This is certainly in the same spirit, but not quite the same, as in principle there could be another homogeneous poly of degree m in m^2 variables with a stabilizer group abstractly isomorphic to $G_P$. (Indeed, any poly equivalent to perm satisfies this, but that's rather trivial. But I don't know how to rule out other polys with abstractly isomorphic stabilizer.) Also, such polys are very special, whereas I thought an important part of the question was about whole categories.

@VladimirDotsenko: The relationship with semi-direct product is closer than your last remark hints at: $\alpha$ must be a left action of $K$ on $H$, and $\beta$ must be a right action of $H$ on $K$. (And then those two actions must satisfy some compatibility assumption.) And indeed, if either of these actions is trivial, then one recovers the semi-direct product.

Not exactly this question, but related: Barvinok showed how to approximate the permanent of real/complex matrices close to J math.lsa.umich.edu/~barvinok/stab.pdf

In 2006 Schonhage disproved Collins's conjecture for degree 3: sciencedirect.com/science/article/pii/S0747717106000472 (though it could still be true asymptotically in the degree).

@StellaBiderman: I suppose it may not seem particularly deep in the sense of having impact on many other seemingly unrelated problems. But from a different standpoint, I think it is a deep, fundamental question, whose techniques rely on and have inspired decades worth of research into the theory of permutation groups...

The fact that 4 is the first dimension n at which every f.p. group can be realized as the fundamental group of a smooth n-manifold is interesting - and since it implies that homeomorphism of 4-manifolds is uncomputable, that implies a certain amount of "wildness" at 4 and beyond - but does this really distinguish 4 from higher numbers? That is, why its its status as the first such dimension relevant? (By analogy: 3 is the first k such that k-SAT is NP-complete, but 3-SAT is not particularly special compared to 4-SAT, 5-SAT, etc.)

Nice! I think I take this as saying that, while truncated ideals make sense and can be very useful, the notion of a truncated ideal being radical isn't really a good "truncated" version of an ideal being radical...

I seem to recall reading somewhere that cobordism (as described in this answer) was actually Poincare's original motivation, but for a long time the world had to settle for homology because that's all that they could calculate. (The OQ asked for history as well, so I thought it worth mentioning.)

Just to clarify: It's not exactly equivalent to just using equations and inequations; without additional boolean operations, this just gives you the affine locally closed sets, but a general constructible set is a union of locally closed sets.

@RobertIsrael: Sorry, of course I was being silly. I was thinking "diagonalizable", not normal. But I think the point of my previous comment is still valid: the normal case (or even the diagonalizable case) seems to miss some of the essential difficulties of the problem.

A closed form solution for the max ent distribution with given marginals (beyond 1-marginals) is fairly well-unknown. (By "well-unknown" I mean "well-known that it's not known." It might even be known that no such thing exists... From a quick google, see p. 10 here.) Using Lagrange multipliers you can always write it in an exponential form, but the terms appearing in that exponential formula don't have simple closed formulae...

Although normal matrices are dense, the real difficulty in this problem stems from non-normal matrices (as your answer shows). I don't think there is any such simple criterion for general matrices...