Definition. A representation datum for a field [fdip-007H]

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A representation datum specifies a finite discrete or compact Hausdorff group \(G\), a scalar field \(\mathbb K\in \{\mathbb R,\mathbb C\}\), a finite-dimensional fiber \(F\) over \(\mathbb K\), a continuous representation \(\rho :G\to GL_{\mathbb K}(F)\), and a separately chosen action \(\alpha :G\to \operatorname {Aut}(X)\) on the finite substrate \(X\). The substrate automorphisms preserve the graph, weights and any supplied oriented cell data, with orientation changes interpreted by the established inverse-transport convention. One must also specify a positive invariant real or Hermitian inner product on \(F\). The field itself is a map \(\phi :V\to F\).

An invariant inner product always exists for these groups. From any positive inner product \(h_0\), define \[h(v,w)=\int _G h_0(\rho (g)v,\rho (g)w)\,d\mu _G(g).\] The normalized Haar measure makes this invariant by a change of variable. The integral is finite by compactness, and \(h(v,v)>0\) for \(v\ne 0\), since \(\rho (g)v\ne 0\) for every \(g\). This proves existence of an invariant metric; it does not make that metric a canonical choice. A selected metric remains part of the background data.

The global action associated to the datum is \[(g\cdot \phi )(v)=\rho (g)\phi (\alpha (g)^{-1}v).\] The two actions have different roles. For a purely internal group, \(\alpha \) is trivial: the group acts on values without moving vertices. A local gauge transformation instead assigns an element \(g_v\) independently at each vertex and transforms links by \(U_e\mapsto g_{h(e)}U_eg_{s(e)}^{-1}\). It fixes the substrate; a global substrate action \(\alpha \) cannot simply be applied independently at each vertex.

If a field has an additional spin space \(S\), an internal action on \(S\otimes F\) is \(1_S\otimes \rho (g)\). Clifford operators \(\gamma _j\otimes 1_F\) then commute with that action because they act on different factors. An identification of internal and spin symmetries, or an action moving the substrate, needs extra compatible data; it is not implied by the abstract group name or by an inclusion in \(SU(2)\).