MATH.9:4.2 - Find the permitted answers fixed by the input’s stabilizer

The stabilizer of d is the subgroup

K_d = {g in G | g*d=d}.

For every g in K_d, equivariance requires f(d)=g*f(d). Thus the permitted answers for an equivariant rule at d are

B(d) = {y in A(d) | g*y=y for every g in K_d}.

To construct B(d), first obtain the transformations fixing the supplied data, then solve their fixed-point conditions together with membership in A(d). For a finite supplied group and a finite decidable answer set, enumerate those elements and test them. When either is infinite, use an algebraic description or another available construction; a failed finite search leaves the unsearched range open.

If B(d) is empty, no deterministic equivariant rule can answer this input under these requirements. Return the input symmetry and the fixed-output condition that permitted answers fail. This result concerns the supplied data and output requirement; it identifies what a repair must change.

If the original problem has a unique permitted answer y, its preservation under K_d forces every g*y to equal y. The fixed-point equations can therefore restrict or find that answer. Establish existence and uniqueness on the range used by this deduction. With several permitted answers, use the same fixed-point calculation to find a compatible choice; it need not recover every permitted answer.