Definition. Signed arrows preserve endpoints before incidence [fdip-003B]
AGENTDRAFTED
Reversing an oriented edge changes the sign of its difference, but
does not erase its endpoints. Fix a commutative unital ring \(k\) and
two finite incidence presentations
\((V,E,s,h)\) and \((V',E',s',h')\). Equip them with weights
\(w:E\to k\), \(w':E'\to k\), scalars \(\mu ,\mu '\in k\), and
retained subsets \(R\subseteq V\), \(R'\subseteq V'\).
Empty carriers, loops and distinct parallel edge labels are allowed.
Write \(C_2=\{+,-\}\) for the abstract group with \(-\cdot -=+\),
and \(\sigma :C_2\to k^\times \) for \(\sigma (+)=1\),
\(\sigma (-)=-1\). A signed endpoint arrow consists of
bijections \(p:V\to V'\), \(q:E\to E'\) and signs
\(\varepsilon :E\to C_2\) such that
\[\begin {aligned}
\varepsilon (e)=+&:\quad s'q(e)=ps(e),\quad h'q(e)=ph(e),\\
\varepsilon (e)=-&:\quad s'q(e)=ph(e),\quad h'q(e)=ps(e),\\
w'q&=w,\qquad \mu '=\mu ,\qquad pR=R'.
\end {aligned}\]
The signs are abstract orientation data, even when \(1=-1\) in \(k\).
An arrow with all signs \(+\) is directed. For the
real Gaussian backgrounds, directed arrows are their exact
relabelings, with \(\mu =m^2\); positivity of mass then recovers \(m'=m\).
Composition applies the right arrow first. The composite with
\((p',q',\varepsilon ')\) is
\[\bigl (p'p,q'q,\ e\mapsto \varepsilon '(q(e))\varepsilon (e)\bigr ).\]
Identity arrows have identity bijections and all signs \(+\). The inverse
of \((p,q,\varepsilon )\) has bijections \(p^{-1},q^{-1}\) and the sign
\(\varepsilon (e)\) at \(q(e)\). These operations make the declared objects
and arrows a groupoid: composition is associative, identities exist,
and every arrow is invertible, as for an action groupoid.
Let \(M=k^V\), \(N=k^E\) and similarly \(M',N'\), with incidence
maps \((df)(e)=f(h(e))-f(s(e))\). The induced ordinary linear maps are
\[\begin {aligned}
(Pf)(p(v))&=f(v),&P&:M\to M',\\
(Ja)(q(e))&=\sigma (\varepsilon (e))a(e),&J&:N\to N'.
\end {aligned}\]
They are isomorphisms: \(P^{-1}f'=f'\circ p\) and
\((J^{-1}a')(e)=\sigma (\varepsilon (e))a'(q(e))\).
Evaluation at \(q(e)\), in the two endpoint cases, gives
\[d'P=Jd:M\longrightarrow N'.\]
Both differences vanish on a loop; the endpoint condition still
remembers where that loop is attached.
For a left action of a group \(\Gamma \),
take actions \(p_g,q_g\) on the vertex and edge sets. A
signed endpoint action additionally requires
\[\varepsilon _1(e)=+,\qquad
\varepsilon _{gk}(e)=\varepsilon _g(q_k(e))\varepsilon _k(e).\]
Thus the arrow assigned to \(gk\) is the composite of those assigned to
\(g\) and \(k\). The induced maps satisfy
\(P_gP_k=P_{gk}\) and \(J_gJ_k=J_{gk}\). Indeed evaluation at
\(p_{gk}(v)\) gives \(f(v)\) on both sides of the first equation;
at \(q_{gk}(e)\) the second has coefficient
\(\sigma (\varepsilon _g(q_k(e)))\sigma (\varepsilon _k(e))a(e)\).
Identity signs and maps give the identity operators. These are linear
representations on two distinct carriers; no faithfulness is assumed.
A weaker signed-incidence symmetry requires the same
actions, abstract sign cocycle and weight preservation, but only the
incidence equation \(dP_g=J_gd\), instead of the endpoint equations.
It is an incidence intertwiner and suffices for
operator transport. It need not be a graph automorphism;
loops and characteristic two distinguish the requirements.
The concrete half-turn satisfies the stronger endpoint
conditions.
The left-action and equivariance conventions are
[milne2025group, Definition 4.1, pp. 57-58].
[milne2025group, Example 7.1(b), p. 101] gives a permutation
representation over a field; the ring and signed endpoint construction
above is proved directly. The incidence convention in
[spielman2009effective, Section 2.1, pp. 4-5] concerns a connected
undirected positive-weight graph with a chosen orientation; the
endpoint-difference definition here also allows loops, parallel labels
and arbitrary coefficients.
Mathlib's Representation.ofMulAction sends the free
basis vector at \(v\) to that at \(g\cdot v\).
Finsupp.linearEquivFunOnFinite identifies its coefficients
with functions on a finite set, yielding inverse precomposition for a
group action. Representation.IntertwiningMap bundles an
intertwiner between already supplied representations. Neither supplies
the endpoint equations or abstract sign cocycle above.