TFPT scientific source excerpt
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000157 | Let
000158 | \[
000159 | E=E_-\oplus E_+
000160 | \]
000161 | be the two-factor carrier and let
000162 | \[
000163 | G_{\mathrm{car}}:=S\bigl(U(E_-)\times U(E_+)\bigr),
000164 | \qquad
000165 | S^+=\Lambda^{\mathrm{even}}E.
000166 | \]
000167 | Let $\mathbb Y_{\mathrm{br}}$ denote the branch Yukawa tensor induced by the canonical transport
000168 | kernel and rigid family holonomy of
000169 | \TFPTxref{thm:exact-transport-closure,thm:rigid-family-local-system}. Decompose its carrier-homogeneous
000170 | components as trilinear maps of the form
000171 | \[
000172 | B_{p,q;r,t}:
000173 | (\Lambda^p E_-\otimes \Lambda^q E_+)
000174 | \otimes
000175 | (\Lambda^r E_-\otimes \Lambda^t E_+)
000176 | \otimes E_+
000177 | \longrightarrow
000178 | \mathbb C.
000179 | \]
000180 | Let the primitive component of $\mathbb Y_{\mathrm{br}}$ be the unique indecomposable generator of
000181 | \Cref{def:primitive-yukawa-generator,lem:lowest-primitive-yukawa-generator} with two nontrivial
000182 | fermionic legs. Then, up to exchanging the two fermionic legs, that primitive component is
000183 | necessarily of type
000184 | \[
000185 | (E_-\otimes E_+) \otimes \Lambda^2 E_- \otimes E_+
000186 | \longrightarrow
000187 | \Lambda^3 E_-\otimes \Lambda^2 E_+\cong\mathbb C,
000188 | \]