Theorem. Creation and contraction generate all endomorphisms [meinrenken2013clifford, Section 3.2.4, Theorem 3.3, pp. 59--60] [fcap-001L]
✍️source
creation_contraction_adjoin_eq_top spinAction_surjective
L∃∀N

Assume that \(W\) is finite and free over the commutative ring \(K\). The subalgebra of \(\operatorname {End}_K(\bigwedge W)\) generated by all creation operators and all contractions is the whole endomorphism algebra. Since the polar pairing identifies \(W'\) with \(W^*\), these generators lie in the image of the exterior Clifford action. Hence \[\rho :\mathcal {C}\kern -2pt\ell (Q)\twoheadrightarrow \operatorname {End}_K(\bigwedge W)\] is surjective.

Meinrenken proves generation first in rank one by four matrix units and then by tensor decomposition. Over a split field he combines surjectivity with a dimension calculation to obtain an isomorphism and irreducibility. TauCeti formalizes the generation and surjectivity statements over a finite free module. This card does not claim that \(S\) is irreducible, or that \(\rho \) is injective.

Theorem. Creation and contraction generate all endomorphisms [meinrenken2013clifford, Section 3.2.4, Theorem 3.3, pp. 59--60] [fcap-001L]
✍️source
creation_contraction_adjoin_eq_top spinAction_surjective
L∃∀N

Reference. Clifford algebras and lie theory [meinrenken2013clifford]

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]

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]
✍️source
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]