Three invariants and one relation

Six of the ten plane classes have invariant rings generated by two polynomials with nothing tying them together. The other four — the rotation groups 2, 4, 3 and 6, which contain no reflection — need three.

Three functions of two variables cannot be independent. A map from a two-dimensional space into three dimensions has a two-dimensional image, and that image sits inside some surface: there is a polynomial identity the three generators satisfy, and it is not optional. The ring of invariants is then not a polynomial ring but a hypersurface — polynomials in three variables, modulo one equation.

This essay computes that equation for all four classes. Its point is less the answer than the method: the relation is found as the kernel of a linear map, and then checked at points where every quantity in it is an integer, so that neither the finding nor the checking rests on recognising a familiar formula.

The simplest case first #

The half-turn group, 2, has two operations: the identity and minus the identity. A polynomial is invariant exactly when every term has even degree, so the invariants are spanned by x², xy, y² and their products.

Call those three u, v and w. They are invariant, they generate — any even-degree monomial is a product of them — and they are not independent, because

$$v^2 = uw$$

(xy)² = x² · y², which is true for the plainest possible reason and is the relation. There is exactly one, and every other identity among the three follows from it.

The surface v² = uw is a cone, and the correspondence is exact: the quotient of the plane by the half-turn is a cone, with the fixed point of the rotation at its apex. The orbifold picture of this quotient and the algebraic one are the same object, and the apex is where the orbit is shorter than the group.

How the relation is found #

Recognising v² = uw takes no machinery. The other three cases are not recognisable, and the method has to be general.

The generators are found first, degree by degree, as in the previous essay: at each degree the products of what has already been chosen span a subspace, and anything outside it is new. For 4 that gives u = x² + y² at degree two, then v = x⁴ + y⁴ and w = x³y − xy³ at degree four.

Then the relation. At each degree D in turn, every monomial u^a v^b w^c with 2a + 4b + 4c = D is written out as an actual polynomial in x and y, and the map sending the coefficient vector to that polynomial is formed. The relation is a vector in the kernel of that map, and the first degree at which the kernel is non-trivial is the degree of the relation.

For 4 the kernel first appears at degree eight, and the relation it returns is

$$u^4 - 3u^2v + 2v^2 + 2w^2 = 0$$

which nobody would have guessed, and which is correct. The coefficients are integers because the whole computation is: the generators have integer coefficients, the products do, and the elimination is fraction-free.

Checking it where the arithmetic is exact #

A relation found by elimination has to be checked by something that is not elimination.

The check used here is evaluation at lattice points. At a point with integer coordinates every generator takes an integer value — u, v and w are integer polynomials — so the relation becomes an identity between integers, with no rounding anywhere in it. At (2, 1), for instance, 4’s generators take the values u = 5, v = 17, w = 6, and 625 − 3·25·17 + 2·289 + 2·36 is 625 − 1275 + 578 + 72, which is zero.

Five such points are checked in the figure and more in the site’s gate. A relation that was nearly right — a coefficient out by one — would fail every one of them, and a relation that was right for a wrong reason would have to be right at all of them by coincidence.

This is the same test the round trip applies to a pattern, moved into algebra: a claim gets an input it must handle, chosen so that a wrong answer cannot pass.

The threefold case, and the shape of its surface #

For 3 the generators come out as u = x² − xy + y² at degree two, and two cubics at degree three. That there are two independent cubic invariants is the fact that decides the whole of the Landau ladder later: a threefold group leaves cubic polynomials alone, and no other plane rotation group does.

The relation appears at degree six, and its form is

$$w^2 - 3vw + 9v^2 = u^3$$

after clearing signs — a cubic surface, and the quotient of the plane by a threefold rotation. The apex of that surface is the rotation centre, exactly as the cone’s apex was.

The two sides of this identity have different characters and it is worth saying which: the left is quadratic in the two cubics, the right is cubic in the quadratic, and both are degree six. A relation always has that shape, because the only way three generators of degrees d₁, d₂, d₃ can be dependent is at a degree that is a sum of multiples of each of them.

Why the sixfold case looks worse and is not #

The sixfold group’s generators are of degrees two, six and six, and the two degree-six ones have coefficients running to twenty. The relation is correspondingly heavy.

Nothing has gone wrong. The invariant of degree six for a sixfold rotation is a form with six lobes, and written on the hexagonal basis its coefficients are the binomial-like numbers that appear when the sixth power of a linear form is expanded. What matters is that the identity holds exactly at every tested point, and that the count of generators is three rather than four.

The ugliness is a fact about coordinates, and this collection has made that point before about a physical property. It bears repeating here because the invariant ring is where the temptation to prefer a prettier basis is strongest: the hexagonal basis makes the group’s matrices integers and its invariants ugly, and Cartesian axes make the invariants pretty and the matrices irrational. Only one of those choices keeps the arithmetic exact, and exactness is what this site trades on.

What the numerator counts #

The Molien series of a hypersurface ring has a shape as definite as a free one’s:

$$\frac{1 + t^{e}}{(1 - t^{d_1})(1 - t^{d_2})}$$

with e the degree of the relation. The numerator is the correction for the overcounting: without the relation, products of the three generators would be counted independently, and the relation says that one combination of them is zero.

Reading the numerator that way turns the sequence into a description of the ring rather than a list of numbers. For 2 it is (1 + t²)/(1 − t²)²; for 4, (1 + t⁴)/((1 − t²)(1 − t⁴)); for 3, (1 + t³)/((1 − t²)(1 − t³)); for 6, (1 + t⁶)/((1 − t²)(1 − t⁶)). In each case the exponent in the numerator is the degree of the third generator, which is not a coincidence: for these rings — Gorenstein, in the language of the subject — the relation’s degree and the generators’ degrees are tied together.

None of that is proved here. What is computed is the sequence and the relation, and the shape of the rational function is then checked against them term by term.

The surface is the quotient, and the apex is a special position #

The relation is an equation, and an equation defines a surface. What that surface is has a name in this collection already: it is the plane folded up along its own symmetries, the quotient in which every orbit has become one point.

For the half-turn group the surface v² = uw is a cone. Its apex is the single point where the map from the plane fails to be two-to-one — the rotation centre, whose orbit has one member instead of two — and that failure is what makes the surface singular there rather than smooth. For the threefold group the surface has an apex of a different kind, sharper, matching an orbit of one where the generic orbit has three.

So the singularities of the invariant surface are exactly the special positions of the group, and their type records how short the orbit is. That correspondence runs both ways and is the reason invariant theory and the geometry of quotients are the same subject seen from two ends: one starts from polynomials that do not move, the other from points that do not move.

Counting the relation, and counting the group #

The relation’s degree is not free. For the four classes here it is 4, 8, 6 and 12 — which are, in order, twice the order of 2, twice the order of 4, twice the order of 3, and twice the order of 6.

That pattern is worth stating carefully, because it is the kind of observation that is either a theorem or a coincidence over four cases and this collection is obliged to say which. It is a consequence of a theorem: for a hypersurface invariant ring the numerator of the Molien series is 1 + t^e with e = d₁ + d₂ + d₃ − ... in a form that need not be spelled out here, and the four values above follow. What is computed here is the four degrees; the pattern is read off them afterwards, and no claim is made that four cases establish it.

A useful sanity check comes from the abstract group. The four rings with a relation belong to the four cyclic groups, and cyclic groups are exactly the plane classes whose abelianisation is the group itself. That is not a proof of anything about invariants, but it is a second place the same four-way split appears, computed by a route — Smith normal form on a presentation — with no polynomial in it.

The four rings, side by side #

Setting the four out together makes the pattern in them visible and the difference from the free rings sharp.

Every one has a quadratic generator — the metric — and then a pair of generators of the same higher degree, whose degree is the rotation order except for the half-turn where it is two again. The pair is what the corresponding reflection group does not have: adding a mirror to any of these groups kills one of the two, leaving the free ring of the previous essay with degrees 2 and n.

That is the cleanest way to say what a reflection does to an invariant ring. It does not add invariants; it removes one, and in removing it removes the relation as well.

The relation is not a defect #

It is easy to read the four rows with a relation as the awkward cases and the six free ones as the well-behaved ones. That is backwards in one respect worth naming.

The rotation groups are the chiral ones — they contain no operation that reverses handedness — and their invariant rings carry a generator that a reflection group’s does not: for 4 it is w = x³y − xy³, which changes sign under every mirror the square lattice has. A quantity transforming like w is one that distinguishes a left-handed arrangement from a right-handed one, and it exists precisely because the group is free of reflections.

So the third generator is not an obstruction. It is the invariant that carries the handedness, and it is present exactly in the groups where handedness is a meaningful property of an orbit. Its square is a polynomial in the other two — the relation says so — which is the algebraic form of the statement that squaring a chirality forgets it.

What the pictures cannot show #

Two things.

A surface in three dimensions is not drawn here. The quotient of the plane by a rotation group is a two-dimensional object sitting inside a three-dimensional space of invariant values, and drawing it would need a projection this collection has no convention for. What the figures show instead is the relation as an identity between integers, checked at points, which is the part that can be seen exactly.

Independence is invisible. That two generators of a free ring satisfy no relation is a statement about every polynomial identity there could be, and no picture of any finite number of curves carries it. The absence of a relation is established by the degree-by-degree search terminating without finding one, up to the bound Noether’s theorem provides — which is a proof, and not a drawing.

The four surfaces have names #

The four relations are computed here from kernels and checked at lattice points, and the surfaces they define are not new objects. They are among the most studied singularities in algebraic geometry, and recognising them says why the pattern in the table is a theorem rather than four coincidences.

Complexify the plane and a rotation by 2π/n becomes a diagonal matrix with entries ω and ω̄, whose determinant is one. So the four rotation groups act inside the special linear group, and the four reflection groups do not — a reflection has determinant −1 whatever the coordinates.

That single fact has a consequence. A finite group acting with determinant one has an invariant ring that is Gorenstein, which in two variables forces it to be a hypersurface: three generators and exactly one relation, never more. So the shape of the table — six free rings and four with a single relation — is decided by which groups sit inside the special linear group, and the four with a relation are precisely the four that do.

The surfaces themselves are the Kleinian singularities, classified since Klein’s work on the icosahedron. The cyclic group of order n gives the surface uv = wⁿ, and that is the half-turn’s cone at n = 2 and the threefold’s cubic at n = 3, in coordinates chosen to make it obvious rather than the ones the elimination happens to produce.

That is worth knowing for what it says about the two cases meeting. The reflection groups’ free rings are Chevalley’s theorem; the rotation groups’ hypersurfaces are Klein’s classification; and the boundary between them is the determinant. A reader who has followed the free case and this one has met two large theorems from two centuries, and the only thing separating their domains is a sign.

There is one caution to attach. The classification of Kleinian singularities is by the complex group and this page’s computations are over the reals, so the correspondence is exact for the identification and not for the coordinates: the relation printed here for the threefold group and the textbook’s uv = w³ are the same surface described in two bases, and comparing them means finding the change of variables rather than comparing coefficients. That is the same distinction the essay already draws about ugliness — the surface is an object and its equation is a description of it.

Where this lands #

Four rings, four relations, each computed from a kernel and checked at integer points. The classification of the plane classes by their invariant rings is complete, and it agrees with the classification by reflections that has nothing to do with polynomials.

The count of cubic invariants that fell out along the way — two for 3, one for 3m, none for anything else — is the input to a question about matter rather than about polynomials: whether a crystal may lose a symmetry gradually or must lose it all at once. That is Landau’s condition, and the number it turns on is a Molien coefficient computed here for an entirely different reason.

There is a small moral in the two theorems sitting either side of one determinant. Neither was proved for crystallography and neither mentions a lattice; both were answers to questions about polynomials, asked seventy years apart in different subjects; and between them they decide the shape of every invariant ring a plane crystal class has. That is the ordinary situation on this site rather than a remarkable one, and it is worth noticing chiefly because the alternative — a classification proved case by case over ten groups — would have looked equally convincing and would have said nothing about why.