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

Let \(V=W\oplus W'\) be a split nondegenerate complex quadratic space with \(\dim W=3\). For polar-dual bases \(e_i,e'_i\), use \[S^+:(1,e_{12},e_{13},e_{23}),\qquad S^-:(e_{123},-e_3,e_2,-e_1).\] For \(\langle s,t\rangle =[e_{123}]\bigl ( \operatorname {rev}_{\wedge }(s)\wedge t\bigr )\), the second displayed basis is dual to the first, and the full pairing matrix on \(S^+\oplus S^-\) is \[\Omega =\begin {pmatrix}0&I_4\\-I_4&0\end {pmatrix}.\] Clifford generators are self-adjoint for this pairing. Under the paired even-algebra equivalence \(\Phi \) of Theorem [fcap-002V], reversal therefore sends \[(A,D)\longmapsto (D^{\mathsf T},A^{\mathsf T}).\] Thus the even reverse-norm-one group is \[\{(A,A^{-\mathsf T}):A\in GL_4(\mathbb C)\}.\] At this stage it is \(GL_4\), not yet the actual Spin group.

The vector-normalization condition cuts out the determinant-one subgroup. If \(v=\sum _i x_i e_i+y_i e'_i\), its odd Clifford action has block form \[\rho (v)=\begin {pmatrix}0&B_v\\B_v^\sharp &0\end {pmatrix},\] where \[B_v=\begin {pmatrix} 0&-y_3&y_2&-y_1\\ y_3&0&x_1&x_2\\ -y_2&-x_1&0&x_3\\ y_1&-x_2&-x_3&0 \end {pmatrix}.\] These matrices run through all skew \(4\times 4\) matrices. For a skew matrix \(B\), define \(B^\sharp \) by \[\begin {aligned} (B^\sharp )_{01}&=B_{23},& (B^\sharp )_{02}&=-B_{13},& (B^\sharp )_{03}&=B_{12},\\ (B^\sharp )_{12}&=B_{03},& (B^\sharp )_{13}&=-B_{02},& (B^\sharp )_{23}&=B_{01}, \end {aligned}\] with skew extension. Then \((B^\sharp )^\sharp =B\), \(BB^\sharp =-\operatorname {Pf}(B)I\), and \(\operatorname {Pf}(B_v)=-\sum _i x_i y_i\). The last two identities give \(\rho (v)^2=Q(v)I\), matching the Clifford convention.

The complementary-minor identity is \[(ABA^{\mathsf T})^\sharp =\det (A)A^{-\mathsf T}B^\sharp A^{-1}.\] One derivation identifies a skew matrix with an alternating two-form: the operation \(B\mapsto B^\sharp \) is the complementary two-form determined by the ordered volume basis, while \(\bigwedge ^4 A\) multiplies that volume by \(\det A\). Evaluating on basis pairs gives the six coordinate rules above and the displayed identity.

Conjugation by \(\operatorname {diag}(A,A^{-\mathsf T})\) sends the two off-diagonal blocks to \[ABA^{\mathsf T},\qquad A^{-\mathsf T}B^\sharp A^{-1}.\] By the complementary-minor identity, it preserves the vector graph \(\{(B,B^\sharp ):B^{\mathsf T}=-B\}\) exactly when \(\det A=1\). Sufficiency is immediate. For necessity, choose any nonzero \(B^\sharp \); equality of the two lower blocks forces \(\det A=1\). Faithfulness of the full Clifford action in this even dimension lifts this matrix statement to normalization of the actual vector subspace.

Every Spin element therefore acts on \(S^+\) through \(SL_4(\mathbb C)\). Conversely, for \(A\in SL_4(\mathbb C)\), let \(x\) be the unique even Clifford preimage of \((A,A^{-\mathsf T})\). It has reverse norm one and, by the graph calculation, normalizes \(V\). The carrier-completion argument of Proposition [fcap-0032] places \(x\) in actual Spin. The mutually inverse maps are \[g\longmapsto \rho _+(g),\qquad A\longmapsto \Phi ^{-1}(A,A^{-\mathsf T}).\] Hence \[\operatorname {Spin}_6(\mathbb C)\simeq SL_4(\mathbb C).\] In the displayed bases \(S^+\) is the standard four-dimensional module and \(S^-\) is its dual.

[fulton2004representation, Exercise 20.39, p. 311] give the group and standard \(S^+\) module. [goodman2009symmetry, Section 6.3.3, Exercise 1, p. 322] independently give the same complex group. Here the cross-pairing proves \(S^-\) is dual, and the vector-graph calculation proves the actual image.

[meinrenken2013clifford, Section 3.7.6, p. 84] gives the compact real form \(\operatorname {Spin}(6)=SU(4)\).

TauCeti formalizes the paired half-spin algebra and matrix equivalences TauCeti.SpinPolarizationData.evenCliffordEquivProdEnd TauCeti.SpinPolarizationData.evenCliffordEquivProdMatrix. The vector graph, determinant-one group image and inverse are proved above, not by these declarations.

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

Reference. Clifford algebras and lie theory [meinrenken2013clifford]

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]

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]
✍️source

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]

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

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