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a856717 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 | \section{Table of Affine Reflection Supergroups and Secondary Invariant}
\label{sec:sec_inv_table}
The symmetric continuous representation in \cref{sec:formulation} relies on the Hironaka-type decomposition: a $G$-invariant field can be written as a linear combination of a finite set of fixed $G$-invariant basis functions (secondary invariants) with coefficient fields that enjoy a higher affine reflection symmetry $W_a$ (cf. \cref{eq:hironaka_param,thm:hironaka_decomposition_func}). For practical use, the only group-dependent ingredient is the explicit choice of these basis functions $\{\eta_i\}_{i=1}^r$, where $r=[W_a\!:\!G]$. This section tabulates non-trivial $\eta_i$ for all planar groups (we omit trivial $\eta_1 = 1$y), together with a compatible embedding $G\subset W_a$ and the associated lattice generators $(\mathbf a,\mathbf b)$ of $W_a$. The table serves as a plug-in recipe: once the target symmetry group $G$ and lattice are fixed, we directly obtain $(W_a,\mathbf a,\mathbf b)$ and the corresponding $\eta_i$, and then parameterize $G$-symmetric continuous fields via \cref{eq:hironaka_param}.
All notations in \cref{sec:sec_inv_table} follow the same conventions as in
\cref{sec:formulation}. In particular, $c_i$ and $s_i$ denote the cosine and sine
generators associated with the fundamental lattice directions. For the hexagonal lattice, the secondary invariants appearing in the table are defined as
\begin{align*}
\phi_1^{-+} &= s_1 + s_2 - (c_1 s_2 + c_2 s_1), \\
\phi_2^{--} &= s_1 - s_2 + c_1 s_2 - c_2 s_1 + 2(c_1 - c_2)(c_1 s_2 + c_2 s_1).
\end{align*}
\begin{table}[h]
\centering
\caption{Affine Reflection Supergroups and Secondary Invariants}
\renewcommand{\arraystretch}{1.2}
\resizebox{0.9\linewidth}{!}{
\begin{tabular}{llllllllc}
\toprule
Lattice & $G$ & $W_a$ & $\mathbf{a}$ & $\mathbf{b}$ & $\eta_2$ & $\eta_3$ & $\eta_4$ & $r$ \\
\midrule
Oblique & $p1$ & \multirow{9}{*}{$p2mm$} & $\mathbf{a}$ & $\mathbf{b}$ & $s_1$ & $s_2$ & $s_1s_2$ & $4$ \\
Oblique & $p2$ & & $\mathbf{a}$ & $\mathbf{b}$ & $s_1s_2$ & $-$ & $-$ & $2$ \\
Rectangular & $pm$ & & $\mathbf{a}$ & $\mathbf{b}$ & $s_{1}$ & $-$ & $-$ & $2$ \\
Rectangular & $pg$ & & $\mathbf{a}$ & $\mathbf{b}/2$ & $s_{1}c_{2}$ & $s_{1}s_{2}$ & $c_{2}s_{2}$ & $4$ \\
Rectangular & $cm$ & & $\mathbf{a}/2$ & $\mathbf{b}/2$ & $c_{1}c_{2}$ & $c_{1}s_{2}$ & $c_{2}s_{2}$ & $4$ \\
Rectangular & $p2mm$ & & $\mathbf{a}$ & $\mathbf{b}$ & $-$ & $-$ & $-$ & $1$ \\
Rectangular & $p2mg$ & & $\mathbf{a}/2$ & $\mathbf{b}$ & $s_{1}s_{2}$ & $-$ & $-$ & $2$ \\
Rectangular & $p2gg$ & & $\mathbf{a}/2$ & $\mathbf{b}/2$ & $c_{1}c_{2}$ & $c_{1}s_{1}s_{2}$ & $s_{1}c_{2}s_{2}$ & $4$ \\
Rectangular & $c2mm$ & & $\mathbf{a}/2$ & $\mathbf{b}/2$ & $c_{1}c_{2}$ & $-$ & $-$ & $2$ \\
\midrule
Square & $p4$ & \multirow{3}{*}{$p4mm$} & $\mathbf{a}$ & $\mathbf{b}$ & $(c_1-c_2)s_1s_2$ & $-$ & $-$ & $2$ \\
Square & $p4gm$ & & $\mathbf{a}/2$ & $\mathbf{b}/2$ & $c_{1}c_{2}$ & $(c_{1}-c_{2})s_{1}s_{2}$ & $(c_{1}-c_{2})c_{1}c_{2}s_{1}s_{2}$ & $4$ \\
Square & $p4mm$ & & $\mathbf{a}$ & $\mathbf{b}$ & $-$ & $-$ & $-$ & $1$ \\
\midrule
Hexagonal & $p3$ & \multirow{5}{*}{$p6mm$} & $\mathbf{a}$ & $\mathbf{b}$ & $\phi_1^{-+}$ & $\phi_2^{--}$ & $\phi_3^{+-}=\phi_1^{-+}\phi_2^{--}$ & $4$ \\
Hexagonal & $p3m1$ & & $\mathbf{a}$ & $\mathbf{b}$ & $\phi_1^{-+}$ & $-$ & $-$ & $2$ \\
Hexagonal & $p31m$ & & $\mathbf{a}$ & $\mathbf{b}$ & $\phi_2^{--}$ & $-$ & $-$ & $2$ \\
Hexagonal & $p6$ & & $\mathbf{a}$ & $\mathbf{b}$ & $\phi_3^{+-}$ & $-$ & $-$ & $2$ \\
Hexagonal & $p6mm$ & & $\mathbf{a}$ & $\mathbf{b}$ & $-$ & $-$ & $-$ & $1$ \\
\bottomrule
\end{tabular}
}
\end{table}
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