Theorem. The top exterior word is the type-B spin highest vector [fcap-002Z]
✍️source

Let \(V=W\oplus W'\oplus \mathbb C z\) be a nondegenerate quadratic space of dimension \(2\ell +1\), with \(\ell \geq 1\). Assume \(W,W'\) are complementary polar-dual isotropic spaces, \(z\) is orthogonal to both, and \(Q(z)=1\). Choose bases \(e_i,e_i'\) with \(B_Q(e_i,e_j')=\delta _{ij}\) and the plus odd action on \(S=\bigwedge W\), \[\rho (\iota _Q(z))=\alpha ,\] where \(\alpha \) is exterior parity. This sign is part of the choice: \(Q(z)=1\) alone says only that the line coordinate squares to \(1\). Write \(c_i,a_i,N_i,H_i,\epsilon _i\) as in Theorem [fcap-0025].

Choose the positive roots \[\Phi ^+=\{\epsilon _i-\epsilon _j,\ \epsilon _i+\epsilon _j:i<j\} \cup \{\epsilon _i:1\leq i\leq \ell \}.\] The chain simple roots use \(E_i=c_i a_{i+1}\) and \(F_i=c_{i+1}a_i\). At the terminal short root \(\alpha _\ell =\epsilon _\ell \), take \[E_\ell =c_\ell \alpha ,\qquad F_\ell =\alpha a_\ell ,\qquad [E_\ell ,F_\ell ]=2N_\ell -1.\] The matrix short root has a coefficient \(2\) because \(B_Q(z,z)=2\); the Clifford bivector is nevertheless exactly \(e_\ell z\), with no extra factor \(\frac 12\) TauCeti.SpinPolarizationData.typeBQuadraticEquiv_typeBShortRootGenerator. On an increasing word, the terminal creation or removal sign cancels the parity sign, so the short operators insert or remove coordinate \(\ell \) with coefficient \(1\) TauCeti.SpinPolarizationData.typeBSpinRep_shortRootGenerator_apply TauCeti.SpinPolarizationData.typeBSpinRep_shortNegativeRootGenerator_apply.

This convention gives the same flag Borel as in Proposition [fcap-002X]. For \(i<j\), put \(X_{ij}=c_i a_j\), \(Y_{ij}=c_i c_j\), and \(K_i=c_i\alpha \). The chain commutators generate every \(X_{ij}\), while \[[X_{ij},K_j]=K_i,\qquad [K_i,K_j]=-2Y_{ij}.\] Because \(2\) is invertible over \(\mathbb C\), the simple chain operators and \(K_\ell \) generate the root spaces of every positive short and long root. They therefore generate the full positive nilradical.

The top word \[\Omega _B=e_1\wedge \cdots \wedge e_\ell \] is killed by each positive chain root and by \(E_\ell \), since the relevant creation coordinate is already occupied. The chain coroots have eigenvalue \(0\), and the terminal coroot \(2N_\ell -1\) has eigenvalue \(1\). Hence \[\operatorname {wt}(\Omega _B) =\omega _\ell =\tfrac 12(\epsilon _1+\cdots +\epsilon _\ell ).\] The chain moves together with the terminal toggle connect all exterior basis words. As in Theorem [fcap-002Y], the distinct joint \(H_i\)-weights first project any nonzero invariant subspace onto a basis line; the root moves then reach every line. Thus \(S\) is the irreducible type-\(B_\ell \) module \(L(\omega _\ell )\). The short root changes exterior degree, so neither \(S^+\) nor \(S^-\) is a type-B submodule.

[fulton2004representation, Proposition 20.20, p. 307] independently identifies the odd-dimensional spin representation as the irreducible module of highest weight \(\frac 12(\epsilon _1+\cdots +\epsilon _\ell )\). The preceding short-root and nilradical calculations give that conclusion in the fixed FCAP operator convention.

[goodman2009symmetry, Proposition 6.2.4, p. 315] independently gives the same irreducible type-B module and its half-sign weights. As in Theorem [fcap-0025], its occupied Cartan sign is opposite ours; negating its Cartan axes and positive system gives the displayed highest weight. TauCeti formalizes the short and difference-root actions and the integral simple-coroot eigenvalues TauCeti.SpinPolarizationData.typeBSpinRep_differenceRootGenerator_apply TauCeti.SpinPolarizationData.spinAction_typeBQuadraticEquiv_typeBSimpleCorootGenerator_basis.

The top word's weight and annihilation data are available over any field with \(2\) invertible, for a supplied polarization, basis of \(W\), and residual vector \(z\) with \(Q(z)=1\). The Cartan acts on \(\Omega _B\) through the terminal fundamental weight TauCeti.SpinPolarizationData.typeBSpinLieRep_apply_cartan_exteriorBasis_univ, and the Lie subalgebra generated by all positive simple-root generators annihilates it TauCeti.SpinPolarizationData.typeBSpinLieRep_lieSpan_range_simpleRootGenerator_le_lieAnnihilator_exteriorBasis_univ. The declarations index the rank by \(n+1=\ell \); their terminal \(\operatorname {Fin.last}(n)\) labels our \(\omega _\ell \).

The general highest-vector criterion is available separately: a nonzero Cartan weight vector is highest exactly when every raising generator kills it LieAlgebra.Basis.isHighestWeightVector_iff_forall_e. It assumes a characteristic-zero field, a finite-dimensional Lie algebra with nondegenerate Killing form, and a Lie basis. For type B, the Killing condition follows when every endomorphism in the full standard action is triangularizable, in particular over \(\mathbb C\) TauCeti.isKilling_typeB. Triangularizability of the diagonal Cartan alone is not this hypothesis. The selected spin-module declarations give the weight and annihilation data, rather than a separately packaged highest-vector conclusion. The Borel-generation and irreducibility arguments above complete the displayed module identification.

Theorem. The top exterior word is the type-B spin highest vector [fcap-002Z]
✍️source

Reference. Symmetry, Representations, and Invariants [goodman2009symmetry]

Reference. Representation theory: A first course [fulton2004representation]

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

Convention. Clifford generators and polarized operators [fcap-0027]

Definition. The three polarized Clifford operators [fcap-001I]

wedge contract lineOperator cliffordOperator
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. Rank one: matrix units and occupation [fcap-001P]

Theorem. Matrix units give the even-dimensional structure theorem [fcap-002V]

Proposition. An odd spinor action selects one Clifford summand [fcap-002W]

Theorem. Exterior pairings determine compact spinor reality [fcap-0041]

Proposition. Real matrix blocks control Spin module complexification [fcap-0042]

Theorem. Occupation signs split the half-spin carrier [fcap-0025]

spinAction_diagonalBivector_basis iSup_spinWeightSpace_even_eq_spinPlus iSup_spinWeightSpace_odd_eq_spinMinus
L∃∀N

Proposition. The simple type-D operators generate the positive nilradical [fcap-002X]

Theorem. The two top exterior words are the type-D fork highest vectors [fcap-002Y]

Theorem. The top exterior word is the type-B spin highest vector [fcap-002Z]
✍️source

Proposition. Exterior weights give the finite spin characters [fcap-0030]

Proposition. Complex Spin in dimension three is \(SL_2\) on the spin module [fcap-0031]

Proposition. Complex Spin in dimension four is two copies of \(SL_2\) [fcap-0032]

Proposition. Complex Spin in dimension five is the symplectic group on spinors [fcap-0033]

Proposition. Complex Spin in dimension six is \(SL_4\) on a half-spin module [fcap-0034]

Proposition. The outer nodes of \(D_4\) permute the three eight-dimensional classes [fcap-003B]

Theorem. The actual compact \(\operatorname {Spin}(8)\) has outer Lie-automorphism group \(S_3\) [fcap-003C]

Proposition. Clifford multiplication gives the triality trilinear [fcap-003D]

Theorem. A pinned three-cycle integrates to compact triality [fcap-003T]

Proof.

Theorem. Triality cycles three real modules and preserves the trilinear [fcap-003U]

Proof.

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]

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. Occupation signs split the half-spin carrier [fcap-0025]

spinAction_diagonalBivector_basis iSup_spinWeightSpace_even_eq_spinPlus iSup_spinWeightSpace_odd_eq_spinMinus
L∃∀N

Proposition. The simple type-D operators generate the positive nilradical [fcap-002X]

Theorem. The two top exterior words are the type-D fork highest vectors [fcap-002Y]