Definition. Finite symmetries of a Gaussian background [fdip-004S]

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Fix finite incidence data \((V,E,s,h)\), weights \(w_e>0\), a constant mass \(m>0\), and counting pairings on \(H=\mathbb R^V\) and \(\mathbb R^E\). A finite symmetry of a Gaussian background is a finite group \(G\) with a weight-preserving signed endpoint action \(g\mapsto (p_g,q_g,\varepsilon _g)\) on these data. No retained subset is selected here: in the signed-arrow definition one may take \(R=V\). An additional retained subset would have to be \(G\)-stable. The signs belong to the abstract \(C_2\) of that definition, with \(\sigma (+)=1\) and \(\sigma (-)=-1\). This unpartitioned data does not assert preservation of a nonempty proper retained set as in Definition [fdip-000G].

The vertex and edge representations are \[\begin {aligned} (\rho (g)f)(p_g(v))&=f(v),\\ (J_ga)(q_g(e))&=\sigma (\varepsilon _g(e))a(e). \end {aligned}\] Thus \(\rho (g)f=f\circ p_g^{-1}\); in particular \(\rho (g)\delta _v=\delta _{p_g(v)}\). We call \(H\) with this action the vertex permutation module. With \((df)(e)=f(h(e))-f(s(e))\), the endpoint equations give \(d\rho (g)=J_gd\). The sign cocycle gives \(\rho (gk)=\rho (g)\rho (k)\) and \(J_{gk}=J_gJ_k\). These conventions are the real specialization of Definition [fdip-003B], whose proof includes loops and parallel edge labels.

Write \(W=\operatorname {diag}(w_e)\) and \(Q=d^*Wd+m^2I\) for the massive precision. A reflection may reverse the chosen edge orientations; it is then a signed symmetry, not necessarily a directed relabeling in Definition [fdip-000G]. Forgetting those orientations preserves the energy, while still retaining incidence, weights and distinct edge labels. The underlying permutation-representation convention is [milne2025group, Example 7.1(b), p. 101]; the signed extension and its endpoint requirements are proved in Definition [fdip-003B].