Definition. The group-algebra action of the half-turn [fdip-000L]
AGENTDRAFTED
The half-turn acts on fields before a precision or
probability law is chosen. Let \(k\) be a nonzero commutative ring and
\(C_2=\{1,r\}\), with \(r^2=1\). In its
Group Algebra, write \(g=[r]\) and \(1=[1]\).
Every element has unique coefficients \(a+bg\), and
\[\begin {aligned}
(a+bg)(c+dg)&=(ac+bd)\\
&\quad +(ad+bc)g.
\end {aligned}\]
[sengupta2010representations, Sections 3.1-3.2, pp. 39-41]
gives the group basis and linear extension over a field. The
ring-module extension supplies the general ring theorem;
the following formulas identify its particular faithful actions.
Put \(V_k=k^4\), in vertex order \((0,1,2,3)\), and
\(W_k=k^R\), with \(R=(0,2)\) in that order. Let
\(\delta _i(j)=\mathbf 1_{j=i}\) be the coordinate basis of \(V_k\);
let \(\delta _i^R(j)=\mathbf 1_{j=i}\) for \(i,j\in R\) be the
coordinate basis of \(W_k\). These coordinate vectors differ from the
real spectral vectors \(u_a\) and from the edge labels
\(e_i\). The half-turn and its retained action are
\[\begin {aligned}
P(\phi _0,\phi _1,\phi _2,\phi _3)&=(\phi _2,\phi _3,\phi _0,\phi _1),\\
J(x_0,x_2)&=(x_2,x_0).
\end {aligned}\]
Applying either permutation twice returns the original coordinates,
so \(P^2=I\) and \(J^2=I\). Their unique unital \(k\)-algebra extensions are
\[\begin {aligned}
\widehat \rho _V:k[C_2]&\longrightarrow \operatorname {End}_k(V_k),\\
\widehat \rho _V(a+bg)&=aI+bP,\\
\widehat \rho _W:k[C_2]&\longrightarrow \operatorname {End}_k(W_k),\\
\widehat \rho _W(a+bg)&=aI+bJ.
\end {aligned}\]
The associated left module action is \(z\cdot v=\widehat \rho _V(z)v\),
and similarly on \(W_k\). Both algebra actions are faithful, meaning
that their defining maps are injective.
Indeed, \(P^2=I\) gives
\[\begin {aligned}
(aI+bP)(cI+dP)
&=(ac+bd)I\\
&\quad +(ad+bc)P.
\end {aligned}\]
The same calculation uses \(J^2=I\) on \(W_k\). Addition and scalar
multiplication are preserved term by term, and \(1\) maps to \(I\).
These facts prove the module laws. A linear extension is unique because
\(1,g\) are a basis. If \(aI+bP=0\), applying it to \(\delta _0\)
gives \(a\delta _0+b\delta _2=0\); coordinates \(0,2\) give
\(a=b=0\). Applying \(aI+bJ\) to \(\delta _0^R\) proves injectivity
on \(W_k\) in the same way. No absence of zero divisors is required.
Coordinate projection
\(\pi _R(\phi )=(\phi _0,\phi _2)\) sends \(\delta _0,\delta _2\) to
\(\delta _0^R,\delta _2^R\) and kills \(\delta _1,\delta _3\). Since
\(\pi _RP=J\pi _R\), it intertwines the whole algebra action:
\[\pi _R\widehat \rho _V(z)=\widehat \rho _W(z)\pi _R
\qquad (z\in k[C_2]).\]
Thus it is a \(k[C_2]\)-module map by the
all-element intertwining theorem. Both coordinate
faithfulness witnesses above work with zero divisors; they use more
than faithfulness of the group action alone.
Suppose now that \(2\) is a unit in \(k\), and put \(h=2^{-1}\).
The element \(g\) is a central involution.
Its two idempotents and algebra product are
\[e_+=h(1+g),\qquad e_-=h(1-g).\]
Their images under the two algebra actions are the following concrete
module projectors. Abstract centrality in \(k[C_2]\) supplies commutation
with the represented algebra; it does not make \(P\) central in all
\(\operatorname {End}_k(V_k)\).
The represented projectors are
\[E_\pm =h(I\pm P),\qquad F_\pm =h(I\pm J).\]
They give the eigenspace decompositions
\[\begin {aligned}
V_k&=V_+\oplus V_-,&V_\pm &=\operatorname {im}E_\pm ,\\
W_k&=W_+\oplus W_-,&W_\pm &=\operatorname {im}F_\pm .
\end {aligned}\]
For either sign \(\varepsilon \in \{1,-1\}\), the concrete
half-turn sector is
\[\begin {aligned}
V_\varepsilon &=\{(x,y,\varepsilon x,\varepsilon y):x,y\in k\},\\
W_\varepsilon &=\{(x,\varepsilon x):x\in k\}.
\end {aligned}\]
Their free bases are
\[\begin {aligned}
b_{\varepsilon ,0}&=\delta _0+\varepsilon \delta _2,\\
b_{\varepsilon ,1}&=\delta _1+\varepsilon \delta _3,\\
c_\varepsilon &=\delta _0^R+\varepsilon \delta _2^R.
\end {aligned}\]
The first two coordinates of \(V_\varepsilon \), and the first
coordinate of \(W_\varepsilon \), recover the coefficients uniquely.
Consequently the free ranks are two and one.
To verify the splitting, \(E_++E_-=I\) gives
\(v=E_+v+E_-v\). An element in both images is fixed and annihilated
by \(E_+\), so it is zero. Further, \(PE_\varepsilon =\varepsilon
E_\varepsilon \), while \(Pv=\varepsilon v\) gives
\(E_\varepsilon v=h(v+\varepsilon Pv)=v\). This identifies the
images with the displayed coordinate sets; the proof for \(W_k\) is
identical. Mathlib's LinearMap.IsIdempotentElem.isCompl
supplies the abstract complement between an idempotent's range and
kernel. The coordinate bases and ranks above identify its concrete
summands here.
Projection respects these summands:
\[\pi _RE_\varepsilon =F_\varepsilon \pi _R.\]
Its restriction \(\pi _\varepsilon :V_\varepsilon \to W_\varepsilon \)
sends \(b_{\varepsilon ,0}\) to \(c_\varepsilon \) and
\(b_{\varepsilon ,1}\) to zero. It is surjective, with
\[\ker \pi _\varepsilon =k b_{\varepsilon ,1}.\]
The first coefficient of a sector vector proves the kernel formula
and the displayed image of \(b_{\varepsilon ,0}\) proves surjectivity.
For each sign, the square commutes, with the horizontal codomains
restricted to the indicated sectors:
The horizontal maps select a sector; the right map removes one
free coordinate within it. The algebra factors \(k\times k\),
the full rank-two sectors, and the retained rank-one sectors are
different levels of decomposition.