How many invariants of each degree

The previous ladder asked what a group does to a function, and answered it with characters: the space of functions splits into pieces the group keeps separate, and the sizes of those pieces are integers computed from traces. This one asks the smaller and older question inside it. Which functions does the group not move at all?

Such a function is an invariant: a polynomial f with f(Mv) = f(v) for every operation M of the group. There are always some — constants, for a start, and the sum of the squares of the coordinates for any group of rotations and reflections — and the useful question is how many independent ones there are at each degree. That count is a sequence of integers, one per degree, and almost every statement this collection makes about what symmetry decides for a physical property is a statement about its first few terms.

The counting is a rational function #

Molien’s theorem gives the whole sequence at once, as the expansion of

$$\Phi(t) = \frac{1}{|G|}\sum_{g \in G} \frac{1}{\det(I - tM_g)}$$

whose coefficient at tᵈ is the number of independent invariants of degree d. Nothing about the statement is obvious, and the shape of it is worth pausing on: a sum over the group of one over a determinant, and out of it comes a count of polynomials.

In two dimensions that determinant is small enough to write down. For a two-by-two matrix, det(I − tM) = 1 − (tr M)t + (det M)t², so every term of the sum is one over a quadratic whose two coefficients are the trace and the determinant of a single operation. Both are integers, because every operation of a plane point group is an integer matrix in the lattice’s own basis.

That makes the expansion an integer recursion. Writing aₙ for the coefficients of one term,

$$a_n = (\operatorname{tr} M)\,a_{n-1} - (\det M)\,a_{n-2}, \qquad a_0 = 1$$

and the answer for the group is the sum of those sequences divided by the order. The division has to come out exactly, and that is the first thing this machinery checks: a remainder is not a rounding error but a statement that the input was not a group.

What a non-group produces #

The check earns its place, because the failure it catches is quiet. Hand the recursion a set of matrices that is not closed — a four-fold rotation without its own square, say — and nothing goes wrong. The traces exist, the determinants exist, the recursion runs, and the sum divides by two at every degree, giving a sequence of perfectly respectable whole numbers that are the dimensions of nothing whatever.

So the closure is tested by multiplication before any series is computed, rather than inferred afterwards from the answer looking plausible. That is the same discipline the round trip applies to a pattern: a computation whose failure mode is a believable wrong answer needs a test that is not the answer.

The second route, which produces the polynomials #

The trace formula counts invariants without ever producing one. The other route produces them, and it is the averaging projector applied to polynomials rather than to a metric.

Take a monomial of degree d, push it through every operation of the group, and add the results. The sum is invariant by construction — applying another operation permutes the terms — and it is |G| times the projection of the monomial onto the space of invariants. Do that to every monomial of that degree, take the rank of what comes back, and the rank is the dimension of the invariant space.

That is exact integer linear algebra on polynomials, sharing no line of code with the recursion above: one substitutes linear forms into a polynomial and eliminates, the other multiplies traces. They agree at every degree of every one of the ten classes, and the second hands back a basis a figure can draw.

The arithmetic is done in arbitrary-precision integers, and not out of caution. Averaging a degree-twelve monomial over the twelve operations of 6mm on a hexagonal basis produces coefficients in the tens of thousands, and the elimination that takes the rank multiplies them again. A rank computed in floating point is a rank that can be wrong without saying so, and the failure would be a dimension — a whole number, believable, and out by one.

Reading the sequence #

A few features of the sequence say something immediately.

The value at degree two is one for every class but two. A group with a rotation of order three or more fixes exactly one quadratic form up to scale, and that form is the metric — the statement that a finite group of matrices preserves a metric, which this collection proves by averaging, read off a Molien coefficient. For 1, 2, m and 2mm the value is larger, and those are exactly the classes whose lattices have a shape a symmetry argument does not fix.

A zero is a prohibition. Where the coefficient is zero there is no invariant of that degree at all, so a physical quantity whose symmetry is that of a degree-d form is required to vanish. Every statement of the form this class cannot be piezoelectric is a zero in a sequence of this kind, and Neumann’s principle is the assertion that the physics has to obey the arithmetic.

The sequence grows, and how fast is a fact about the order. Far out, the number of invariants of degree d is about d/|G|: the space of forms of degree d has dimension d + 1, the group acts on it, and the invariants are one orbit’s worth. The largest groups have the sparsest sequences, which is the sense in which more symmetry means fewer permitted quantities.

The polynomials are not the counts #

The counts are a fact about the group; the polynomials are a fact about the coordinates, and this collection has made the distinction before about elastic constants.

Molien’s formula uses each matrix only through its trace and its determinant, and both survive a change of basis. So the counts are the same whatever coordinates the group is written in — including the lattice basis, where a threefold rotation is the integer matrix with entries 0, −1, 1, −1 and looks nothing like a rotation.

The invariants themselves are not basis-free at all. On a hexagonal basis the quadratic invariant of 3 is x² − xy + y², not x² + y², and a reader who expects the second and finds the first has met the basis rather than an error. The two polynomials describe the same curves; the coordinates differ.

The whole census #

Doing this for all ten plane classes gives the table the rest of this ladder argues from. It is worth setting out here in full, because several later claims are single columns of it.

Two facts stand out of it and each gets an essay of its own. The six classes with a free ring are exactly the six generated by reflections, which is Chevalley’s theorem and is checked here rather than quoted. And the four without reflections need a third generator, so a relation appears — an identity the three generators satisfy, found as the kernel of a map rather than looked up.

What the invariants are for #

The reason to count invariants rather than to admire them is that a physical property with a symmetry is an invariant of that symmetry, and the count of independent invariants of the right degree is the count of numbers a measurement can return.

A fully symmetric property tensor of rank r is a form of degree r in disguise: its components are indexed by a multiset of r directions and the polynomial’s coefficients are indexed the same way. So the number of independent components a class permits for such a property is a Molien coefficient, and the table of permitted components is a table of these sequences read down a column.

That correspondence stops where a property’s symmetry is not that of a form, and the elastic tensor is the standard place it stops. Its indices are symmetric within each pair and under exchanging the pairs, which is not the symmetry of a degree-four polynomial, and its counts are not in the table. A property with its own symmetries needs its own average.

One case, worked by hand #

The recursion is short enough to run without a computer, and doing it once makes the rest of this ladder concrete.

Take 2mm, which has four operations: the identity, the half-turn, and two mirrors. Their traces are 2, −2, 0 and 0; their determinants are 1, 1, −1 and −1. So the four sequences are

• the identity, aₙ = 2aₙ₋₁ − aₙ₋₂, giving 1, 2, 3, 4, 5, …;
• the half-turn, aₙ = −2aₙ₋₁ − aₙ₋₂, giving 1, −2, 3, −4, 5, …;
• each mirror, aₙ = aₙ₋₂, giving 1, 0, 1, 0, 1, ….

Averaging the four at each degree gives 1, 0, 2, 0, 3, 0, 4 — one invariant at degree zero, none at degree one, two at degree two, and so on up by one every second degree. Those two quadratic invariants are x² and y², which is the statement that a rectangular lattice’s shape is not fixed by its symmetry: the two axes may be scaled independently and 2mm does not object.

The identity always contributes the sequence 1, 2, 3, 4, …, since its term is 1/(1 − t)², and every other operation contributes something smaller. So the leading behaviour of every one of these sequences comes from a single operation, and the rest of the group is a correction. That is the arithmetic form of the statement that a group of order |G| divides the polynomials into orbits of size at most |G|.

A sequence that is a rational function #

The sequences here are not arbitrary. Every one of them is the expansion of a rational function whose denominator is a product of factors 1 − tᵈ — the generating function of a graded ring — and the exponents in that denominator are the degrees of a set of generators.

That is a strong statement and it is checked rather than assumed: for each class, every pair of degrees whose product is the order of the group is tried, and the pair whose series matches the Molien coefficients term by term is reported. Where no pair matches, the ring is not free and the routine says so instead of returning a plausible pair.

Reading a denominator as a list of degrees is the move that makes the next two essays possible, and it is also where the crystallographic restriction turns up in a form nobody would have predicted: the degrees of the reflection group with an n-fold rotation are 2 and n, for every n, and only five of those groups belong to a lattice.

What is proved and what is checked #

The distinction matters here as much as anywhere in this collection, because two of the three statements above are theorems and the third is an agreement between computations.

Molien’s theorem is a theorem, proved in 1897 and not proved here; what is done here is to evaluate it. The claim that the coefficients count invariants is imported.

The agreement of the two routes is a measurement. The averaged-monomial computation is an independent implementation of the same count, run at every degree up to six on every class, and it agrees. That is evidence about the code rather than about mathematics: it would have caught a sign error, a mis-transposed matrix, or an element list missing an operation, and it did catch the last of those while this was being written.

The closure test is a refusal. It is the assertion that runs when the input is wrong, and it is fed a set that is not a group as part of the site’s gate, so that it cannot quietly stop working.

The habit this collection runs on is that every claim gets a test it could fail. For a count of invariants the test cannot be a picture — a polynomial’s dimension count is not visible in a drawing — so it has to be a second computation, and the second computation has to share nothing with the first. Two implementations of one formula agreeing proves that the formula was typed twice.

Why there is a finite list of generators at all #

The census prints, for each class, the degrees of the generators of its invariant ring. That there is a finite list is not obvious from anything above — the counts grow without limit, so there are invariants of every large degree — and the fact that finitely many of them generate the rest is a theorem with a history worth telling.

Hilbert proved it in 1890, for a much wider class of groups than these, and the proof was famously indirect: it establishes that a finite generating set exists without producing one, by an argument about ideals in a polynomial ring. Paul Gordan, who had spent years computing generating sets by hand for particular cases, is supposed to have said that it was not mathematics but theology.

The remark is unfair and the complaint underneath it is not. An existence proof does not tell a computation when to stop. A search for generators that has found several and is looking for more has no way, from Hilbert’s theorem alone, to know whether it is finished — which is the same difficulty the site’s other searches have wherever a bound is not available.

The answer for a finite group came later. Noether’s bound, from 1916, says the invariant ring of a finite group of order |G| is generated in degrees at most |G|. That is a computable stopping rule: search up to degree |G|, and anything not generated by what has been found by then does not exist.

For the plane classes that bound is small. The largest of them has order twelve, so the search runs to degree twelve, and the generators all turn up far below it — degrees two, four and six for the largest classes, and the census reports them. So the finiteness is Hilbert’s, the stopping rule is Noether’s, and the degrees are this page’s, and only the last of those three is computed here.

What the counting cannot do #

One boundary is worth stating plainly, because the sequence looks like more of an answer than it is.

The Molien series counts invariants of each degree, and it counts them as a vector space — how many linearly independent polynomials of that degree are fixed by the group. The ring structure is invisible to it: the series does not say which invariants are products of smaller ones and which are new.

That distinction is the whole of the generator question. An invariant of degree four may be the square of a quadratic invariant, in which case it is nothing new, or it may be independent of everything below it. The counts are the same either way, so the sequence alone cannot distinguish a class whose invariants are all built from one quadratic from a class with a genuinely new invariant at every even degree.

Separating them takes the second route, the one that produces the polynomials rather than counting them: build the products of the known generators at each degree, compare their span against the count, and any shortfall is a new generator. That is a computation over polynomials rather than over integers, it is what fills the census’s degree column, and it is the reason two independent implementations are worth having — the recursion says how many there should be, and the linear algebra says which ones they are.

Where the ladder goes next #

Counting is the first rung and the least of it. The sequence has structure — it is the expansion of a rational function whose denominator is a product of terms 1 − t^d, and those d are the degrees of a set of generators — and that structure is where the subject becomes interesting.

The next two essays take it apart: which groups have generators that are algebraically independent, and what happens in the four that do not. After them the same arithmetic decides two things that look nothing like polynomial algebra — whether a transition may be continuous, and which rotation orders a lattice permits, read off the degrees rather than off the traces.