Example. The half-turn on vertex and edge fields [fdip-000J]
AGENTDRAFTED
The alternating four-cycle has a symmetry for every
\(t>0\), including unequal weights. Let \(C_2=\{1,r\}\) with \(r^2=1\).
Its left actions on the distinct vertex and edge sets
\(V=\mathbb Z/4\mathbb Z\) and \(E=\{e_0,e_1,e_2,e_3\}\) are
\[r\cdot i=i+2,\qquad r\cdot e_i=e_{i+2}.\]
Indices are modulo \(4\). These permutations square to the identity,
preserve the endpoints \(e_i:i\to i+1\), and preserve the weights
\((1,t,1,t)\). Milne's [milne2025group, Definition 4.1, p. 57]
gives the left-action convention; his
[milne2025group, Example 7.1(b), p. 101] constructs the associated
permutation representation.
The half-turn exchanges \(0,2\) and \(1,3\), and exchanges
\(e_0,e_2\) and \(e_1,e_3\). Directed arrows and their weights agree
after rotation. Solid vertices form the retained set \(\{0,2\}\);
the dashed vertices form its complement. Each set is preserved.
The induced maps on real fields use inverse precomposition:
\[\begin {aligned}
P:\mathbb R^V&\to \mathbb R^V,&(P\phi )_i&=\phi _{i-2},\\
U:\mathbb R^E&\to \mathbb R^E,&(Ua)(e_i)&=a(e_{i-2}).
\end {aligned}\]
Both maps have the coordinate array
\[\begin {pmatrix}
0&0&1&0\\0&0&0&1\\1&0&0&0\\0&1&0&0
\end {pmatrix},\]
but their domains differ. They satisfy \(P^2=I\) and \(U^2=I\).
Sending \(1\) to the identity and \(r\) to these maps gives two
real linear representations of \(C_2\).
Write \(\delta _i(j)=\mathbf 1_{j=i}\) for the vertex coordinate
basis and \(\eta _i(e_j)=\mathbf 1_{j=i}\) for the edge coordinate basis.
Then \(P\delta _0=\delta _2\ne \delta _0\) and
\(U\eta _0=\eta _2\ne \eta _0\). Since \(r\) is the only nonidentity
element of \(C_2\), both representations are
faithful. Finite coordinate reindexing preserves
the counting pairings; weight preservation gives
\[\begin {aligned}
P^TP&=I,\qquad U^TU=I,\\
U^TWU&=W,
\end {aligned}\]
where \(W=\operatorname {diag}(1,t,1,t)\). The last identity also
preserves the weighted edge pairing.
The two representations interact through the
incidence matrix. With its edge-by-vertex convention
from [spielman2009effective, Section 2.1, p. 4],
\[\begin {aligned}
(BP\phi )(e_i)&=\phi _{i+1-2}-\phi _{i-2}\\
&=(UB\phi )(e_i).
\end {aligned}\]
Thus \(BP=UB\): the incidence map is
equivariant between the vertex and edge actions.
The same formulas define linear representations over any nonzero
commutative unital ring, with any period-two weights in that ring.
The delta witnesses still prove faithfulness because \(1\ne 0\).
Over the zero ring the two field modules have one element, so this
faithfulness assertion is excluded. The positive real weights above
are the specialization used for Gaussian laws.
This is the all-positive case of a
signed endpoint action. The general endpoint
composition, abstract orientation cocycle and weaker
signed-incidence symmetry have that defining
address. The loop and reversal examples explain why
endpoint and sign coherence cannot be inferred from the incidence
intertwiner alone.
Mathlib's Representation.ofMulAction sends a coordinate
basis vector at \(v\) to the one at \(g\cdot v\). Its finitely supported
coefficient module is identified with all functions on a finite set by
Finsupp.linearEquivFunOnFinite, giving the inverse-precomposition
convention above. Representation.IntertwiningMap bundles
a linear map with the equation \(B P_g=U_gB\). These interfaces do not
provide the particular incidence identity or the coherent signed-edge
action without their proofs.