Proposition. An odd spinor action selects one Clifford summand [fcap-002W]
Proposition. An odd spinor action selects one Clifford summand [fcap-002W]
Work over \(\mathbb C\) with \(V=W\oplus W'\oplus \mathbb C z\), where \(W,W'\) are polar-dual isotropic spaces of dimension \(\ell \), \(z\) is orthogonal to them, and \(Q(z)=1\). On \(S=\bigwedge W\), the two Clifford actions agree on \(W\oplus W'\) and differ on the last generator: \[\rho _+(\iota _Q(z))=\alpha ,\qquad \rho _-(\iota _Q(z))=-\alpha ,\] where \(\alpha \) is exterior parity. For polar-dual bases, normalize the hyperbolic chirality element by \[\Gamma =\prod _{i=1}^{\ell } \bigl (1-2\iota _Q(e_i)\iota _Q(e_i')\bigr ).\] The even-dimensional action sends \(\Gamma \) to \(\prod _i(1-2N_i)=\alpha \). Moreover \(\Gamma ^2=1\), and \(\Gamma \) anticommutes with every hyperbolic vector. Consequently \[\chi =\Gamma \iota _Q(z),\qquad \chi ^2=1\] is central in \(\mathcal {C}\kern -2pt\ell (Q)\). The complementary central idempotents \[p_+=\frac {1+\chi }{2},\qquad p_-=\frac {1-\chi }{2}\] satisfy \[\rho _+(p_+)=1,\quad \rho _+(p_-)=0,\qquad \rho _-(p_+)=0,\quad \rho _-(p_-)=1.\] Thus each odd action has a visible nonzero kernel.
The central idempotents also identify the whole algebra, rather than only
showing a dimension mismatch. Given
\(A\in \operatorname {End}_{\mathbb C}(S)\), the preceding theorem supplies a
hyperbolic Clifford element \(a\) with common image \(A\). Therefore the joint
map obeys
\[(\rho _+,\rho _-)(p_+a)=(A,0),\qquad
(\rho _+,\rho _-)(p_-a)=(0,A).\]
It is surjective. Since
\[\dim _{\mathbb C}\mathcal {C}\kern -2pt\ell (Q)=2^{2\ell +1}
=2\dim _{\mathbb C}\operatorname {End}_{\mathbb C}(S),\]
it is an isomorphism
\[\mathcal {C}\kern -2pt\ell (Q)\simeq
\operatorname {End}_{\mathbb C}(S)\oplus
\operatorname {End}_{\mathbb C}(S).\]
The corresponding abstract two-block existence theorem is
CliffordAlgebra.nonempty_algEquiv_matrix_prod_of_finrank_eq_two_mul_add_one.
The explicit joint-map proof determines how the two actions select its
summands.
The restrictions \(\rho _+|_{\mathcal {C}\kern -2pt\ell (Q)^0}\) and
\(\rho _-|_{\mathcal {C}\kern -2pt\ell (Q)^0}\) are parity-conjugate. When \(\ell \geq 1\) they are
unequal, as the Spin element below shows; for \(\ell =0\) both are the same
scalar action. For every even Clifford word, the number of hyperbolic factors
has the same parity as the number of \(z\)-factors. Conjugation by \(\alpha \)
contributes exactly that sign, so
\[\rho _-(x)=\alpha \rho _+(x)\alpha ^{-1}
\qquad (x\in \mathcal {C}\kern -2pt\ell (Q)^0).\]
If \(\rho _+(x)=0\), conjugacy makes both coordinates of the joint isomorphism
zero, hence \(x=0\). The even source and target have the same dimension, so
either restriction is an isomorphism
\[\mathcal {C}\kern -2pt\ell (Q)^0\simeq \operatorname {End}_{\mathbb C}(S).\]
This faithful range is formalized by
TauCeti.evenSpinAction_surjective and
TauCeti.SpinPolarizationData.evenCliffordEquivEnd.
When \(\ell \geq 1\), exterior parity is not a decomposition into invariant
odd-dimensional Spin modules. Fix \(i\), put \(u=e_i+e_i'\), and note
\(Q(u)=Q(z)=1\). The even product
\[g=\iota _Q(z)\iota _Q(u)\in \operatorname {Spin}(Q)\]
belongs to the actual Spin carrier by the norm-one pair criterion
CliffordAlgebra.ι_mul_ι_mem_spinGroup_of_norm_mul_norm_eq_one.
Its plus action is \(\alpha (c_i+a_i)\) and satisfies
\[1\longmapsto -e_i,\qquad e_i\longmapsto 1.\]
Its minus action is the negative of this operator. Thus the two even
restrictions are unequal, and this actual Spin element exchanges even and
odd exterior degree. For \(\ell =0\), the carrier \(S\) is one-dimensional and
\(S^-=0\); no two-summand parity claim is made.
[fulton2004representation, Lemma 20.16 and (20.18), p. 306] give the odd complex product algebra and the faithful even restriction. Their second full-Clifford module is written on \(\bigwedge W'\), rather than as a second sign on the same carrier.
[goodman2009symmetry, Theorem 6.1.3, proof, pp. 304--305] construct the exterior carrier. [goodman2009symmetry, Lemma 6.1.4(3), p. 306] put the two normalized odd-vector signs on this common carrier. This supplies the common-carrier translation used above.
[meinrenken2013clifford, Section 3.2.4, (3.5), p. 60] supplies the normalized chirality product on the hyperbolic block. His Theorem 3.3 is an even split theorem, not an odd product theorem. The two-idempotent conclusion here is restricted to the split complex setting; an odd Clifford center over a general real field need not split.