private CubeAction DetermineBelongingCoset(CubeAction e)
{
var eState = new BlockSet(ToStablize);
e.Act(eState.State);
if (!OrbitToCoset.ContainsKey(eState))
{
return(null);
}
var cosetRepresentative = OrbitToCoset[eState];
if (Utils.ShouldVerify())
{
{
var cosetReprState = new BlockSet(ToStablize);
cosetRepresentative.Act(cosetReprState.State);
Utils.DebugAssert(cosetReprState.Equals(eState));
}
{
// States in orbit 1-to-1 maps to each *left* coset (gH). I.e.
// iff. e^(-1) * cosetRepresentative stablizes the BlockSet being
// observed. This deduces that, group actions in same *left*
// coset, always act the BlockSet being observed to the same state.
var reCosetRep = e.Reverse().Mul(cosetRepresentative);
Utils.DebugAssert(Stablized.IsStablizedBy(reCosetRep));
}
{
// Iff. e * cosetRepresentative^(-1) stablizes the BlockSet being
// observed. This is the condition for *right* coset. It is not what
// we need here, and group actions in same *right* coset, may act the
// BlockSet being observed to different states.
var eRCosetRep = e.Mul(cosetRepresentative.Reverse());
// Utils.DebugAssert(observed.IsStablizedBy(eRCosetRep)); // Doesn't hold
}
}
return(cosetRepresentative);
}