| \section{Preliminaries} |
| \label{sec:preliminaries} |
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| \textbf{Planar groups.} |
| In $n$-dimensional space, a Euclidean transformation is defined as the composition of an orthogonal transformation and a translation. The set of all Euclidean transformations constitutes the \textbf{Euclidean group}, denoted as $E(n)$. A \textbf{crystallographic group} $G$ is a discrete subgroup of $E(n)$ that contains $n$ linearly independent translations. Specifically, crystallographic groups are referred to as \textbf{planar groups} when $n=2$, and \textbf{space groups} when $n=3$. We primarily focus on the case where $n=2$. |
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| A classical result states that planar groups are classified into 17 types up to affine coordinate transformations. The simplest planar group is $p1$, generated by translations along two linearly independent directions. For any other planar group $G$, its elements contain nontrivial orthogonal components in addition to translations, and the quotient $G/T(G)$ is finite, where $T(G)$ denotes the translation subgroup of $G$. An important subclass is formed by \textbf{affine reflection groups}, which are generated by affine reflections. There are four such groups: $p2mm$, $p4mm$, $p3m1$, and $p6mm$. |
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| \textbf{Symmetric functions.} |
| For a planar group $G$, a function $f$ on $\mathbb{R}^2$ is called $G$-invariant if $f(g(\mathbf{x}))=f(\mathbf{x})$ for all $g\in G$. We denote by $C_G(\mathbb{R}^2)$ the space of continuous $G$-invariant functions. Consider a $p1$-invariant function $f$ periodic with respect to two fundamental translations $\mathbf{a}_1$ and $\mathbf{a}_2$. Let $S = \{m\mathbf{b}_1+n\mathbf{b}_2\}$ be the reciprocal lattice, where $\langle\mathbf{a}_i,\mathbf{b}_j\rangle=2\pi\delta_{ij}$. We choose one representative from each non-zero pair $(\mathbf{k},-\mathbf{k})$. Denoting such a half reciprocal lattice by $S^+$, then $f$ can be written as a Fourier series |
| \begin{equation} |
| f(\mathbf{x}) = \frac{a_{0}}{2}+\sum_{\mathbf{k} \in S^+} [a_{\mathbf{k}}\cos(\langle \mathbf{k}, \mathbf{x}\rangle) + b_{\mathbf{k}}\sin(\langle \mathbf{k}, \mathbf{x}\rangle)]. |
| \end{equation} |
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| \textbf{Asymmetric unit.} |
| A symmetric function is determined by its values on the asymmetric unit. The asymmetric unit represents the minimal, non-redundant portion of space from which the full periodic structure is obtained through the symmetry operations of the group. For $p1$, any fundamental period constitutes an asymmetric unit. However, for other planar groups, the existence of additional transformations implies that the asymmetric unit is only a portion of the period. Specifically for affine reflection groups, any point in space can be mapped into the asymmetric unit via a finite sequence of reflection transformations. |
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