\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}
