}{2},0,\frac{1}{2}\right)$
$\displaystyle\quad{\bm{v}}_{5b}$
$\displaystyle={\bm{v}}_{3}+{\bm{e}}_{9}=\left(1,0,0\right)$

The dual crystal structure and crystal structure obtained from this dual
periodic graph are shown in Figure 6c and d, respectively. The obtained
crystal structure is consistent with the BCC structure. Therefore, the BCC
structure can be reproduced from the dual periodic graph using our method.

###  3.4 Challenges of Structure Generation by Standard Realizations

In this work, we generated FCC, HCP, and BCC structures from each
corresponding dual periodic graph using our calculation method. Our method can
be applied to generate undiscovered crystal structures from target polyhedra
if a dual periodic graph is obtained from the type and number of the
polyhedra. For example, by generating crystal structures using only
combinatorial information of tetrahedra as input, it is possible to generate
new tetrahedrally packed structures other than BCC structures. However, for
more complex structures, it is not easy to select an appropriate basis of
closed paths from a huge number of combinations of closed paths. To generate
crystal structures from graphs using standard realization theory, we should
select an appropriate $3$D vector space from the $b$-dimensional vector space.
In addition, it is necessary to select appropriate closed paths that define
the $(b-3)$-dimensional subspace for the basis of the closed paths.

For example, in a dual periodic graph with a BCC structure, there are eight
closed paths consisting of three nodes (Figure S4), 15 closed paths consisting
of four nodes (Figure S5), 24 closed paths consisting of five nodes (Figure
S6), and 16 closed paths consisting of six nodes (Figure S7), giving a total
of 63 closed paths. To generate the dual BCC structure, seven closed paths
corresponding to $b$ must be selected. We consider three closed paths
consisting of three nodes, three consisting of four nodes, and one consisting
of six nodes as the seven closed paths. The number of such closed path
combinations is $\binom{8}{3}$ $\times$ $\binom{15}{3}$ $\times$
$\binom{16}{1}$ (=407,680). Among them, we have identified 128 combinations of
closed paths that can generate the structure of the most symmetric
$Im\overline{3}m$ space group. It is noted that the combination of closed
paths in Figure S3 is a representative example of these 128 combinations.
Other combinations of closed paths either generated crystal structures that
are less symmetric than the $Im\overline{3}m$ space group or failed to
generate crystal structures. We found the closed path combination shown in
Figure S3 by trial and error, but the rule for selecting the appropriate
closed path combination is still unclear. Even for structures with a small
number of atoms in the unit cell and high symmetry, such as BCC structures, it
is difficult to generate the correct structure because of the large number of
closed path combinations. This problem needs to be solved to be able to apply
this method to more complex crystal structures. In the future, we plan to
apply this method to more complex crystal structures and develop a method to
efficiently select the appropriate closed path combination.

When using this method to predict an unknown structure, it is necessary to
define a dual periodic graph from the given polyhedra. Given a set of
polyhedra, the number of vertices and the number of edges joining between
vertices in the dual periodic graph are determined, and the dual periodic
graph can be defined by determining which edges are connected to each other.
For example, a dual periodic graph can be obtained by determining the
connectable faces between polyhedra based on the given shape of the polyhedra.
By applying our method to all combinations of such obtained dual periodic
graphs, it is possible, in principle, to generate crystal structures from any
polyhedron.

Our structure generation method, when integrated with other techniques, should
allow the prediction of practical structures. For instance, combining our
approach with the atomic-site configuration optimization method should enable
the prediction of complex crystal structures composed of various elemental
species. Recent reports indicate that quantum annealing can be used to solve
atomic-site optimization problems at low cost 19, facilitating the prediction
of existing multicomponent compounds 20.

The integration of our method with phonon analysis can be used to predict low-
symmetry structures. Our method generates the most symmetrical structure for a
given graph, so it cannot produce structures with low symmetry. Conducting a
phonon calculation of the derived structure, using either first-principles
calculations or machine learning potentials, can generate a low-symmetry
structure along the direction of structural instability 21. These optimization
methods require the input of the target crystal structure, so their
combination with our crystal structure generation method can facilitate
exploration of crystalline materials.

Finally, we compare our method based on discrete geometry with crystal
structure prediction methods by generative models such as GAN, VAE, and
diffusion models. While a large amount of training data is required to
accurately predict crystal structures with generative models 6, our method,
which does not use machine learning techniques, needs no training data. In the
generative model approach, the lattice vectors and atomic coordinates of the
crystal structure are predicted by independent models or processes 4. In
contrast, our method can directly transform graphs into corresponding lattice
vectors and atomic coordinates in a unified process. A notable limitation of
generative models is that they tend to predict structures with low symmetry 6.
Our approach, however, consistently generates structures with high symmetry.
The inherent capability of our method to facilitate the bidirectional
transformation between graphs and crystal structures presents a promising
avenue for the discovery of new crystal structures that would be difficult
with conventional methods, for example when it is employed in conjunction with
a generative model designed to generate graphs.

##  4 Conclusion

In this study, we proposed a method for crystal structure generation based on
the discrete geometry of target polyhedra. Most conventional methods of
crystal structure prediction use random structure generation, which produces
many unimportant structures and contributes to inefficiency in material
exploration. In the developed method, the shape and connectivity of space-
filling polyhedra are represented as a dual periodic graph and crystal
structures are generated from this graph based on standard realization theory.
We demonstrated that this method can generate FCC, HCP, and BCC structures
from dual periodic graphs. This work is a first step toward generating
undiscovered crystal structures based on the target polyhedra. Thus, this work
paves the way for structure-driven materials development and has the potential
to identify highly functional materials that have been difficult to discover
with conventional composition-driven development.

Basis for closed paths of FCC, HCP, and BCC structures used in this study, all
closed paths of BCC structures, and example of python code to generate a dual
structure.

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On the Equivalence of Module Categories over a Group-Theoretical Fusion
Category

On the Equivalence of Module Categories
over a Group-Theoretical Fusion Category

Sonia NATALE

S. Natale

Facultad de Matemática, Astronomía, Física y Computación, Universidad Nacional
de Córdoba,
CIEM-CONICET, Córdoba, Argentina<EMAIL_ADDRESS>http://www.famaf.unc.edu.ar/~natale/

Received April 28, 2017, in final form June 14, 2017; Published online June
17, 2017

We give a necessary and sufficient condition in terms of group cohomology for
two indecomposable module categories over a group-theoretical fusion category
${\mathcal{C}}$ to be equivalent. This concludes the classification of such
module categories.

fusion category; module category; group-theoretical fusion category

18D10; 16T05

##  1 Introduction

Throughout this paper we shall work over an algebraically closed field $k$ of
characteristic zero. Let $\mathcal{C}$ be a fusion category over $k$. The
notion of a ${\mathcal{C}}$-module category provides a natural
categorification of the notion of representation of a group. The problem of
classifying module categories plays a fundamental role in the theory of tensor
categories.

Two fusion categories ${\mathcal{C}}$ and ${\mathcal{D}}$ are called
_categorically Morita equivalent_ if there exists an indecomposable
$\mathcal{C}$-module category $\mathcal{M}$ such that $\mathcal{D}^{\rm op}$
is equivalent as a fusion category to the category
$\operatorname{Fun}_{\mathcal{C}}({\mathcal{M}},{\mathcal{M}})$ of
${\mathcal{C}}$-module endofunctors of ${\mathcal{M}}$. This defines an
equivalence relation in the class of all fusion categories.

Recall that a fusion category ${\mathcal{C}}$ is called _pointed_ if every
simple object of ${\mathcal{C}}$ is invertible. A basic class of fusion
categories consists of those which are categorically Morita equivalent to a
pointed fusion category; a fusion category in this class is called _group-
theoretical_. Group-theoretical fusion categories can be described in terms of
finite groups and their cohomology.

The purpose of this note is to give a necessary and sufficient condition in
terms of group cohomology for two indecomposable module categories over a
group-theoretical fusion category to be equivalent. For this, it is enough to
solve the same problem for indecomposable module categories over pointed
fusion categories.

Let ${\mathcal{C}}$ be a pointed fusion category. Then there exist a finite
group $G$ and a 3-cocycle $\omega$ on $G$ such that
${\mathcal{C}}\cong{\mathcal{C}}(G,\omega)$, where ${\mathcal{C}}(G,\omega)$
is the category of finite-dimensional $G$-graded vector spaces with
associativity constraint defined by $\omega$ (see Section 2.3 for a precise
definition). Let ${\mathcal{M}}$ be an indecomposable right
${\mathcal{C}}$-module category. Then there exists a subgroup $H$ of $G$ and a
2-cochain $\psi\in C^{2}(H,k^{\times})$ satisfying

$\displaystyle d\psi=\omega|_{H\times H\times H},$ (1.1)

such that ${\mathcal{M}}$ is equivalent as a ${\mathcal{C}}$-module category
to the category ${\mathcal{M}}_{0}(H,\psi)$ of left $A(H,\psi)$-modules in
${\mathcal{C}}$, where $A(H,\psi)=k_{\psi}H$ is the group algebra of $H$ with
multiplication twisted by $\psi$ [8], [1, Example 9.7.2].

The main result of this paper is the following theorem.

######  Theorem 1.1.

Let $H,L$ be subgroups of $G$ and let $\psi\in C^{2}(H,k^{\times})$ and
$\xi\in C^{2}(L,k^{\times})$ be $2$-cochains satisfying condition (1.1). Then
${\mathcal{M}}_{0}(H,\psi)$ and ${\mathcal{M}}_{0}(L,\xi)$ are equivalent as
${\mathcal{C}}$-module categories if and only if there exists an element $g\in
G$ such that $H={}^{g}L$ and the class of the $2$-cocycle

$\displaystyle{\xi}^{-1}{\psi}^{g}\Omega_{g}|_{L\times L}$ (1.2)

is trivial in $H^{2}(L,k^{\times})$.

Here we use the notation ${}^{g}x=gxg^{-1}$ and ${}^{g}L=\\{{}^{g}x\colon x\in
L\\}$. The 2-cochain ${\psi}^{g}\in C^{2}(L,k^{\times})$ is defined by
${\psi}^{g}(g_{1},g_{2})={\psi}({}^{g}g_{1},{}^{g}g_{2})$, for all
$g_{1},g_{2}\in L$, and $\Omega_{g}\colon G\times G\to k^{\times}$ is given by

$\displaystyle\Omega_{g}(g_{1},g_{2})=\frac{\omega({}^{g}g_{1},{}^{g}g_{2},g)\omega(g,g_{1},g_{2})}{\omega({}^{g}g_{1},g,g_{2})}.$

Observe that [8, Theorem 3.1] states that the indecomposable module categories
considered in Theorem 1.1 are parameterized by conjugacy classes of pairs
$(H,\psi)$. However, this conjugation relation is not described loc. cit.
(compare also with [7] and [1, Section 9.7]).

Consider for instance the case where ${\mathcal{C}}$ is the category of
finite-dimensional representations of the 8-dimensional Kac Paljutkin Hopf
algebra. Then ${\mathcal{C}}$ is group-theoretical. In fact,
${\mathcal{C}}\cong{\mathcal{C}}(G,\omega,C,1)$, where $G\cong D_{8}$ is a
semidirect product of the group $L=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ by
$C=\mathbb{Z}_{2}$ and $\omega$ is a certain (nontrivial) 3-cocycle on $G$
[9]. Let $\xi$ represent a nontrivial cohomology class in
$H^{2}(L,k^{\times})$. According to the usual conjugation relation among pairs
$(L,\psi)$, the result in [8, Theorem 3.1] would imply that the pairs $(L,1)$
and $(L,\xi)$, not being conjugated under the adjoint action of $G$, give rise
to two inequivalent ${\mathcal{C}}$-module categories. These module categories
both have rank one, whence they give rise to non-isomorphic fiber functors on
${\mathcal{C}}$. However, it follows from [4, Theorem 4.8(1)] that the
category ${\mathcal{C}}$ has a unique fiber functor up to isomorphism. In
fact, in this example there exists $g\in G$ such that $\Omega_{g}|_{L\times
L}$ is a 2-cocycle cohomologous to $\xi$. See Example 3.6.

Certainly, the condition given in Theorem 1.1 and the usual conjugacy relation
agree in the case where the 3-cocycle $\omega$ is trivial, and it reduces to
the conjugation relation among subgroups when they happen to be cyclic.

As explained in Section 3.1, condition (1.2) is equivalent to the condition
that $A(L,\xi)$ and ${}^{g}A(H,\psi)$ be isomorphic as algebras in
${\mathcal{C}}$, where
$\underline{G}\to\underline{\text{Aut}}_{\otimes}{\mathcal{C}}$,
$g\mapsto{}^{g}(\ )$, is the adjoint action of $G$ on ${\mathcal{C}}$ (see
Lemma 3.2).

Theorem 1.1 can be reformulated as follows.

######  Theorem 1.2.

Two ${\mathcal{C}}$-module categories ${\mathcal{M}}_{0}(H,\psi)$ and
${\mathcal{M}}_{0}(L,\xi)$ are equivalent if and only if the algebras
$A(H,\psi)$ and $A(L,\xi)$ are conjugated under the adjoint action of $G$ on
${\mathcal{C}}$.

Theorem 1.1 is proved in Section 3.3. Our proof relies on the fact that, as
happens with group actions on vector spaces, the adjoint action of the group
$G$ in the set of equivalence classes of ${\mathcal{C}}$-module categories is
trivial (Lemma 3.1). In the course of the proof we establish a relation
between the 2-cocycle in (1.2) and a 2-cocycle attached to $g$, $\psi$ and
$\xi$ in [8] (Remark 3.4 and Lemma 3.5).

We refer the reader to [1] for the main notions on fusion categories and their
module categories used throughout.

##  2 Preliminaries and notation

### 2.1

Let $\mathcal{C}$ be a fusion category over $k$. A (_right_)
${\mathcal{C}}$-_module category_ is a finite semisimple $k$-linear abelian
category $\mathcal{M}$ equipped with a bifunctor
$\bar{\otimes}\colon\mathcal{M}\times\mathcal{C}\rightarrow\mathcal{M}$ and
natural isomorphisms

$\displaystyle\mu_{M,X,Y}\colon\ M\bar{\otimes}(X\otimes
Y)\rightarrow(M\bar{\otimes}X)\bar{\otimes}Y,\qquad r_{M}\colon\
M\bar{\otimes}\textbf{1}\rightarrow M,$

$X,Y\in{\mathcal{C}}$, $M\in\mathcal{M}$, satisfying the following conditions:

$\displaystyle\mu_{M\bar{\otimes}X,Y,Z}\mu_{M,X,Y\otimes
Z}(\operatorname{id}_{M}\bar{\otimes}a_{X,Y,Z})=(\mu_{M,X,Y}\otimes\operatorname{id}_{Z})\mu_{M,X\otimes
Y,Z},$ (2.1)
$\displaystyle(r_{M}\otimes\operatorname{id}_{Y})\mu_{M,\textbf{1},Y}=\operatorname{id}_{M}\bar{\otimes}l_{Y},$
(2.2)

for all $M\in{\mathcal{M}}$, $X,Y\in{\mathcal{C}}$, where
$a\colon\otimes\circ(\otimes\times\operatorname{id}_{\mathcal{C}})\to\otimes\circ(\operatorname{id}_{\mathcal{C}}\times\otimes)$
and $l\colon\textbf{1}\otimes?\to\operatorname{id}_{\mathcal{C}}$, denote the
associativity and left unit constraints in ${\mathcal{C}}$, respectively.

Let $A$ be an algebra in ${\mathcal{C}}$. Then the category
${}_{A}{\mathcal{C}}$ of left $A$-modules in ${\mathcal{C}}$ is a right
${\mathcal{C}}$-module category with action
$\bar{\otimes}\colon_{A}{\mathcal{C}}\times{\mathcal{C}}\to{\mathcal{C}}_{A}$,
given by $M\bar{\otimes}X=M\otimes X$ endowed with the left $A$-module
structure $(m_{M}\otimes\operatorname{id}_{X})a_{A,M,X}^{-1}\colon
A\otimes(M\otimes X)\to M\otimes X$, where $m_{M}\colon A\otimes M\to M$ is
the $A$-module structure in $M$. The associativity constraint of
${}_{A}{\mathcal{C}}$ is given by $a^{-1}_{M,X,Y}\colon
M\bar{\otimes}(X\otimes Y)\to(M\bar{\otimes}X)\bar{\otimes}Y$, for all
$M\in{}_{A}{\mathcal{C}}$, $X,Y\in{\mathcal{C}}$.

A ${\mathcal{C}}$-module functor ${\mathcal{M}}\to{\mathcal{M}}^{\prime}$
between right ${\mathcal{C}}$-module categories
$({\mathcal{M}},\bar{\otimes})$ and
$({\mathcal{M}}^{\prime},\bar{\otimes}^{\prime})$ is a pair $(F,\zeta)$, where
$F\colon{\mathcal{M}}\to{\mathcal{M}}^{\prime}$ is a functor and
$\zeta_{M,X}\colon F(M\bar{\otimes}X)\to F(M)\bar{\otimes}^{\prime}X$ is a
natural isomorphism satisfying

$\displaystyle(\zeta_{M,X}\otimes\operatorname{id}_{Y})\zeta_{M\bar{\otimes}X,Y}F(\mu_{M,X,Y})={\mu^{\prime}}_{F(M),X,Y}\zeta_{M,X\otimes
Y},$ (2.3) $\displaystyle{r^{\prime}}_{F(M)}\zeta_{M,\textbf{1}}=F(r_{M}),$
(2.4)

for all $M\in{\mathcal{M}}$, $X,Y\in{\mathcal{C}}$.

Let ${\mathcal{M}}$ and ${\mathcal{M}}^{\prime}$ be ${\mathcal{C}}$-module
categories. An _equivalence_ of ${\mathcal{C}}$-module categories
${\mathcal{M}}\to{\mathcal{M}}^{\prime}$ is a ${\mathcal{C}}$-module functor
$(F,\zeta)\colon{\mathcal{M}}\to{\mathcal{M}}^{\prime}$ such that $F$ is an
equivalence of categories. If such an equivalence exists, ${\mathcal{M}}$ and
${\mathcal{M}}^{\prime}$ are called _equivalent ${\mathcal{C}}$-module
categories_. A ${\mathcal{C}}$-module category is called _indecomposable_ if
it is not equivalent to a direct sum of two nontrivial $\mathcal{C}$-submodule
categories.

Let ${\mathcal{M}},{\mathcal{M}}^{\prime}$ be indecomposable
${\mathcal{C}}$-module categories. Then
$\operatorname{Fun}_{\mathcal{C}}({\mathcal{M}},{\mathcal{M}})$ is a fusion
category with tensor product given by composition of functors and the category
$\operatorname{Fun}_{\mathcal{C}}({\mathcal{M}},{\mathcal{M}}^{\prime})$ is an
indecomposable module category over
$\operatorname{Fun}_{\mathcal{C}}({\mathcal{M}},{\mathcal{M}})$ in a natural
way. If $A$ and $B$ are indecomposable algebras in ${\mathcal{C}}$ such that
${\mathcal{M}}\cong_{A}\\!{\mathcal{C}}$ and
${\mathcal{M}}^{\prime}\cong{}_{B}{\mathcal{C}}$, then
$\operatorname{Fun}_{\mathcal{C}}({\mathcal{M}},{\mathcal{M}})^{\rm op}$ is
equivalent to the fusion category ${}_{A}{\mathcal{C}}_{A}$ of
$(A,A)$-bimodules in ${\mathcal{C}}$ and there is an equivalence of
${}_{A}{\mathcal{C}}_{A}$-module categories
${}_{B}{\mathcal{C}}_{A}\cong\operatorname{Fun}_{\mathcal{C}}({\mathcal{M}},{\mathcal{M}}^{\prime})$,
where ${}_{B}{\mathcal{C}}_{A}$ is the category of $(B,A)$-bimodules in
${\mathcal{C}}$.

### 2.2

Let ${\mathcal{M}}$ be a ${\mathcal{C}}$-module category. Every tensor
autoequivalence $\rho\colon{\mathcal{C}}\to{\mathcal{C}}$ induces a
${\mathcal{C}}$-module category structure ${\mathcal{M}}^{\rho}$ on
${\mathcal{M}}$ in the form $M\bar{\otimes}^{\rho}X=M\bar{\otimes}\rho(X)$,
with associativity constraint

$\displaystyle\mu^{\rho}_{M,X,Y}=\mu_{M,\rho(X)\otimes\rho(Y)}\big{(}\operatorname{id}_{M}\bar{\otimes}{\rho^{2}_{X,Y}}^{-1}\big{)}\colon\
M\bar{\otimes}\rho(X\otimes Y)\to(M\bar{\otimes}\rho(X))\bar{\otimes}\rho(Y),$

for all $M\in{\mathcal{M}}$, $X,Y\in{\mathcal{C}}$, where
$\rho^{2}_{X,Y}\colon\rho(X)\otimes\rho(Y)\to\rho(X\otimes Y)$ is the monoidal
structure of $\rho$. See [7, Section 3.2].

Suppose that $A$ is an algebra in ${\mathcal{C}}$. Then $\rho(A)$ is an
algebra in ${\mathcal{C}}$ with multiplication

$\displaystyle m_{\rho(A)}=\rho(m_{A})\rho^{2}_{A,A}\colon\
\rho(A)\otimes\rho(A)\to\rho(A).$

The functor $\rho$ induces an equivalence of ${\mathcal{C}}$-module categories
${}_{\rho(A)}{\mathcal{C}}\to(_{A}{\mathcal{C}})^{\rho}$ with intertwining
isomorphisms

$\displaystyle{\rho^{2}_{M,X}}^{-1}\colon\
\rho(M\bar{\otimes}X)\to\rho(M)\bar{\otimes}^{\rho}X.$

### 2.3

Let $G$ be a finite group. Let $X$ be a $G$-module. Given an $n$-cochain $f\in
C^{n}(G,X)$ (where $C^{0}(G,M)=M$), the coboundary of $f$ is the
$(n+1)$-cochain $df=d^{n}f\in C^{n+1}(G,X)$ defined by

$\displaystyle
d^{n}f(g_{1},\dots,g_{n+1})=g_{1}.f(g_{2},\dots,g_{n+1})+\sum_{i=1}^{n}f(g_{1},\dots,g_{i}g_{i+1},\dots,g_{n})$
$\displaystyle\hphantom{d^{n}f(g_{1},\dots,g_{n+1})=}{}+(-1)^{n+1}f(g_{1},\dots,g_{n}),$

for all $g_{1},\dots,g_{n+1}\in G$. The kernel of $d^{n}$ is denoted
$Z^{n}(G,M)$; an element of $Z^{n}(G,M)$ is an $n$-cocycle. We have
$d^{n}d^{n-1}=0$, for all $n\geq 1$. The $n$th cohomology group of $G$ with
coefficients in $M$ is $H^{n}(G,M)=Z^{n}(G,M)/d^{n-1}(C^{n-1}(G,M))$. We shall
write $f\equiv f^{\prime}$ when the cochains $f,f^{\prime}\in
C^{n}(G,k^{\times})$ differ by a coboundary.

We shall assume that every cochain $f$ is _normalized_ , that is,
$f(g_{1},\dots,g_{n})=1$, whenever one of the arguments $g_{1},\dots,g_{n}$ is
the identity. If $H$ is a subgroup of $G$ and $f\in C^{n}(H,k^{\times})$, we
shall indicate by $f^{g}$ the $n$-cochain in ${}^{g^{-1}\\!}H$ given by
$f^{g}(h_{1},\dots,h_{n})=f({}^{g}h_{1},\dots,{}^{g}h_{n})$,
$h_{1},\dots,h_{n}\in H$.

Let $\omega\colon G\times G\times G\to k^{\times}$ be a 3-cocycle on $G$. Let
${\mathcal{C}}(G,\omega)$ denote the fusion category of finite-dimensional
$G$-graded vector spaces with associativity constraint defined, for all
$U,V,W\in{\mathcal{C}}(G,\omega)$, as

$\displaystyle a_{X,Y,Z}((u\otimes v)\otimes
w)=\omega^{-1}(g_{1},g_{2},g_{3})u\otimes(v\otimes w),$

for all homogeneous vectors $u\in U_{g_{1}}$, $v\in V_{g_{2}}$, $w\in
W_{g_{3}}$, $g_{1},g_{2},g_{3}\in G$. Any pointed fusion category is
equivalent to a category of the form ${\mathcal{C}}(G,\omega)$.

A fusion category ${\mathcal{C}}$ is called _group-theoretical_ if it is
categorically Morita equivalent to a pointed fusion category. Equivalently,
${\mathcal{C}}$ is group-theoretical if and only if there exist a finite group
$G$ and a $3$-cocycle $\omega\colon G\times G\times G\to k^{\times}$ such that
${\mathcal{C}}$ is equivalent to the fusion category
${\mathcal{C}}(G,\omega,H,\psi)=_{A(H,\psi)\\!}{\mathcal{C}}(G,\omega)_{A(H,\psi)}$,
where $H$ is a subgroup of $G$ such that the class of $\omega|_{H\times
H\times H}$ is trivial and $\psi\colon H\times H\to k^{\times}$ is a 2-cochain
on $H$ satisfying condition (1.1).

Let
${\mathcal{C}}(G,\omega,H,\psi)\cong{\mathcal{C}}(G,\omega)^{*}_{{\mathcal{M}}_{0}(H,\psi)}$
be a group-theoretical fusion category. Then there is a bijective
correspondence between equivalence classes of indecomposable
${\mathcal{C}}(G,\omega,H,\psi)$-module categories and equivalence classes of
indecomposable ${\mathcal{C}}(G,\omega)$-module categories. This
correspondence attaches to every indecomposable
${\mathcal{C}}(G,\omega)$-module category ${\mathcal{M}}$ the
${\mathcal{C}}(G,\omega,H,\psi)$-module category

$\displaystyle{\mathcal{M}}(H,\psi)=\operatorname{Fun}_{{\mathcal{C}}(G,\omega)}({\mathcal{M}}_{0}(H,\psi),{\mathcal{M}}).$

##  3 Indecomposable module categories over
$\boldsymbol{{\mathcal{C}}(G,\omega)}$

Throughout this section $G$ is a finite group and $\omega\colon G\times
G\times G\to k^{\times}$ is a 3-cocycle on $G$.

### 3.1

Let $g\in G$. Consider the 2-cochain $\Omega_{g}\colon G\times G\to
k^{\times}$ given by

$\displaystyle\Omega_{g}(g_{1},g_{2})=\frac{\omega({}^{g}g_{1},{}^{g}g_{2},g)\omega(g,g_{1},g_{2})}{\omega({}^{g}g_{1},g,g_{2})}.$

For all $g\in G$ we have the relation

$\displaystyle d\Omega_{g}=\frac{\omega}{\omega^{g}}.$ (3.1)

Let ${\mathcal{C}}={\mathcal{C}}(G,\omega)$ and let $g\in G$. For every object
$V$ of ${\mathcal{C}}$ let ${}^{g}V$ be the object of ${\mathcal{C}}$ such
that ${}^{g}V=V$ as a vector space with $G$-grading defined as
$({}^{g}V)_{x}=V_{{}^{g}x}$, $x\in G$. For every $g\in G$, we have a functor
$\operatorname{ad}_{g}\colon{\mathcal{C}}\to{\mathcal{C}}$, given by
$\operatorname{ad}_{g}(V)={}^{g}V$ and $\operatorname{ad}_{g}(f)=f$, for every
object $V$ and morphism $f$ of ${\mathcal{C}}$. Relation (3.1) implies that
$\operatorname{ad}_{g}$ is a tensor functor with monoidal structure defined by

$\displaystyle\big{(}\operatorname{ad}_{g}^{2}\big{)}_{U,V}\colon\
{}^{g}U\otimes{}^{g}V\to{}^{g}(U\otimes
V),\qquad\big{(}\operatorname{ad}_{g}^{2}\big{)}_{U,V}(u\otimes
v)=\Omega_{g}(h,h^{\prime})^{-1}u\otimes v,$

for all $h,h^{\prime}\in G$, and for all homogeneous vectors $u\in U_{h}$,
$v\in V_{h^{\prime}}$.

For every $g,g_{1},g_{2}\in G$, let $\gamma(g_{1},g_{2})\colon G\to
k^{\times}$ be the map defined in the form

$\displaystyle\gamma(g_{1},g_{2})(g)=\frac{\omega(g_{1},g_{2},g)\omega({}^{g_{1}g_{2}}g,g_{1},g_{2})}{\omega(g_{1},{}^{g_{2}}g,g_{2})}.$

The following relation holds, for all $g_{1},g_{2}\in G$:

$\displaystyle\Omega_{g_{1}g_{2}}=\Omega_{g_{1}}^{g_{2}}\Omega_{g_{2}}d\gamma(g_{1},g_{2}).$
(3.2)

In this way,
$\operatorname{ad}\colon\underline{G}\to\underline{\text{Aut}}_{\otimes}{\mathcal{C}}$,
$\operatorname{ad}(g)=\big{(}\operatorname{ad}_{g},\operatorname{ad}_{g}^{2}\big{)}$,
gives rise to an action by tensor autoequivalences of $G$ on ${\mathcal{C}}$
where, for every $g,x\in G$, $V\in{\mathcal{C}}(G,\omega)$, the monoidal
isomorphisms
${\operatorname{ad}^{2}}_{V}\colon{}^{g}({}^{g^{\prime}}V)\to{}^{gg^{\prime}}V$
are given by

$\displaystyle\operatorname{ad}^{2}_{V}(v)=\gamma(g,g^{\prime})(x)v,$

for all homogeneous vectors $v\in V_{x}$, $h\in G$. The equivariantization
${\mathcal{C}}^{G}$ with respect to this action is equivalent to the category
of finite-dimensional representations of the twisted quantum double
$D^{\omega}G$ (see [5, Lemma 6.3]).

For each $g\in G$, and for each ${\mathcal{C}}$-module category
${\mathcal{M}}$, let ${\mathcal{M}}^{g}$ denote the module category induced by
the functor $\operatorname{ad}_{g}$ as in Section 2.2. Recall that the action
of ${\mathcal{C}}$ on ${\mathcal{M}}^{g}$ is defined by
$M\bar{\otimes}^{g}V=M\bar{\otimes}({}^{g}V)$, for all objects $V$ of
${\mathcal{C}}$.

######  Lemma 3.1.

Let $g\in G$ and let ${\mathcal{M}}$ be a ${\mathcal{C}}$-module category.
Then ${\mathcal{M}}^{g}\cong{\mathcal{M}}$ as ${\mathcal{C}}$-module
categories.

###### Proof.

For each $g\in G$, let $\\{g\\}$ denote the object of ${\mathcal{C}}$ such
that $\\{g\\}=k$ with degree $g$. In what follows, by abuse of notation, we
identify $\\{g\\}\otimes\\{h\\}$ and $\\{gh\\}$, $g,h\in G$, by means of the
canonical isomorphisms of vector spaces.

Let $R_{g}\colon{\mathcal{M}}^{g}\to{\mathcal{M}}$ be the functor defined by
the right action of $\\{g\\}$: $R_{g}(M)=M\bar{\otimes}\\{g\\}$. Consider the
natural isomorphism $\zeta\colon
R_{g}\circ\bar{\otimes}^{g}\to\bar{\otimes}\circ(R_{g}\times\operatorname{id}_{\mathcal{C}})$,
defined as

$\displaystyle\zeta_{M,V}=\mu_{M,\\{g\\},V}\mu^{-1}_{M,{}^{g}V,\\{g\\}}\colon\
R_{g}(M\bar{\otimes}^{g}V)\to R_{g}(M)\bar{\otimes}V,$

for all objects $M$ of ${\mathcal{M}}$ and $V$ of ${\mathcal{C}}$, where $\mu$
is the associativity constraint of ${\mathcal{M}}$.

The functor $R_{g}$ is an equivalence of categories with quasi-inverse given
by the functor $R_{g^{-1}}\colon$ ${\mathcal{M}}\to{\mathcal{M}}^{g}$.

A direct calculation, using the coherence conditions (2.1) and (2.2) for the
module category ${\mathcal{M}}$, shows that $\zeta$ satisfies conditions (2.3)
and (2.4). Hence $(R_{g},\zeta)$ is a ${\mathcal{C}}$-module functor.
Therefore ${\mathcal{M}}^{g}\cong{\mathcal{M}}$ as ${\mathcal{C}}$-module
categories, as claimed. ∎

######  Lemma 3.2.

Let $H$ be a subgroup of $G$ and let $\psi$ be a $2$-cochain on $H$ satisfying
(1.1). Let $A(H,\psi)$ denote the corresponding indecomposable algebra in
${\mathcal{C}}$. Then, for all $g\in G$, ${}^{g}A(H,\psi)\cong
A({}^{g}H,\psi^{g^{-1}}\Omega_{g^{-1}})$ as algebras in ${\mathcal{C}}$.

###### Proof.

By definition,
${}^{g}A(H,\psi)=A\big{(}{}^{g}H,\psi^{g^{-1}}\big{(}\Omega_{g}^{g^{-1}}\big{)}^{-1}\big{)}$.
It follows from formula (3.2) that $\big{(}\Omega_{g}^{g^{-1}}\big{)}^{-1}$
and $\Omega_{g^{-1}}$ differ by a coboundary. This implies the lemma. ∎

### 3.2

Let $H$, $L$ be subgroups of $G$ and let $\psi\in C^{2}(H,k^{\times})$,
$\xi\in C^{2}(L,k^{\times})$, be 2-cochains such that $\omega|_{H\times
H\times H}=d\psi$ and $\omega|_{L\times L\times L}=d\xi$.

Let $B$ be an object of the category ${}_{A(H,\psi)}{\mathcal{C}}_{A(L,\xi)}$
of $(A(H,\psi),A(L,\xi))$-bimodules in ${\mathcal{C}}$. For each $z\in G$, let
$\pi_{l}(h)\colon B_{z}\to B_{hz}$ and $\pi_{r}(s)\colon B_{z}\to B_{zs}$,
denote the linear maps induced by the actions of $h\in H$ and $s\in L$,
respectively. Then the following relations hold, for all $h,h^{\prime}\in H$,
$s,s^{\prime}\in L$:

$\displaystyle\pi_{l}(h)\pi_{l}(h^{\prime})=\omega(h,h^{\prime},z)\psi(h,h^{\prime})\pi_{l}(hh^{\prime}),$
(3.3)
$\displaystyle\pi_{r}(s^{\prime})\pi_{r}(s)=\omega(z,s,s^{\prime})^{-1}\xi(s,s^{\prime})\pi_{r}(ss^{\prime}),$
(3.4) $\displaystyle\pi_{l}(h)\pi_{r}(s)=\omega(h,z,s)\pi_{r}(s)\pi_{l}(h).$
(3.5)

######  Lemma 3.3.

Let $g\in G$ and let $B_{g}$ denote the homogeneous component of degree $g$ of
$B$. Then the map $\pi\colon H\cap{}^{g}L\to\text{\rm GL}(B_{g})$, defined as
$\pi(x)=\pi_{r}\big{(}{}^{g^{-1}}x\big{)}^{-1}\pi_{l}(x)$ is a projective
representation of $H\cap{}^{g}L$ with cocycle $\alpha_{g}$ given, for all
$x,y\in H\cap{}^{g}L$, as follows:

$\displaystyle\alpha_{g}(x,y)=\psi(x,y)\xi^{-1}\big{(}{}^{g^{-1}}x,{}^{g^{-1}}y\big{)}\frac{\omega(x,y,g)\omega\big{(}x,yg,{}^{g^{-1}}\big{(}y^{-1}\big{)}\big{)}}{\omega\big{(}xyg,{}^{g^{-1}}\big{(}y^{-1}\big{)},{}^{g^{-1}}\big{(}x^{-1}\big{)}\big{)}}du_{g}(x,y)$
$\displaystyle\hphantom{\alpha_{g}(x,y)=}{}\times\frac{\omega\big{(}{}^{g^{-1}}y,{}^{g^{-1}}\big{(}y^{-1}\big{)},{}^{g^{-1}}\big{(}x^{-1}\big{)}\big{)}}{\omega\big{(}{}^{g^{-1}}x,{}^{g^{-1}}y,{}^{g^{-1}}\big{(}y^{-1}x^{-1}\big{)}\big{)}},$

where the $1$-cochain $u_{g}$ is defined as
$u_{g}(x)=\omega\big{(}xg,{}^{g^{-1}}x,{}^{g^{-1}}\big{(}x^{-1}\big{)}\big{)}$.

###### Proof.

It follows from (3.4) that
$\pi_{r}(s)^{-1}=\omega\big{(}z,s,s^{-1}\big{)}\xi\big{(}s,s^{-1}\big{)}^{-1}\pi_{r}\big{(}s^{-1}\big{)}$,
for all $z\in G$, $s\in L$. In addition, for all $h,h^{\prime}\in L$, we have
the following relation:

$\displaystyle\xi\big{(}{h^{\prime}}^{-1},h^{-1}\big{)}\xi(h,h^{\prime})=df(h,h^{\prime})\frac{\omega\big{(}h^{\prime},{h^{\prime}}^{-1},h^{-1}\big{)}}{\omega\big{(}h,h^{\prime},{h^{\prime}}^{-1}h^{-1}\big{)}},$

where $f$ is the 1-cochain given by $f(h)=\xi\big{(}h,h^{-1}\big{)}$. A
straightforward computation, using this relation and conditions (3.3), (3.4)
and (3.5), shows that $\pi(x)\pi(y)=\alpha_{g}(x,y)\pi(xy)$, for all $x,y\in
H\cap{}^{g}L$. This proves the lemma. ∎

######  Remark 3.4.

Lemma 3.3 is a version of [8, Proposition 3.2], where it is shown that $B$ is
a simple object of ${}_{A(H,\psi)}{\mathcal{C}}_{A(L,\xi)}$ if and only if $B$
is supported on a single double coset $HgL$ and the projective representation
$\pi$ in the component $B_{g}$ is irreducible.

For all $g\in G$, $\psi^{g}\Omega_{g}$ is a 2-cochain in ${}^{g^{-1}\\!}\\!H$
such that
$\omega|_{{}^{g^{-1}}H\times{}^{g^{-1}}H\times{}^{g^{-1}}H}=d(\psi^{g}\Omega_{g})$.
Then the product $\xi^{-1}\psi^{g}\Omega_{g}$ defines a 2-cocycle of
${}^{g^{-1}}H\cap L$.

######  Lemma 3.5.

The class of the $2$-cocycle
$\big{(}\xi^{-1}\psi^{g}\Omega_{g}\big{)}^{g^{-1}}$ in
$H^{2}(H\cap{}^{g}L,k^{\times})$ coincides with the class of the $2$-cocycle
$\alpha_{g}$ in Lemma 3.3.

###### Proof.

A direct calculation shows that for all $x,y\in G$,

$\displaystyle\frac{\omega\big{(}y,y^{-1},x^{-1}\big{)}}{\omega\big{(}x,y,y^{-1}x^{-1}\big{)}}\frac{\omega\big{(}{}^{g}x,{}^{g}y,g\big{)}\omega\big{(}{}^{g}x,{}^{g}yg,{}y^{-1}\big{)}}{\omega\big{(}{}^{g}x^{g}yg,{}y^{-1},x^{-1}\big{)}}=\Omega_{g}(x,y)d\theta_{g}(x,y),$

where the 1-cochain $\theta_{g}$ is defined as
$\theta_{g}(x)=\omega\big{(}g,x,x^{-1}\big{)}^{-1}$. This implies that
$\alpha_{g}^{g}\equiv\xi^{-1}\psi^{g}\Omega_{g}$, as was to be proved. ∎

### 3.3

In this subsection we give a proof of the main result of this paper.

###### Proof of Theorem 1.1.

Let $H,L$ be subgroups of $G$ and let $\psi\in C^{2}(H,k^{\times})$ and
$\xi\in C^{2}(L,k^{\times})$ be 2-cochains satisfying condition (1.1). Let
$A(H,\psi)$, $A(L,\xi)$ be the associated algebras in ${\mathcal{C}}$ and let
${\mathcal{M}}_{0}(H,\psi)$, ${\mathcal{M}}_{0}(L,\xi)$ be the corresponding
${\mathcal{C}}$-module categories.

Let ${\mathcal{M}}={\mathcal{M}}_{0}(L,\xi)$. For every $g\in G$, let
${\mathcal{M}}^{g}$ denote the module category induced by the autoequivalence
$\operatorname{ad}_{g}\colon{\mathcal{C}}\to{\mathcal{C}}$. The
${\mathcal{C}}$-module category ${\mathcal{M}}^{g}$ is equivalent to
${}_{{}^{g}A(L,\xi)}{\mathcal{C}}$. Hence, by Lemma 3.2,
${\mathcal{M}}^{g}\cong{\mathcal{M}}_{0}\big{(}{}^{g}L,\xi^{g^{-1}}\Omega_{g^{-1}}\big{)}$.

Suppose that there exists an element $g\in G$ such that $H={}^{g}L$ and the
class of the cocycle $\xi^{-1}\psi^{g}\Omega_{g}$ is trivial on $L$. Relation
(3.2) implies that $\Omega_{g}^{g^{-1}}=\Omega_{g^{-1}}^{-1}$, and thus the
class of $\psi^{-1}\xi^{g^{-1}}\Omega_{g^{-1}}$ is trivial on $H$. Then
$\psi=\xi^{g^{-1}}\Omega_{g^{-1}}df$, for some 1-cochain $f\in
C^{1}(H,k^{\times})$. Therefore
${}^{g}A(L,\xi)=A\big{(}H,\xi^{g^{-1}}\Omega_{g^{-1}}\big{)}\cong A(H,\psi)$
as algebras in ${\mathcal{C}}$. Thus we obtain equivalences of
${\mathcal{C}}$-module categories

$\displaystyle{\mathcal{M}}_{0}(L,\xi)\cong{\mathcal{M}}_{0}(L,\xi)^{g}\cong{}_{{}^{g}A(L,\xi)}{\mathcal{C}}\cong{\mathcal{M}}_{0}(H,\psi),$

where the first equivalence is deduced from Lemma 3.1.

Conversely, suppose that
$F\colon{\mathcal{M}}_{0}(L,\xi)\to{\mathcal{M}}_{0}(H,\psi)$ is an
equivalence of ${\mathcal{C}}$-module categories. Recall that there is an
equivalence

$\displaystyle\operatorname{Fun}_{\mathcal{C}}\left({\mathcal{M}}_{0}(L,\xi),{\mathcal{M}}_{0}(H,\psi)\right)\cong{}_{A(H,\psi)}{\mathcal{C}}_{A(L,\xi)}.$

Under this equivalence, the functor $F$ corresponds to an object $B$ of
${}_{A(H,\psi)}{\mathcal{C}}_{A(L,\xi)}$ such that there exists an object
$B^{\prime}$ of ${}_{A(L,\xi)}{\mathcal{C}}_{A(H,\psi)}$ satisfying

$\displaystyle B\otimes_{A(L,\xi)}B^{\prime}\cong A(H,\psi),$ (3.6)

as $A(H,\psi)$-bimodules in ${\mathcal{C}}$, and

$\displaystyle B^{\prime}\otimes_{A(H,\psi)}B\cong A(L,\xi),$ (3.7)

as $A(L,\xi)$-bimodules in ${\mathcal{C}}$.

Let $\operatorname{FPdim}_{A(H,\psi)}M$ denote the Frobenius–Perron dimension
of an object $M$ of ${}_{A(H,\psi)}{\mathcal{C}}_{A(H,\psi)}$. Then we have

$\displaystyle\dim M=\dim
A(H,\psi)\operatorname{FPdim}_{A(H,\psi)}M=|H|\operatorname{FPdim}_{A(H,\psi)}M.$

Taking Frobenius–Perron dimensions in both sides of (3.6) and using this
relation we obtain that $\dim\left(B\otimes_{A(L,\xi)}B^{\prime}\right)=|H|$.

On the other hand, $\dim(B\otimes_{A(H,\psi)}B^{\prime})=\frac{\dim B\dim
B^{\prime}}{\dim{A(L,\xi)}}=\frac{\dim B\dim B^{\prime}}{|L|}$. Thus

$\displaystyle\dim B\dim B^{\prime}=|H||L|.$ (3.8)

Since $A(H,\psi)$ is an indecomposable algebra in ${\mathcal{C}}$, then it is
a simple object of ${}_{A(H,\psi)}{\mathcal{C}}_{A(H,\psi)}$. Then (3.7)
implies that $B$ is a simple object of
${}_{A(H,\psi)}{\mathcal{C}}_{A(L,\xi)}$ and $B^{\prime}$ is a simple object
of ${}_{A(L,\xi)}{\mathcal{C}}_{A(H,\psi)}$.

In view of [8, Proposition 3.2], the support of $B$ is a two sided
$(H,L)$-double coset, that is, $B=\bigoplus_{(h,h^{\prime})\in H\times
L}B_{hgh^{\prime}}$, where $g\in G$ is a representative of the double coset
that supports $B$. Moreover, the homogeneous component $B_{g}$ is an
irreducible $\alpha_{g}$-projective representation of the group ${}^{g}L\cap
H$, where the 2-cocycle $\alpha_{g}$ satisfies
$\alpha_{g}\equiv\big{(}\xi^{-1}\psi^{g}\Omega_{g}\big{)}^{g^{-1}}$; see
Remark 3.4 and Lemmas 3.3 and 3.5.

Notice that the actions of $h\in H$ and $h^{\prime}\in L$ induce isomorphisms
of vector spaces $B_{g}\cong B_{hg}$ and $B_{g}\cong B_{gh^{\prime}}$. Hence

$\displaystyle\dim B=|HgL|\dim B_{g}=\frac{|H||L|}{|H\cap{}^{g}L|}\dim
B_{g}=[H:H\cap{}^{g}L]|L|\dim B_{g}.$ (3.9)

In particular, $\dim B\geq|L|$. Reversing the roles of $H$ and $L$, the same
argument implies that $\dim B^{\prime}\geq|H|$. Combined with relations (3.8)
and (3.9) this implies

$\displaystyle|H||L|=\dim B\dim B^{\prime}\geq|H|[H:H\cap{}^{g}L]|L|\dim
B_{g}.$

Hence $[H\colon H\cap{}^{g}L]\dim B_{g}=1$, and therefore $[H\colon
H\cap{}^{g}L]=1$ and $\dim B_{g}=1$. The first condition means that
$H\subseteq{}^{g}L$, while the second condition implies that the class of
$\alpha_{g}$ is trivial in $H^{2}(H\cap{}^{g}L,k^{\times})$. Since the rank of
${\mathcal{M}}_{0}(H,\psi)$ equals the index $[G:H]$ and the rank of
${\mathcal{M}}_{0}(H,\xi)$ equals the index $[G:L]$, then $|H|=|L|$. Thus we
get that $H={}^{g}L$ and that the class of the 2-cocycle (1.2) is trivial in
$H^{2}(L,k^{\times})$. This finishes the proof of the theorem. ∎

######  Example 3.6.

Let $B_{8}$ be the 8-dimensional Kac Paljutkin Hopf algebra. The Hopf algebra
$B_{8}$ fits into an exact sequence

$\displaystyle k\longrightarrow k^{C}\longrightarrow B_{8}\longrightarrow
kL\longrightarrow k,$

where $C=\mathbb{Z}_{2}$ and $L=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. See [3].
This exact sequence gives rise to mutual actions by permutations

$\displaystyle C\overset{\vartriangleleft}{\longleftarrow}C\times
L\overset{\vartriangleright}{\longrightarrow}L,$

and compatible cocycles $\tau\colon L\times L\to\big{(}k^{C}\big{)}^{\times}$,
$\sigma\colon C\times C\to(k^{L})^{\times}$, such that $B_{8}$ is isomorphic
to the bicrossed product $kC{}^{\tau}\\#_{\sigma}kL$. The data $\lhd$, $\rhd$,
$\sigma$ and $\tau$ are explicitly determined in [4, Proposition 3.11] as
follows. Let $C=\langle x\colon x^{2}=1\rangle$, $L=\langle z,t\colon
z^{2}=t^{2}=ztz^{-1}t^{-1}=1\rangle$. Then $\lhd\colon C\times L\to C$ is the
trivial action of $L$ on $C$, $\rhd\colon C\times L\to L$ is the action
defined by $x\rhd z=z$ and $x\rhd t=zt$,

$\displaystyle\tau_{x^{n}}\big{(}z^{i}t^{j},z^{i^{\prime}}t^{j^{\prime}}\big{)}=(-1)^{nji^{\prime}},$

for all $0\leq n,i,i^{\prime},j,j^{\prime}\leq 1$, and

$\displaystyle\sigma_{z^{i}t^{j}}\big{(}x^{n},x^{n^{\prime}}\big{)}=(\sqrt{-1})^{j\big{(}\frac{n+n^{\prime}-\langle
n+n^{\prime}\rangle}{2}\big{)}},$

for all $0\leq i,j,n,n^{\prime}\leq 1$, where $\langle n+n^{\prime}\rangle$
denotes the remainder of $n+n^{\prime}$ in the division by $2$. Here we use
the notation $\tau(a,a^{\prime})(y)=:\tau_{y}(a,a^{\prime})$ and, similarly,
$\sigma(y,y^{\prime})(a)=:\sigma_{a}(y,y^{\prime})$, $a,a^{\prime}\in L$,
$y,y^{\prime}\in C$.

In view of [9, Theorem 3.3.5] (see [6, Proposition 4.3]), the fusion category
of finite-dimensional representations of $B_{8}^{\rm op}\cong B_{8}$ is
equivalent to the category ${\mathcal{C}}(G,\omega,L,1)$, where $G=L\rtimes C$
is the semidirect product with respect to the action $\rhd$, and $\omega$ is
the 3-cocycle arising from the pair $(\tau,\sigma)$ under one of the maps of
the so-called _Kac exact sequence_ associated to the matched pair.

In this example $G$ is isomorphic to the dihedral group $D_{8}$ of order 8.
The 3-cocycle $\omega$ is determined by the formula

$\displaystyle\omega\big{(}x^{n}z^{i}t^{j},x^{n^{\prime}}z^{i^{\prime}}t^{j^{\prime}},x^{n^{\prime\prime}}z^{i^{\prime\prime}}t^{j^{\prime\prime}}\big{)}=\tau_{x^{n}}\big{(}z^{i^{\prime}}t^{j^{\prime}},x^{n^{\prime}}\rhd
z^{i^{\prime\prime}}t^{j^{\prime\prime}}\big{)}\sigma_{z^{i^{\prime\prime}}t^{j^{\prime\prime}}}\big{(}x^{n},x^{n^{\prime}}\big{)},$
(3.10)

for all $0\leq
i,j,i^{\prime},j^{\prime},i^{\prime\prime},j^{\prime\prime},n,n^{\prime},n^{\prime\prime}\leq
1$.

Notice that $\omega|_{L\times L\times L}=1$. Hence, for every 2-cocycle $\xi$
on $L$, the pair $(L,\xi)$ gives rise to an indecomposable
${\mathcal{C}}$-module category ${\mathcal{M}}(L,\xi)$. Formula (3.10) implies
that $\Omega_{x}|_{L\times L}$ is given by

$\displaystyle\Omega_{x}\big{(}z^{i}t^{j},z^{i^{\prime}}t^{j^{\prime}}\big{)}=(-1)^{ji^{\prime}},\qquad
0\leq i,i^{\prime},j,j^{\prime}\leq 1.$

Then $\Omega_{x}$ is a 2-cocycle representing the unique nontrivial cohomology
class in $H^{2}(L,k^{\times})$. By Theorem 1.1, for any 2-cocycle $\xi$ on
$L$, ${\mathcal{M}}_{0}(L,1)$ and ${\mathcal{M}}_{0}(L,\xi)$ are equivalent as
${\mathcal{C}}(G,\omega)$-module categories, and therefore so are the
corresponding ${\mathcal{C}}$-module categories ${\mathcal{M}}(L,1)$ and
${\mathcal{M}}(L,\xi)$. This implies that indecomposable
${\mathcal{C}}$-module categories are in this example parameterized by
conjugacy classes of subgroups of $D_{8}$ on which $\omega$ has trivial
restriction, as claimed in [2, Section 6.4].

### Acknowledgements

This research was partially supported by CONICET and SeCyT - Universidad
Nacional de Córdoba, Argentina.

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