Example. The rank-one polarized model [meinrenken2013clifford, Section 3.2.4, Theorem 3.3, p. 59] [fcap-001P]
Example. The rank-one polarized model [meinrenken2013clifford, Section 3.2.4, Theorem 3.3, p. 59] [fcap-001P]
Let \(W=Ke\) and choose \(e'\in W'\) with \(B_Q(e,e')=1\). In the ordered
basis \((1,e)\) of \(S=\bigwedge W\), creation and contraction are
\[\varepsilon _e=
\begin {pmatrix}0&0\\1&0\end {pmatrix},
\qquad
\iota _{e'}=
\begin {pmatrix}0&1\\0&0\end {pmatrix}.\]
Both are nilpotent, and
\[\varepsilon _e\iota _{e'}+\iota _{e'}\varepsilon _e=I_2.\]
Together with their products they give the four matrix units of
\(\operatorname {End}_K(S)\). This is the rank-one calculation in Meinrenken's
generation proof and a specialization of the general relation in
Lemma [fcap-001J]. The formalized ingredients are
TauCeti.SpinPolarizationData.contract_wedge and
TauCeti.ExteriorAlgebra.creation_contraction_adjoin_eq_top;
neither declaration formalizes this displayed choice of basis.