Example. Quaternion and coordinate selectors [fdip-005G]
AGENTDRAFTED
Let \(\Gamma =Q_8=\{\pm 1,\pm i,\pm j,\pm k\}\) and take
the simple undirected Cayley graph with edges
\(\{g,gs\}\), \(s\in \{\pm i,\pm j\}\), counted once each.
Give each edge weight one, choose any orientation, and set \(m=1\).
The two parts
\[A=\{\pm 1,\pm k\},\qquad B=\{\pm i,\pm j\}\]
give \(K_{4,4}\): every vertex in one part has the 4 distinct
vertices of the other as its neighbors. In part order,
\[L=\begin {pmatrix}4I_4&-J_4\\-J_4&4I_4\end {pmatrix},
\qquad Q=L+I_8,\]
where \(J_4\) is the all-ones matrix. The two constant-on-part
vectors give Laplacian eigenvalues \(0,8\); the two partwise
zero-sum spaces give eigenvalue \(4\), of total multiplicity 6.
Left multiplication gives a signed graph action \(\rho \) on
\(\mathbb R^{Q_8}\), independent of the arbitrary edge orientations.
For the central involution \(z=-1\), put
\[e_-=(1-[z])/2\in \mathbb R[Q_8],\qquad
P_-=\rho (e_-)=(I-\rho (z))/2.\]
Since \(z^2=1\) and \(z^{-1}=z\), this is a self-adjoint
idempotent. Its image is the four-dimensional space of fields
\(f(-g)=-f(g)\). The group-algebra decomposition
\(\mathbb R[Q_8]\cong \mathbb R^4\times \mathbb H\)
in Proposition [fgap-001B] sends \(e_-\) to \((0,1)\): each real character
takes \(z\) to \(1\), while the quaternionic realization takes it
to \(-1\). Thus this selector retains precisely the quaternion factor
in the regular module, without assigning it a physical interpretation.
Compare the coordinate selector on the subgroup
\(R=\langle i\rangle =\{1,-1,i,-i\}\), which retains two vertices
from each part. In the full graph, decompose fields into the two
constant-on-part lines and the two partwise zero-sum spaces. On the
normalized part constants, the precision and covariance are
\[\begin {pmatrix}5&-4\\-4&5\end {pmatrix},\qquad
\frac 19\begin {pmatrix}5&4\\4&5\end {pmatrix};\]
on the zero-sum spaces the covariance is \(1/5\). It follows that
full covariance entries are \(13/45\) on the diagonal,
\(4/45\) between distinct vertices in the same part, and \(1/9\)
between parts. For example the first entry is
\((1-1/4)/5+(5/9)/4=13/45\).
In retained order \((1,-1,i,-i)\), compression therefore gives
\[C_R=\frac 1{45}\begin {pmatrix}
13&4&5&5\\4&13&5&5\\5&5&13&4\\5&5&4&13
\end {pmatrix}.\]
The two within-part difference vectors have covariance eigenvalue
\((13-4)/45=1/5\). On normalized retained part constants the
covariance block is
\(\frac 1{45}\begin {pmatrix}17&10\\10&17\end {pmatrix}\),
with eigenvalues \(3/5,7/45\). Inversion gives the retained
precision spectrum
\[\operatorname {spec}(S_R)=\{5/3,5,5,45/7\}.\]
It differs from \(\{5,5,5,5\}\) for the quaternion selector despite
equal rank and the same full background precision.
The chosen \(Q_8\) action is auxiliary data used to construct
\(P_-\); it is not another slot of the selected tuple.
Indeed the underlying graph has \(2(4!)^2=1152\) automorphisms:
its connected bipartition is unique up to exchange, and every
permutation within either part, as well as their exchange, preserves
all edges. The selected-background groupoid remembers the graph and
the resulting projector, not a named subgroup of this automorphism
group. Remembering the action would require an enhanced tuple and
arrows intertwining that action.