Twelve of the thirty-two are free

Six of the ten plane classes have an invariant ring generated by two polynomials with no relation between them, and the six are exactly those generated by reflections. That is Chevalley’s theorem — with Shephard and Todd — and it is stated in every dimension, which means the interesting thing to do with it is to run it somewhere the answer is not already known by hand.

Three dimensions and thirty-two classes is that place. It is the case a crystallographer actually works in, it is small enough to enumerate completely, and the answer is a number this collection can produce rather than look up.

Twelve of the thirty-two, and the twelve are exactly the classes generated by their mirror planes. The proportion is close to the plane’s — three in five there, three in eight here — and the mechanism is identical.

What “free” means, and why it is worth having #

The polynomials a group leaves alone form a ring: sums and products of invariants are invariants. Every such ring is generated by finitely many polynomials, and the question is whether the generators are independent — whether the only polynomial relation among them is the trivial one.

When they are, the ring is a polynomial ring in the generators and everything about it is arithmetic. The number of invariants of degree n is the number of ways of writing n as a non-negative combination of the generators’ degrees; a physical quantity invariant under the class is a polynomial in three named things; and there is nothing else to know.

When they are not, there is a relation — a syzygy — and the counting stops being a partition problem. Four of the ten plane classes need three generators where two variables can only support two, so exactly one identity ties them together, and the relation is as much a part of the answer as the generators are.

Molien’s series, as integers #

The dimension of the invariants of degree n is the coefficient of t to the n in the average over the group of one over the determinant of 1 − tM. For a three-by-three matrix that determinant is a cubic in t whose coefficients are the trace, the sum of the three principal two-by-two minors, and the determinant — all integers for the integer matrices a lattice supplies.

So each element’s contribution expands by an integer recursion, the contributions are summed as integers, and the average over the group has to divide exactly. A remainder is a bug and never a rounding error, which is a property worth designing for: the whole computation runs in integers and has nothing in it that could be nearly right.

The comparison worth pausing on is 432 against 4̅3m. Both have twenty-four elements and both are the rotation groups of a familiar solid, and one has a free ring and the other does not. No single coefficient of the series distinguishes them; what distinguishes them is whether the whole series factors, and that is a question about the series and not about any of its terms.

Finding the degrees, without assuming them #

A ring is free exactly when its Molien series is the reciprocal of a product of three factors 1 − t^d, one per generator degree. Those degrees satisfy two identities that a reader of the subject knows:

• their product is the order of the group;
• the sum of each degree minus one is the number of reflections.

Both are true and neither is used to find them. The search runs over triples whose product is the order — a short list — and accepts a triple only when its series matches Molien’s term for term. The two identities are then checked on whatever was accepted.

The distinction is not pedantry. A search that used the identities to generate candidates could never report a class where they fail, and would look exactly the same as one that verified them — which is the difference between a computation that could surprise its author and one that cannot.

The degrees themselves are worth reading. The full cubic class m3̅m has degrees two, four and six, which are the three independent isotropic combinations a cubic material’s elastic response is written in. 4̅3m has two, three and four, which is the symmetric group on four letters acting as it does on the tetrahedron. 6mm has one, two and six, and the degree-one invariant is the axis itself — a class with a polar direction has a linear invariant, which is exactly what makes it pyroelectric.

An inversion centre is not a reflection #

The theorem turns on one distinction and it is easy to get wrong.

In three dimensions a reflection is an operation of determinant minus one whose fixed set is a plane — eigenvalues one, one and minus one, which for an integer matrix of finite order is the pair (determinant minus one, trace one). An inversion centre also has determinant minus one, and its trace is minus three, and its fixed set is a point.

So the inversion is not a reflection, and a class containing it is not thereby generated by reflections. 3̅m is the standing example: twelve operations, three of them mirrors, and the mirrors alone generate only 3m of order six. Its ring is not free, and the reason is entirely the inversion.

The general form of the distinction is that a reflection fixes a hyperplane and an inversion fixes a point, and in even dimensions minus one is a product of reflections while in odd dimensions it is not. In the plane the half-turn is a product of two mirrors and in space the inversion is not, which is why the plane’s ten classes and the space’s thirty-two behave the way they do.

What the twelve are #

It is worth naming them, because the list is one a crystallographer half knows already under other headings.

They are the classes whose every operation is a product of mirrors: the trivial class; the single mirror m; mm2 and mmm; 4mm and 4/mmm; 3m; 6mm, 6/mmm and 6̄2m; and the two cubic classes with mirrors, 4̄3m and m3̄m. Twelve, and 3̄m — which has three mirrors — is not among them.

Read as abstract groups they are the finite Coxeter groups that a lattice permits — the reflection groups classified by their diagrams, restricted to the ones whose rotations are crystallographic. The degrees are the Coxeter degrees: two, four and six for m3̄m, which is the diagram B₃; two, three and four for 4̄3m, which is A₃; one, two and six for 6mm, which is G₂ with a free direction.

And the twenty that are missing are missing for two different reasons. Some contain a rotation nothing generates from mirrors — 432, 622, 4, 6, 3, 2 and their kin, the classes with no mirror at all. Others contain mirrors and an inversion, so the mirrors generate a proper subgroup and the inversion is the part left over: 2/m, 4/m, 6/m, 3̄m, m3̄ and 4̄2m among them. The second kind is the interesting one, because a reader counting mirrors would call those classes reflection-rich and the theorem does not.

The two twenty-fours, and what separates them #

432 and 4̄3m are the sharpest pair in the table and worth doing properly, because they are the same abstract group and behave differently.

Both have twenty-four elements and both are isomorphic to the symmetric group on four letters. 432 is the rotation group of the cube; 4̄3m is the full symmetry group of the tetrahedron, which contains six mirrors and the rotations of the tetrahedron and nothing else.

4̄3m has six reflections, they generate it, and its ring is free on degrees two, three and four. 432 has no reflection at all — every one of its twenty-four operations has determinant plus one — so nothing can be generated from its reflections and its ring is not free.

The abstract group cannot see the difference and the invariant ring can. That is the same distinction thirty-two classes, eighteen groups makes from the other side: a class is a group of matrices, an abstract group is a multiplication table, and the classification a crystal responds to is the first. Chevalley’s theorem is a statement about matrices — about which elements fix a hyperplane — and it is not an invariant of the abstract group.

The pair also shows what the reflection count is doing. It is not “how symmetric is this class”; both have twenty-four operations. It is “how much of this class is built out of operations that fix a plane”, and one is built entirely of them and the other not at all.

The two verdicts, computed apart #

Everything above is one computation. The other is a closure.

Take the reflections of a class — the operations with determinant minus one and trace one — and multiply them together in every way until nothing new appears. If what results is the whole class, the class is generated by its reflections. That routine expands a set of matrices and never looks at a polynomial, a series or a degree.

The two verdicts agree on all thirty-two, and the agreement is the point. One route expands a rational function into integers; the other closes a set of matrices under multiplication. They share the list of classes and nothing else, and Chevalley’s theorem is the statement that they must agree — so running both is checking the theorem on a case it was proved for, which is exactly what this collection means by verifying rather than quoting.

A single disagreement would be one of two things, and both are worth stating. Either a counterexample to a theorem proved in 1955, which is very unlikely; or a defect in one of the two computations, which is not. That asymmetry is what makes the check worth running: it is a strong test of the code and a weak test of the mathematics, and the code is what needs testing.

What a degree of one means #

One row of the table has a generator of degree one, and it is worth reading because a linear invariant is a physical statement rather than an algebraic curiosity.

A polynomial of degree one invariant under a class is a direction the class fixes. 6mm has degrees one, two and six, and the degree-one generator is the hexagonal axis: every operation of 6mm leaves that direction alone, so the linear function measuring displacement along it is invariant.

That is exactly the condition for a class to be polar, and the ten polar classes are the ten whose Molien series has a non-zero coefficient at degree one — computed here as ten, from the series, rather than taken from the earlier essay. Six of the ten are also free: 1, m, mm2, 4mm, 3m and 6mm. The other four — 2, 3, 4 and 6 — are polar with no mirror at all, which is why they are polar and not free.

So the degree-one coefficient of the series, which is the cheapest number in the whole computation, is the same quantity that decides whether a crystal can be pyroelectric. A class with a linear invariant may have a permanent dipole; a class without one may not, whatever else it has. That is Neumann’s principle in the one degree where it is easiest to see, and the series carries it as its second term.

What it refuses #

The third is the one that keeps the result from being vacuous. If every class had a free ring, “free exactly when generated by reflections” would be a statement with nothing on one side of it — and twenty of the thirty-two are not free, which is a healthy majority in the interesting direction.

The last is the cheapest and catches the most. The constant polynomials are invariant under everything, so the series must begin at one in every class; and the degree-one coefficient is the dimension of the space of vectors the class fixes, which is three only for the trivial class. Both come out of the same average, and either being wrong means the average was taken wrongly — which is the failure a Molien computation is most likely to have.

Why a free ring is the useful case #

The theorem is stated as a classification and it earns its place by making a calculation short, so it is worth saying which calculation.

A physical property of a crystal is a tensor the class leaves alone — Neumann’s principle — and the count of its independent components is the number of invariants of the appropriate kind. When the ring is free, that count is a partition problem: how many ways to reach degree n from three fixed degrees, which is arithmetic a reader can do without a computer.

When it is not free, the generators satisfy a relation and the count is the partition number minus a correction, and the correction is where the errors live. The plane’s version of this collection reports a class needing three generators in two variables whose single relation has to be found before anything can be counted; in three dimensions the relations are larger.

So the twelve classes are the ones for which “how many invariants of degree n” has an answer somebody can write down, and the twenty are the ones for which it needs the machinery. The difference shows up early: at degree four the free classes’ counts are what three fixed degrees allow, and the others’ are not, which is why the series in the second figure diverge there rather than at degree two. That the twelve are exactly the reflection groups is what makes the distinction predictable rather than a matter of luck: a crystallographer meeting a class with mirrors enough to generate it knows the counting will be easy before doing any of it.

Where the exactness stops #

Computed here: the Molien series of each of the thirty-two classes to eleven terms, as an integer recursion averaged exactly over the group; the reflections of each class by determinant and trace; the closure of those reflections; the degrees of a free ring by matching the series against every triple whose product is the order; and both degree identities on every class the search accepted.

Free is not the same as easy to write down. The theorem says the generators exist and the degrees say what shape they have; it does not produce the polynomials. The plane’s essay does produce them by averaging monomials, and that half is not repeated here — the three-variable version is a larger linear algebra problem and the degrees are what the argument needs.

The classes are the classes and not the groups. A crystal class is a group of matrices up to a change of basis, and the ring of invariants is a ring of polynomials in three variables with a basis chosen. Changing the basis changes the polynomials and not the degrees, so everything reported here is basis-free and nothing that could be written as a formula in x, y and z is.

And the eleven terms are eleven terms. A series matching to eleven and diverging at the twelfth would be reported as free and would not be; nothing suggests that happens, and the two degree identities holding on every accepted triple is the reason to believe it does not, since a spurious match would have no reason to satisfy them.

Where the ladder goes next #

Back, to the plane’s version: the groups whose invariants are free, where six of ten have a polynomial ring and the reflections are what decides it, and how many invariants of each degree, where the same series is computed two ways.

Sideways, to what a relation costs when there is one: three invariants and one relation, where a class needing three generators in two variables is forced into exactly one identity.

Onward, to what the invariants are for: the cubic term that forbids a continuous change, where the third coefficient of a Molien series decides whether a transition can be second order, and neumann’s principle, where a physical property is an invariant of the class and nothing else.