Adequacy and the residual input

Proof of Theorem 4.2

The coordinate map

identifies the displayed matrices with the permutation action on the sum zero subspace of \({\mathbb F}_7^3\). This action is faithful. The four matrices \(1,s,t,st\) span \(M_2\) over every field of characteristic seven: the determinant of their coordinate matrix is \(4\ne0\). A subspace stable under the group is therefore stable under every endomorphism, so the representation is absolutely irreducible. The characteristic polynomials of the identity, transpositions, and three cycles are respectively

Thus all eigenvalues already lie in \({\mathbb F}_7\).

A trace zero matrix fixed by conjugation by both \(s\) and \(t\) is zero, as follows by comparing its four entries. This proves \(H^0(S_3,{\operatorname{ad}}^0V)=0\). For any \({\mathbb F}_7[S_3]\) module \(M\) and one cocycle \(f\), put

Summing \(f(gh)=g f(h)+f(g)\) over \(h\) gives \(gu-u=f(g)\). Hence every one cocycle is a coboundary and \(H^1(S_3,M)=0\), giving both asserted first cohomology vanishings.

Write

The three coordinate columns have determinant \(3\ne0\). Under conjugation,

Consequently \((z,(a,b))\mapsto zZ+aA+bB\) is an equivariant isomorphism from \(\operatorname{sgn}\oplus V\) to \({\operatorname{ad}}^0V\).

The sign witness is \(W_{\mathrm{sgn}}=Z\), while \(W_{\mathrm{std}}=W(6,4,3)=3A+3B\) belongs to the standard summand. The corresponding projectors and trace pairings are

For an operator \(D\) with \((D-\alpha I)(D-\beta I)=0\) and \(\alpha\ne\beta\), the canonical projector is

Its image equals \(\ker((D-\alpha I)^n)\) for every \(n\ge1\), and it kills \(\ker((D-\beta I)^n)\). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.

For detection on every irreducible submodule, one can check the stronger assertion that canonical projectors detect every nonzero matrix. In addition to the two projectors above, use

The four trace pairings with these matrices have trivial common kernel in \(M_2({\mathbb F}_7)\), by row reduction. Every irreducible adjoint submodule contains a nonzero matrix, so some canonical projector has nonzero trace pairing on that submodule. This verifies the last adequacy clause and completes the certificate.

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