Crystallography & Symmetry
patterns derived from their groups
What symmetry decides

The groups whose invariants 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 their own reflections. The degrees of those generators multiply to the order of the group, and their excess counts the reflections.

Assumes How many invariants of each degree, What a symmetry actually is and Three reflections, and never four.

15 min read 8 figures The group acts on functionsExactly this manySymmetry is decidable

Counting invariants degree by degree gives a sequence of integers. The sequence has a shape, and the shape is the interesting object: for six of the ten plane classes the whole of it is the expansion of

1(1−td1)(1−td2)\frac{1}{(1 - t^{d_1})(1 - t^{d_2})}

which says that the invariants are exactly the polynomials in two particular ones, with no further relation between them. The invariant ring is then as simple as a ring can be — a polynomial ring in two variables of its own — and the two degrees are the only data.

The other four classes have no such expression, and the difference between the two lists is not a matter of size or of how much symmetry each has. It is a matter of whether the group contains reflections.

Six of the ten plane classes have a free invariant ring, and four do not. Every plane point group, with the degrees of the generators of its invariant ring, whether the ring is free, and the relation where it is not. The six generated by their own reflections — 1, m, 2mm, 4mm, 3m and 6mm — have two generators whose degrees multiply to the order of the group, which is Chevalley's theorem checked rather than quoted. The four without reflections — 2, 4, 3 and 6 — need three generators in two variables, so one polynomial relation ties them together, and the degree that relation appears at is printed at the right of its row. Nothing here is a lookup: the generators are found degree by degree as the invariants the earlier ones do not reach, and the relation is the kernel of the map back to polynomials.
Fig. 1 Every plane point group with the degrees of its invariant generators, whether its ring is free, and the relation where it is not. The six free rings belong to 1, m, 2mm, 4mm, 3m and 6mm; the four with a relation belong to 2, 4, 3 and 6. The two lists are the groups with and without reflections.

What free means, and why two is the ceiling

A ring of invariants is free — or polynomial — when it has generators that satisfy no polynomial relation at all, so that every invariant is a polynomial in them in exactly one way. For invariants of the plane the number of such generators can never be more than two: three functions of two variables are always algebraically dependent, because a map from a two-dimensional space cannot have a three-dimensional image.

So there are only two possibilities. Either two invariants generate everything, and the ring is free; or three are needed, and then exactly one relation ties them together. There is no third case, and the count of generators is the whole of the classification.

Which case a group falls into is decided here rather than looked up. The generators are found degree by degree: at each degree the products of the generators already chosen span a subspace of the invariants, and anything the products do not reach is new. The search stops at the order of the group, which is Noether’s bound, and for these groups nothing new ever appears above degree six.

Chevalley’s theorem, and the check it becomes

The pattern in the table is a theorem. Chevalley proved in 1955 — with Shephard and Todd’s classification of the groups it applies to — that a finite group acting on a space has a polynomial invariant ring exactly when it is generated by reflections.

In the plane a reflection is easy to recognise: it has determinant −1, and that is the whole of the condition, since such a matrix has eigenvalues +1 and −1 and therefore fixes a line pointwise. So the test is arithmetic. Collect the operations of determinant −1, close them under multiplication, and ask whether the closure is the whole group.

That gives two independent computations to compare, and comparing them is what makes this a check rather than a definition:

  • the algebraic side — how many generators the invariant ring needs, found by the degree-by-degree search;
  • the group side — whether the reflections generate, found by closing a set of matrices.

They agree on all ten classes. The four groups without reflections — 2, 4, 3 and 6 — need three generators; the six with reflections need two. Neither computation was told the other’s answer.

4mm against 4: the same order, different invariants. The invariant dimensions of 4mm and 4 at each degree, side by side. 4mm has a free ring with degrees 2 and 4; 4 carries a relation among three generators. A group with a reflection and one without can have the same order and still differ at every degree, because what decides the count is not how many operations there are but how they act — and the difference is visible from the third degree onwards.
Fig. 2 4mm against 4 — a reflection group and the rotation group inside it. Their sequences differ from degree four onwards, and the difference is exactly the extra invariant that 4mm’s mirrors move. The invariants of the smaller group are more numerous, which is the opposite of the usual intuition about symmetry.

The degrees multiply to the order

The numerical half of Chevalley’s theorem is the part that can be checked at a glance, and it is where the arithmetic becomes surprising: for a free ring,

d1d2=∣G∣d_1 d_2 = |G|

The degrees of the generators multiply to the order of the group. For m that is 1 × 2 = 2; for 2mm, 2 × 2 = 4; for 3m, 2 × 3 = 6; for 4mm, 2 × 4 = 8; for 6mm, 2 × 6 = 12. And for the trivial group, 1 × 1 = 1, which is not a degenerate case but the statement that the coordinates themselves generate.

There is no obvious reason why a product of two polynomial degrees should be the number of operations in a group. The reason it holds is that the map sending a point to the values of the two generators is |G|-to-one — every orbit of the group is a single point of the image, and a generic orbit has |G| members — and the degree of such a map is the product of the degrees of its components. So the identity is the orbit–stabiliser theorem seen through the invariants, and it is checked in the table above for every free case.

The second identity is about reflections rather than about the order. For a free ring,

∑i(di−1)=the number of reflections\sum_i (d_i - 1) = \text{the number of reflections}

which for 6mm reads (2 − 1) + (6 − 1) = 6, and 6mm has exactly six mirrors. For 4mm it is 1 + 3 = 4 against four mirrors; for 3m, 1 + 2 = 3 against three; for m, 0 + 1 = 1 against one. Every class checks.

Which polynomials they are

The generators can be written down and they are recognisable, which makes the theorem less abstract than its statement suggests.

For every class with a rotation of order three or more, the first generator has degree two and is the invariant quadratic form — the metric the group preserves, which is why a finite group of matrices has one at all, written in the lattice basis, so x² + y² for the square lattice and x² − xy + y² for the hexagonal one. The second generator is where the group’s own character shows: it has degree n for the group with an n-fold rotation, and it is the polynomial whose level curves have n lobes.

For 6mm the second generator is a degree-six form, and its level curves look like the outline of a snowflake. For 3m it is a cubic and its curves have three lobes. For 2mm both generators are quadratic — x² and y² — which is the statement that a rectangular lattice’s two axes are independent, and the reason a rectangular cell has a shape while a hexagonal one does not.

The level curves of 6mm's invariant of degree 6. The curves on which the invariant 2x⁶ − 6x⁵y + 15x⁴y² − 20x³y³ + 15x²y⁴ − 6xy⁵ + 2y⁶ takes the values 1, 4 and 12, drawn in the lattice's own basis with the basis vectors marked and the mirror lines of 6mm across them. The middle curve is drawn solid and the others faintly. The polynomial is invariant under every operation of the group — checked by substituting each matrix into it and requiring the coefficients to come back unchanged — so every one of those operations carries each curve onto itself. On a hexagonal basis the quadratic invariant is x² − xy + y² rather than x² + y², and its level curves are still circles: the polynomial is a fact about the coordinates, the curve is a fact about the plane.
Fig. 3 The level curves of 6mm’s degree-six generator, drawn in the hexagonal basis with the group’s six mirror lines across them. Every operation of the group carries each curve onto itself. The invariance is checked by substituting the matrices into the polynomial and requiring the coefficients back unchanged, rather than by looking at the picture.

The point of a free ring

A ring being free is not a technicality about presentation. It decides whether a question about invariants can be answered by writing down a general polynomial.

If the ring is free, every invariant of degree d is a sum of products f₁^a f₂^b with a d₁ + b d₂ = d, and the coefficients are unconstrained. So a physical quantity invariant under the group is a general polynomial in two variables of known degrees, and counting the terms is counting the ways to write d as such a sum. That is a partition count, and it is the reason the Molien series of a free ring is a product of two geometric series.

If the ring is not free, the same list of products is over-complete: some combinations of them are the same invariant, because the relation says so, and a count that ignores it counts some invariants twice. The essay on the relation is about exactly the size of that overcount.

6mm: how many independent invariants there are at each degree. The number of independent polynomial invariants of the plane point group 6mm, one bar per degree from 0 to 12. The heights are the coefficients of the Molien series, computed from an integer recursion on the trace and determinant of each operation. A dot above a bar marks a degree where the same number has been computed a second way, by averaging every monomial of that degree over the group and taking the rank of the result — a computation sharing no code with the first. The two agree at every degree checked. Degrees where the bar is absent hold no invariant at all: for 6mm that is every odd degree below the first invariant, and it is a statement about which functions the group refuses to leave alone.
Fig. 4 6mm’s invariants, from degree zero to twelve. The sequence 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 3 is the expansion of 1/((1 − t²)(1 − t⁶)) and nothing else: invariants of degree twelve are spanned by the quadratic to the sixth, the quadratic cubed times nothing, and the degree-six generator squared, with no relation reducing the count.

4mm, written out

One class in full makes the identities concrete, and 4mm is the right one: it is large enough to be interesting and small enough to write down.

Its eight operations are the four rotations of the square lattice and four mirrors — two through the axes, two through the diagonals. The invariant of degree two is x² + y². The invariant of degree four is x⁴ + y⁴, which the diagonal mirrors permit and which is not the square of the first: (x² + y²)² is a different polynomial, and the two together span the two-dimensional space of quartic invariants the sequence reports.

The degrees are 2 and 4, whose product is 8, which is the order. The excess is (2 − 1) + (4 − 1) = 4, which is the number of mirrors. And the invariant of degree six is spanned by (x² + y²)³ and (x² + y²)(x⁴ + y⁴) — two products, and the sequence says two.

Drop the mirrors and keep the four rotations, and the ring changes character entirely. x⁴ + y⁴ is still invariant, and so now is x³y − xy³, which the mirrors used to forbid. Three invariants where the search expects two, and the ring is no longer free.

The level curves of 4mm's invariant of degree 4. The curves on which the invariant x⁴ + y⁴ takes the values 1, 4 and 12, drawn in the lattice's own basis with the basis vectors marked and the mirror lines of 4mm across them. The middle curve is drawn solid and the others faintly. The polynomial is invariant under every operation of the group — checked by substituting each matrix into it and requiring the coefficients to come back unchanged — so every one of those operations carries each curve onto itself. On a hexagonal basis the quadratic invariant is x² − xy + y² rather than x² + y², and its level curves are still circles: the polynomial is a fact about the coordinates, the curve is a fact about the plane.
Fig. 5 The level curves of 4mm’s degree-four generator x⁴ + y⁴, with the group’s four mirror lines. The curves are the rounded squares a reader would draw for a four-fold pattern, and every operation of 4mm carries each of them onto itself — including the diagonal mirrors, which is the condition that rules out the cubic and quartic terms a lower symmetry would allow.

The map a free ring gives

A free ring is a map. Send each point of the plane to the pair of values its two generators take, and every point of one orbit goes to the same place: the map is constant on orbits, by definition of invariance.

What is not obvious is that it separates them — that two points with the same pair of values are in the same orbit — and that is the statement making the ring a complete description of the quotient rather than a partial one. It is checked here by brute force on a window of the lattice: every pair of points is compared both ways, once by asking whether an operation of the group carries one to the other, once by asking whether the invariants agree, and the two answers must match on every pair.

They do, and dropping a single generator breaks it immediately — which is the check that keeps the first one honest. A test that passes on every input it is ever given is a test that proves nothing, so the same routine is asked for a witness: two points, in different orbits, that a proper subset of the invariants cannot tell apart.

Dropping one invariant of 6mm makes two orbits agree. Every lattice point within four cells of the origin, coloured by the values a proper subset of 6mm's invariants takes on it — the 2 generators with the second one removed, over a window of 8 cells. With the full set, the 45 orbits of the group take 45 distinct sets of values, one each, so the invariants are a complete set of coordinates on the quotient. With one removed, the two circled points — in different orbits, so no operation of the group carries one to the other — take the same values and become indistinguishable. That is the whole content of the statement that a complete set of invariants separates orbits: the completeness is what is doing the work.
Fig. 6 Every lattice point near the origin, coloured by the values a proper subset of 6mm’s invariants takes on it — the degree-two generator alone, with the degree-six one removed. The circled pair lie in different orbits and take the same value, so a single invariant is not a complete set of coordinates on the quotient however symmetric it looks.

Where the reflections come in

The theorem is stated as an equivalence, and the direction that explains itself is the one from reflections to freedom.

A reflection fixes a line — the mirror itself — and the polynomial that vanishes on that line is a linear form. The product of the linear forms of all the reflections is a polynomial that changes sign under each of them and is otherwise invariant — the discriminant — and its degree is the number of reflections. Chevalley’s proof builds the free generators out of the geometry of those fixed lines: each reflection removes one degree of freedom from the invariants in a controlled way, and the count Σ(dᵢ − 1) above is the bookkeeping of exactly that removal.

The other direction is the surprising one. A group with no reflections cannot have a free ring — even though nothing about the definition of a free ring mentions reflections, and even though such a group can be much larger than one that has them. The rotation group 6, of order six, needs three generators; the reflection group m, of order two, needs two. Size is not the variable.

What the picture cannot show

A figure can draw the level curves of an invariant, and it cannot draw the statement that two invariants are algebraically independent. Independence is a statement about all polynomial relations at once, and no finite drawing carries it.

What the figures here do instead is show a consequence: the level curves of the two generators of a free ring cross transversally almost everywhere, so the pair of values locates a point up to its orbit. Where the curves are tangent, the pair does not — and those points are exactly the special positions, where the orbit is shorter than the group. That is the geometric content of the |G|-to-one map above, and it is visible; the algebra behind it is not.

The other thing no drawing here shows is that this is a plane story. Chevalley’s theorem holds in any dimension, and the classification of the groups it applies to — the finite reflection groups, and beyond them the complex reflection groups of Shephard and Todd — is a large piece of mathematics that this collection does not enter. In three dimensions the crystallographic reflection groups are the ten holohedries and their kin, and the degrees are triples rather than pairs. Those triples are computed, in twelve of the thirty-two are free, which runs both sides of the theorem over all thirty-two crystal classes: twelve have a free ring and twelve are generated by their reflections, and they are the same twelve. What stays outside this collection is the classification itself, not the arithmetic on the crystallographic cases.

How many components a property may have, at each rank and in each class. One row per plane point group, one column per rank of a fully symmetric property tensor, with the number of independent components in each cell — counted by averaging the tensor over the group index by index, and equal at every entry to the Molien coefficient of that degree. A symmetric property of rank r is a form of degree r, so Neumann's principle and the invariant ring are the same arithmetic in two notations. The last column is the elastic tensor, which is not fully symmetric — symmetric within each pair of indices and under exchanging the pairs — and its counts are not in the table to its left. A property with its own symmetries needs its own average, and that is why the elastic constants are not read off a degree.
Fig. 7 Why the degrees matter outside algebra: the components each class permits for a symmetric property of each rank. A class with a free ring of degrees 2 and n permits components exactly where a partition of the rank into those degrees exists, so the sparse rows are the classes with a large second degree. The last column, the elastic tensor, does not follow the pattern, and that is because it is not a form.

The exception that is not one

The trivial group deserves a line, because its row in the table looks like a mistake. Its ring is free with degrees 1 and 1, its product is 1, which is its order, and its reflection count is zero, which is (1 − 1) + (1 − 1). Every identity holds. The generators are x and y: with no symmetry at all, every polynomial is invariant, and the invariant ring is the whole polynomial ring.

That is worth stating because it is the boundary case the general statements have to survive, and because the group with no symmetry is the one this site had to work hardest to draw. A figure illustrating p1 has to have a motif with no accidental symmetry of its own; an invariant-theoretic statement about the trivial group has to have a ring with no accidental relation. The two difficulties are the same difficulty.

The invariant degrees exist for every n; the lattice permits five of them. The reflection group with an n-fold rotation has an invariant ring generated in degrees 2 and n, for every n whatever — the dimensions on the right are counted by pairing monomials in complex coordinates, which needs no matrix and therefore no lattice. Five of these groups can be written in integer matrices, and those five are named in the middle column; the rest cannot, because a lattice has no five-fold or seven-fold rotation. The crystallographic restriction is usually a statement about traces of matrices. Here it is the statement that only five of these invariant rings belong to a crystal, and the two arguments have nothing in common but their answer.
Fig. 8 The reflection groups by rotation order, crystallographic or not. Every one of them has a free invariant ring with degrees 2 and n — the counting needs no matrix, only a pairing of monomials in complex coordinates — and the five that a lattice permits are named. Freedom of the ring has nothing to do with the restriction; which of these groups is a crystal’s does.

The same two identities, one dimension up

The theorem is stated here for the plane and it is not a plane theorem, and checking the identities in space is the quickest way to be convinced that the numbers are not a coincidence of small cases.

The cubic holohedry m3̅m has order forty-eight, and its invariant ring is free on generators of degrees 2, 4 and 6. The product is 2 × 4 × 6 = 48: the order, exactly. And the excess sum is 1 + 3 + 5 = 9, which is the number of mirrors — three through the cube’s faces and six through its diagonals.

4̅3m has order twenty-four and degrees 2, 3, 4. The product is twenty-four. The excess sum is 1 + 2 + 3 = 6, and the class has six mirrors.

6/mmm has order twenty-four and degrees 2, 2, 6. Product twenty-four again; excess sum 1 + 1 + 5 = 7, and the class has seven mirrors — six vertical and one horizontal.

Three classes, three different sets of degrees, and both identities holding exactly in each. The second is the one worth dwelling on, because it says the degrees know something the order does not: two groups of the same order with different numbers of reflections must have different degrees, and the degrees are what tell them apart.

Where the theorem’s general form was settled

Chevalley’s statement is for real reflection groups, and the complete answer to which groups have free invariant rings is one step further out and worth naming.

Allowing the group to act on complex coordinates enlarges what counts as a reflection: an operation fixing a hyperplane and multiplying the remaining direction by a root of unity, not necessarily −1. Shephard and Todd classified all finite groups generated by such operations in 1954, and the answer is three infinite families together with thirty-four exceptional groups.

Chevalley’s theorem then extends: a finite group acting on complex coordinates has a free invariant ring exactly when it is one of those. So reflection group in the theorem is the complex notion, and the real reflection groups this page’s classes belong to are a sub-case.

That matters here for one reason. Two of the four plane classes without a free ring — the cyclic rotation groups — become reflection groups over the complex numbers, since a rotation of the plane is a multiplication by a root of unity in one complex coordinate. So the failure the census reports is a failure over the reals, and it is the right failure for crystallography, where the coordinates are real and the polynomials describe physical quantities. The theorem’s boundary and the subject’s boundary happen to coincide, and it is worth knowing that they do so by accident rather than by design.

What this settles and what it opens

The table settles which plane classes have a free invariant ring, by two computations that agree, and it recovers both of Chevalley’s numerical identities on every one of them.

What it opens is the other half. Four classes are left over, each needing a third generator and therefore carrying exactly one relation, and that relation is a polynomial identity nobody has yet written down here. Finding it is not a matter of recognising a pattern: it is a kernel, computed from a linear system, and the next essay computes it for all four.

Where this goes next

4mm: how many independent invariants there are at each degree. The number of independent polynomial invariants of the plane point group 4mm, one bar per degree from 0 to 8. The heights are the coefficients of the Molien series, computed from an integer recursion on the trace and determinant of each operation. A dot above a bar marks a degree where the same number has been computed a second way, by averaging every monomial of that degree over the group and taking the rank of the result — a computation sharing no code with the first. The two agree at every degree checked. Degrees where the bar is absent hold no invariant at all: for 4mm that is every odd degree below the first invariant, and it is a statement about which functions the group refuses to leave alone. How many invariants of each degree A group moves the plane about, and some polynomials do not notice. How many independent ones there are at each degree is a sequence of integers, computed here by a recursion on traces and again by averaging every monomial — two routes that share no code and agree everywhere. glide: 3 mirrors. A glide of the plane, drawn together with the mirrors it is a product of. The first shape is the motif; the pale ones are what each mirror in turn produces; the last is the image the motion itself gives. There are 3 mirrors, which is the smallest number that can produce this kind of motion, and their product was formed and compared with the motion before the figure was drawn. Three reflections, and never four Every motion of the plane is a product of mirrors, and the number needed is never more than three. That count is not a curiosity about mirrors — it is the classification of the four motions written as an integer, with the parity of the number deciding handedness and the geometry of the last two mirrors deciding everything else. The invariant degrees exist for every n; the lattice permits five of them. The reflection group with an n-fold rotation has an invariant ring generated in degrees 2 and n, for every n whatever — the dimensions on the right are counted by pairing monomials in complex coordinates, which needs no matrix and therefore no lattice. Five of these groups can be written in integer matrices, and those five are named in the middle column; the rest cannot, because a lattice has no five-fold or seven-fold rotation. The crystallographic restriction is usually a statement about traces of matrices. Here it is the statement that only five of these invariant rings belong to a crystal, and the two arguments have nothing in common but their answer. The degrees that name the restriction The reflection group with an n-fold rotation has invariants of degrees 2 and n — for every n, with no lattice anywhere in the argument. Which of those groups a crystal may have is then the only question left, and its answer is the crystallographic restriction arriving from a direction nobody points it from.

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Algebraic independenceChevalley theoremHypersurfaceInvariant degreesInvariant ringMolien seriesPoint groupReflection groupSyzygy