Definition. The Spin and Pin representations [meinrenken2013clifford, Section 3.2.1, pp. 55--56] [fcap-001M]
✍️source
spinRep pinRep
L∃∀N

The groups \(\operatorname {Spin}(Q)\) and \(\operatorname {Pin}(Q)\) consist of units in the Clifford algebra. Restricting the action \(\rho \) of Theorem [fcap-001K] along these two inclusions gives representations on the same exterior carrier: \[\operatorname {spinRep}:\operatorname {Spin}(Q)\longrightarrow \operatorname {Aut}_K(S), \qquad \operatorname {pinRep}:\operatorname {Pin}(Q)\longrightarrow \operatorname {Aut}_K(S).\] Meinrenken calls the restriction of a Clifford module to the Clifford group the spin representation. The definitions here require only the commutative ring, module, quadratic form, and polarization data already used for \(\rho \); no nondegeneracy or finite-dimensionality is added.

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

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]
✍️source
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]

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

spinAction
L∃∀N