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

✍️sourceAGENTDRAFTED

Let \(K\) be a field of characteristic different from \(2\), and let \(V\) be a finite-dimensional split quadratic space with a chosen polarization \(V=W\oplus W'\) into complementary isotropic subspaces. Put \(\dim _KW=\ell \), choose polar-dual bases \(e_i\in W\) and \(e_i'\in W'\) with \(B_Q(e_i,e_j')=\delta _{ij}\), and set \(S=\bigwedge W\). In the notation of Convention [fcap-0027], write \[c_i=\varepsilon _{e_i},\qquad a_i=a_{e_i'},\qquad N_i=c_i a_i,\qquad P_0=\prod _i(1-N_i)=\prod _i a_i c_i.\] The commuting idempotents \(N_i\) record occupation, so \(P_0\) is the projection onto the vacuum line \(K1\). For increasing subsets \(I=\{i_1<\cdots <i_p\}\) and \(J=\{j_1<\cdots <j_q\}\), put \[c_I=c_{i_1}\cdots c_{i_p},\qquad a_J=a_{j_q}\cdots a_{j_1},\qquad F_{I,J}=c_IP_0a_J.\] The reversed contraction order is essential. If \(e_K\) denotes the increasing exterior basis vector, then \[F_{I,J}(e_K)=\delta _{J,K}e_I,\qquad F_{I,J}F_{K,L}=\delta _{J,K}F_{I,L}.\] Indeed, \(a_J(e_J)=1\). A missing index makes the contraction zero, while a properly larger \(K\) leaves positive exterior degree and is killed by \(P_0\). Finally \(c_I(1)=e_I\). Thus the \(F_{I,J}\) are all matrix units in the image of the Clifford action. This is the multimode extension of the rank-one calculation in Example [fcap-001P].

The matrix units prove surjectivity, and the dimension equality \[\dim _K\mathcal {C}\kern -2pt\ell (Q)=2^{2\ell }=(2^\ell )^2 =\dim _K\operatorname {End}_K(S)\] then proves injectivity. Hence \[\mathcal {C}\kern -2pt\ell (Q)\simeq \operatorname {End}_K(S).\] This full equivalence is formalized by TauCeti.spinAction_bijective and TauCeti.SpinPolarizationData.cliffordEquivEnd. The displayed multi-index matrix units are the basis calculation supplied here, not a separately named declaration.

Every factor in \(P_0\) is even, so the Clifford parity of \(F_{I,J}\) is \(|I|+|J|\) modulo \(2\). The even matrix units are therefore exactly those with \(|I|\equiv |J|\pmod {2}\); they span the two diagonal blocks and give \[\mathcal {C}\kern -2pt\ell (Q)^0\simeq \operatorname {End}_K(S^+)\oplus \operatorname {End}_K(S^-),\qquad S^+=\bigwedge ^{\mathrm {even}}W,\quad S^-=\bigwedge ^{\mathrm {odd}}W.\] The exact formal block equivalence is TauCeti.SpinPolarizationData.evenCliffordEquivProdEnd. If \(\ell \geq 1\), each block has size \(2^{\ell -1}\) and each parity summand is simple for its full endomorphism block TauCeti.SpinPolarizationData.evenCliffordEquivProdMatrix. For \(\ell =0\), the full action still identifies \(K\) with \(\operatorname {End}_K(K)\), but \(S^-=0\) and is not called a simple summand.

At the group level, the block calculation alone does not prove irreducibility after restriction to \(\operatorname {Spin}(Q)\). When \(K\) is additionally separably closed and \(\ell \geq 1\), the formal Spin-spanning argument gives the two irreducible half-spin representations TauCeti.isIrreducible_spinPlusSubrep TauCeti.isIrreducible_spinMinusSubrep. This is stronger input than the occupation weights of Theorem [fcap-0025].

[fulton2004representation, Lemma 20.9 and (20.13), pp. 304--305] derive the full complex action and its two exterior-parity blocks; their bilinear form is the half-polar form \(b_Q\), so their contraction coefficient \(2Q_{\mathrm {FH}}\) is our \(B_Q\).

[meinrenken2013clifford, Section 3.2.4, Theorem 3.3 and (3.5)--(3.6), pp. 59--60] give the split action as a superalgebra isomorphism and prove it by the rank-one tensor calculation; their creation space \(F^*\) is our \(W\) and their bilinear form is \(b_Q\).

[goodman2009symmetry, Theorem 6.1.3, proof, pp. 304--305] independently obtain the complex creation--contraction carrier; their creation space \(W_{\mathrm {GW}}^*\) is our \(W\), and their form \(\beta \) is our unhalved \(B_Q\). The formula \(F_{I,J}=c_IP_0a_J\) synthesizes these arguments in the fixed FCAP order rather than quoting one source verbatim.