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