An orbit is what the invariants cannot tell apart

An invariant takes the same value on every point of an orbit. That is what invariance means, and it makes an invariant a device for forgetting which member of an orbit is in hand.

The question this essay is about is whether the forgetting is complete. Given the values of every invariant at two points, is it possible to tell that the two are in different orbits? If not — if two points in different orbits can share every invariant value — then the invariants are a lossy description of the quotient, and a physical quantity computed from them cannot distinguish two arrangements a crystallographer would call different.

For a finite group acting on real space the answer is that the invariants do separate, always. The interesting part is what the statement costs to check and what a proper subset of the invariants does instead.

What is being asked #

The plane, with a finite group acting on it, has a quotient: the set of orbits, one point of the quotient per orbit of the plane. This collection has met the quotient before, as the plane folded up along the group’s own symmetries, and as the fundamental domain that contains one point of each orbit.

Invariants give a third description of the same object, and an algebraic one. Every invariant is a function on the quotient — constant on orbits, so well defined on the set of them — and the collection of all invariants is a set of coordinates for it. Whether those coordinates are faithful is exactly the separation question.

The distinction matters because the three descriptions are not interchangeable in practice. A fundamental domain requires choosing one, with boundaries that have to be decided; the folded picture is geometric and hard to compute with; the invariants are polynomials, and a computation that uses them needs no choices at all.

One direction is free #

If two points are in the same orbit, every invariant agrees on them. That is the definition of invariance and needs no proof: f(Mv) = f(v) for every operation and every invariant f, so the values at v and at any image of v coincide.

The direction with content is the other one. Two points with the same invariant values — must they be in the same orbit?

For a finite group over the reals the answer is yes, and the argument is short enough to sketch. Suppose p and q are in different orbits. The orbit of p is a finite set of points not containing q, so there is a polynomial vanishing on all of the orbit of p and equal to 1 at q. Average that polynomial over the group: the result is invariant, still zero on the orbit of p — every term vanishes there — and still non-zero at q, because averaging a function that is 1 at q over a group that never sends q into the orbit of p cannot cancel it away. That invariant separates them.

Averaging is doing the work, and it is the same averaging trick that gives a finite group an invariant metric and a fixed point. The argument fails over a finite field, where the average may divide by the characteristic, and this collection makes no claim there.

How it is checked here #

The theorem is imported. What is computed is an instance of it, and computing an instance is worth more than restating a proof, because it is a test the code can fail.

The test runs over the lattice points within a window of the origin. For each point, two keys are computed. The first is the orbit key: the sorted list of images of the point under every operation, which is a complete description of the orbit and is compared as exact integers. The second is the invariant key: the values of the group’s fundamental invariants at the point, again exact integers, since the invariants have integer coefficients and the points have integer coordinates.

Then the two partitions of the window — by orbit key and by invariant key — must be identical. Not merely the same number of parts: the same parts. And they are, on every class, over every window this has been run on.

A test that only passes proves nothing #

The site’s standing habit is that an assertion which has never rejected anything is not evidence. So the same routine is asked a question it must answer differently: drop one of the invariants and separate again.

With a generator removed the partitions come apart, and the routine returns a witness — an explicit pair of points, in different orbits, whose remaining invariant values agree. For 3m with the quadratic removed, that pair is (−4, −4) and (−3, −3); for 4mm with the quartic removed, two points at the same distance from the origin but not related by any operation of 4mm.

And sometimes the witness is a very long way out. Drop 4mm’s quadratic instead, leaving only x⁴ + y⁴, and no pair inside any window a figure can draw fails: two orbits agree on that invariant only when two sums of fourth powers agree, and the smallest instance of that is Euler’s 59⁴ + 158⁴ = 133⁴ + 134⁴, both equal to 635,318,657. The subset separates every orbit within a hundred and forty cells of the origin and fails at a hundred and fifty-eight — which is a useful reminder that a check passing on every case anybody looked at is not a proof, and that the search bound is part of the result.

The witness is the useful output. It says not just that a subset fails but where, and the where is instructive: for a subset consisting of the metric alone, the failure is between any two points at equal distance in different directions, which is the plainest possible statement that distance from the origin is not a complete description of position.

Why the count of invariants is the count that matters #

The number of generators of the invariant ring is not decorative. It is the number of coordinates the quotient needs, and the free rings are exactly those whose quotient needs no more than the plane did.

For a reflection group the ring has two generators, so the quotient is described by two numbers with no relation between them: it is a plane again, and the map from the original plane to it is |G|-to-one away from the special positions. For a rotation group three generators are needed, tied by one relation, so the quotient sits inside three-dimensional space as a surface — the same set of orbits, needing an extra coordinate to be described faithfully.

That difference is invisible in the separation test, which passes for both. It shows in what the invariants are: a free ring gives coordinates, a hypersurface ring gives coordinates plus a constraint.

The threefold group, worked through #

One case in full shows what the values actually look like, and 3 is the right one because its ring is not free and its orbits are small.

Its three generators are the quadratic x² − xy + y² and two cubics. At the point (2, 1) they take the values 3, 2 and 3; at (1, 2) — which is not in the same orbit, since the group has no operation carrying one to the other — they take 3, −2 and −1. The quadratic agrees, and both cubics disagree, which is the separation happening.

The quadratic agreeing is not an accident: (2, 1) and (1, 2) are the same distance from the origin in the hexagonal metric, being related by a mirror the group does not have. So the invariant that captures distance cannot distinguish them, and the two cubics are exactly the part of the ring that notices handedness. Remove them and the group’s quotient collapses onto the quotient by 3m — the larger group that does contain the mirror.

That is the general shape of the phenomenon. A subset of the invariants separates the orbits of some group, and which group is the one whose invariants that subset spans. Dropping invariants does not blur the picture arbitrarily: it replaces the group by a larger one.

Separating a pattern rather than a point #

The whole of this collection’s verification rests on a separation claim of a different kind, and it is worth setting the two side by side.

The round trip takes a point set, forgets the group that made it, and asks which operations map the set to itself. That is a question about a set rather than about a point, and what it separates is patterns: two point sets with the same detected group are indistinguishable to the detector, and two with different groups are told apart.

Invariant separation is the same idea one level down. Two points with the same invariant values are indistinguishable to any symmetric function; two with different values are told apart. The detector is to patterns what the invariant ring is to points, and both are complete: neither loses information the group does not already lose.

Where they meet is in the failure mode the whole site is built around. A motif at a special position produces a pattern with more symmetry than intended — a dot is not a comma — and in invariant language that is a point whose invariant values are those of a larger group’s orbit. The two descriptions of that failure are the same description.

How much of the ring is needed #

If the full set of generators separates and a set with one removed does not, the natural question is whether there is some other small set that does.

For the plane classes the answer is that the generators are minimal: each is needed, and the witness produced when it is dropped proves it. That is not automatic — a ring can have a generating set that is larger than a separating set, and in higher dimensions the two counts genuinely differ — but in two dimensions the generators of these rings are also a minimal separating set, and the drop test establishes it case by case rather than by an argument.

The general question is a live one in invariant theory: the smallest number of polynomials needed to separate the orbits of a group can be smaller than the number needed to generate its invariant ring, and finding it is hard. Nothing here claims otherwise; what is claimed is the two-line statement checked above, that for each of the ten plane classes the computed generating set separates and each of its proper subsets does not.

Where separation is delicate: the special positions #

Away from the special positions every orbit has |G| members and the map to invariant values is |G|-to-one. At a special position the orbit is shorter, and the map behaves differently there: the invariant values still identify the orbit, but nearby orbits crowd together in a way that a numerical version of this test would blur.

That is a real difficulty for anybody doing this in floating point, and it is one of the reasons the arithmetic here is exact. Two orbits near a special position can have invariant values agreeing to many decimal places and still be different orbits, and a comparison with a tolerance will merge them. The exact comparison cannot.

The place this bites in practice is not invariant theory but structure determination, where the analogous question — are these two sites the same site — is asked of measured coordinates with error bars, and the answer depends on a threshold. This collection has an essay about what a tolerance does to a decidable claim, and separation near a special position is the same difficulty in another notation.

Orbits, counted and separated #

Two questions about orbits appear in this collection and they are different questions.

How many orbits are there? That is a count, answered by Burnside’s lemma — the average over the group of the number of points each operation leaves fixed — and it needs no invariant at all.

Which orbit is this? That is a separation question, answered by evaluating invariants, and no count can answer it.

The two live side by side and each is useless for the other’s purpose. A count says a colouring problem has 46 answers; it does not say which of them a given colouring is. A set of separating invariants identifies the colouring; it does not say how many there are without an enumeration.

What a physical quantity computed from invariants can know #

The reason to care is that a physical quantity invariant under a crystal’s symmetry is a function of the invariants, and separation says exactly how much such a quantity can distinguish.

It can distinguish two arrangements that are in different orbits — because some invariant separates them, and any function of the invariants may be built to notice. It can never distinguish two arrangements in the same orbit, since every invariant agrees on them. So the answer to what can a symmetric measurement tell apart is: precisely the orbits, no more and no less.

This is the sharp form of a statement that turns up loosely all over crystallography — that a measurement respecting the symmetry cannot see anything the symmetry moves. It also explains the standing difficulty with diffraction: an experiment measuring intensities is invariant under more operations than the crystal has, so it separates fewer things than the structure does, and what it merges is a fact about the experiment’s group rather than about the crystal’s.

What the pictures cannot show #

The figures colour points by invariant values and circle a witness pair. Three things they do not show:

They do not show every invariant. The separation test uses the fundamental generators, from which every invariant is built; a figure showing “all invariants” is not a thing that exists. The claim being illustrated is about the generated ring, and the generators stand in for it.

They do not show the quotient. The set of orbits is a surface in a space of invariant values, and drawing it needs a projection this collection has no convention for. What is drawn is the domain rather than the image.

They do not show the theorem. Every window is finite, and no finite window establishes a statement about all points. What the figures establish is that the code implements the theorem correctly on the cases shown, and that a proper subset of the invariants fails on a case exhibited by name.

Where it leaves the ladder #

Separation is what makes the invariant ring a description of the quotient rather than a collection of functions that happen to be constant on orbits. With it, the ring is the algebra of the quotient and every question about orbits becomes a question about polynomials.

The next place that pays off is not in this field. It is in what a crystal may do when it loses symmetry: the possible low-symmetry phases are orbits of directions in an order-parameter space, and telling them apart — deciding whether two apparently different distortions are the same one seen twice — is exactly a separation question, asked of a group acting on a space of order parameters rather than on the plane.

Where separation fails, and why finiteness is the hypothesis #

The theorem is stated here for a finite group, and it is worth knowing that the hypothesis is not decoration — because the natural generalisation is false and the failure has a whole subject built on it.

The proof needs the average. Given two points in different orbits, one constructs an invariant separating them by averaging a function over the group, and the average exists because the group is finite and can be summed over.

For an infinite group there is no such sum. The invariants of a continuous group acting on a space generally fail to separate orbits, and the failure is structural rather than accidental: an invariant is a continuous function, so it takes the same value on an orbit and on everything in that orbit’s closure — and if one orbit lies in the closure of another, no invariant can tell them apart.

So the correct general statement is that invariants separate closed orbits, and the non-closed ones are collapsed together. That is the starting point of geometric invariant theory, and it is why the quotient of a space by an infinite group is a subtler object than the quotient by a finite one.

None of that arises here, and stating it says exactly why: every group in this collection is finite, so every orbit is finite, so every orbit is closed, and the theorem applies with nothing to check.