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

✍️sourceAGENTDRAFTED

Let \(V=W\oplus \mathbb Cz\oplus W'\) be a five-dimensional complex quadratic space with a supplied nondegenerate polarization, where \(\dim W=2\), \(Q(z)=1\), and \(\rho (z)=\alpha \). On \(S=\bigwedge W\), use the ordered basis \[(1,e_1\wedge e_2,e_1,e_2).\] Define \[\langle s,t\rangle = [e_1\wedge e_2]\bigl (\operatorname {rev}_{\wedge }(s)\wedge t\bigr ).\] Its matrix is \(\Omega =\operatorname {diag}(J,J)\), where \(J=\left (\begin {smallmatrix}0&1\\-1&0\end {smallmatrix}\right )\). The pairing preserves exterior parity, so parity itself is self-adjoint. Each creation operator and each polar-dual contraction operator is self-adjoint as well. Consequently the faithful even Clifford equivalence from Proposition [fcap-002W] takes reversal to \[A\longmapsto A^{\dagger } =\Omega ^{-1}A^{\mathsf T}\Omega \] TauCeti.SpinPolarizationData.evenCliffordEquivEnd. Reverse norm one is therefore exactly the symplectic equation \(A^{\mathsf T}\Omega A=\Omega \).

This norm calculation alone does not yet prove membership in the vector-generated Spin group. The five vector operators \[c_1,\ c_2,\ a_1,\ a_2,\ \alpha \] are linearly independent, traceless, and \(\Omega \)-self-adjoint. The self-adjoint subspace of \(M_4(\mathbb C)\) has dimension six, and its trace-zero part has dimension five because \(\operatorname {tr}(I)=4\). Hence the Clifford image of the vector space is exactly \[\rho (V)= \{T:T^{\dagger }=T,\ \operatorname {tr}(T)=0\}.\] Conjugation by every \(A\in Sp_{\Omega }(4,\mathbb C)\) preserves this subspace: symplectic conjugation preserves the adjoint condition, and trace is conjugation-invariant.

The odd Clifford action is injective on the vector subspace. More generally, if an odd Clifford value \(y\) acts by zero, then \(zy\) is even and acts by zero. Faithfulness of the even action and invertibility of \(z\) give \(y=0\). Thus preservation of the displayed matrix subspace lifts to preservation of \(V\) inside the Clifford algebra.

For \(A\in Sp_{\Omega }(4,\mathbb C)\), let \(x\) be its unique preimage under the even Clifford equivalence. The adjoint equation gives \(\operatorname {rev}(x)x=1\), while the preceding calculation proves \(xVx^{-1}=V\). The carrier-completion argument of Proposition [fcap-0032] now places \(x\) in actual Spin. Conversely, every Spin element has symplectic matrix because its reverse norm is one. These assignments are inverse group maps, and the chosen exterior basis is the module intertwiner: \[\operatorname {Spin}_5(\mathbb C)\simeq Sp_{\Omega }(4,\mathbb C), \qquad S\simeq \mathbb C^4.\]

Fulton--Harris state the same group and standard spin-module identification [fulton2004representation, Exercise 20.39, p. 311]. Goodman--Wallach independently write the target as \(Sp(2,\mathbb C)\), the rank-two symplectic group on a four-dimensional complex space [goodman2009symmetry, Section 6.3.3, Exercise 1, p. 322]. The trace-zero normalization calculation supplies the actual group inverse rather than inferring the image from a dimension count. By contrast, Meinrenken's \(\operatorname {Spin}(5)=Sp(2)\) is the compact quaternionic real form [meinrenken2013clifford, Section 3.7.6, pp. 83--84].

TauCeti proves that in odd dimension the even Clifford action is an equivalence onto the full endomorphism algebra and is faithful TauCeti.SpinPolarizationData.evenCliffordEquivEnd TauCeti.evenSpinAction_injective. It does not package the invariant alternating form or the group image as \(Sp_4\); those calculations are the explicit argument above.