Definition. Polarization data and its carrier [chevalley1954algebraic, II.2.1--II.2.2, pp. 42--44] [fcap-001H]
✍️source
SpinPolarizationData
L∃∀N

Let \(K\) be a commutative ring, let \(V\) be a \(K\)-module, and let \(Q\) be a quadratic form. Polarization data consists of isotropic submodules \(W,W'\subseteq V\), an orthogonal remainder \(L\), and a linear equivalence \[(W\oplus W')\oplus L\simeq V.\] The polar form identifies \(W'\) with \(W^*\); the pairing is separating in both variables. A coordinate \(\ell :L\to K\) is injective and satisfies \(\ell (z)^2=Q(z)\). Thus \(L\) is an orthogonal remainder equipped with an injective coordinate. Over a field it is at most one-dimensional; it vanishes in the classical even split model. The carrier of the exterior spinor model is \[S=\bigwedge _K W.\]

Chevalley treats complementary totally singular subspaces in the split, even-dimensional field case. Meinrenken uses the equivalent Lagrangian model \(V=F^*\oplus F\) [meinrenken2013clifford, Section 3.2.2, Proposition 3.5, pp. 56--57]. The packaged remainder and commutative-ring hypotheses are TauCeti's stated generalization.

Definition. Polarization data and its carrier [chevalley1954algebraic, II.2.1--II.2.2, pp. 42--44] [fcap-001H]
✍️source
SpinPolarizationData
L∃∀N

Reference. Clifford algebras and lie theory [meinrenken2013clifford]

Reference. The algebraic theory of spinors [chevalley1954algebraic]

Polarization and the exterior spinor model [fcap-001G]

Definition. Polarization data and its carrier [chevalley1954algebraic, II.2.1--II.2.2, pp. 42--44] [fcap-001H]
✍️source
SpinPolarizationData
L∃∀N

Definition. Creation, contraction, and the line operator [meinrenken2013clifford, Section 3.2.2, Proposition 3.5, p. 57] [fcap-001I]

wedge contract lineOperator
L∃∀N

Lemma. The polarized Clifford relation [meinrenken2013clifford, Section 3.2.1, pp. 54--55] [fcap-001J]

contract_wedge cliffordOperator_sq
L∃∀N

Theorem. The exterior Clifford action [meinrenken2013clifford, Section 3.2.2, Proposition 3.5, p. 57] [fcap-001K]

spinAction
L∃∀N

Theorem. Creation and contraction generate all endomorphisms [meinrenken2013clifford, Section 3.2.4, Theorem 3.3, pp. 59--60] [fcap-001L]

creation_contraction_adjoin_eq_top spinAction_surjective
L∃∀N

Definition. The Spin and Pin representations [meinrenken2013clifford, Section 3.2.1, pp. 55--56] [fcap-001M]

spinRep pinRep
L∃∀N

Lemma. The restricted actions are the Clifford action [meinrenken2013clifford, Section 3.2.1, pp. 55--56] [fcap-001N]

spinRep_apply pinRep_apply
L∃∀N

Example. The rank-one polarized model [meinrenken2013clifford, Section 3.2.4, Theorem 3.3, p. 59] [fcap-001P]

Remark. Horizons beyond the exterior model [meinrenken2013clifford, Section 3.2.4, Theorem 3.3, pp. 59--60]; [kostant1997clifford, Section 2.4, Theorem 8, pp. 286--287]; [lawson2016spin, I.2, pp. 13--19] [fcap-0017]