Four root systems, and the same four rotations

The crystallographic restriction is the first thing this collection proves and the thing most of it leans on. A rotation carrying a lattice to itself has an integer matrix on any lattice basis, its trace is therefore an integer, and the trace of a rotation through 2π/n is 2cos(2π/n). Only five values of n make that a whole number, and the answer is one, two, three, four and six.

The argument is three lines long and it has a shape worth noticing: it is about integrality, and the lattice is only how the integrality got there. The matrix is integral because there is a lattice for it to be integral with respect to. Take the lattice away and the argument has nothing to stand on.

So the obvious question is whether the lattice is doing any work beyond that. The restriction with no lattice is one answer — the restriction survives in a setting where the discreteness comes from somewhere else — and this essay is another, from a direction that starts nowhere near a crystal.

Two mirrors, and a whole number between them #

Take two lines through the origin of the plane and the reflections in them. The product of the two reflections is a rotation through twice the angle between the lines, which is the fact three reflections and never four is built on, and the group they generate is finite exactly when that angle is a rational multiple of π.

That is not yet a restriction — every rational multiple works, and the group of order 2n appears for every n. Something more is needed to cut the list down, and in the theory of root systems it is a condition on how the mirrors are measured against each other.

Attach to each mirror a vector perpendicular to it, called a root, and write the reflection in the mirror as

s_α(β) = β − ⟨β, α∨⟩ α, where ⟨β, α∨⟩ = 2 (β · α) / (α · α).

The bracket is called a Cartan integer, and the whole of what follows is the demand that it be one — that for every pair of roots in the collection, that number is a whole number rather than merely a real one.

The reason the demand bites is a product. For two roots α and β at angle θ,

⟨β, α∨⟩ ⟨α, β∨⟩ = 4 (α·β)² / (α·α)(β·β) = 4cos²θ.

The left-hand side is a product of two integers. The right-hand side is at most four, with equality only when the roots are parallel — and a root system is required to contain no multiples of a root other than ±α, so distinct pairs never reach it. So the product is nought, one, two or three, and there are no other cases to consider.

That is the whole restriction, and it has arrived without a lattice, a translation, or a periodic pattern being mentioned. The only things in play are two vectors, an inner product, and the demand that a certain ratio be integral.

What each of the four cases becomes #

A value of the product fixes cosθ up to sign, so it fixes the angle. Taking the obtuse choice — which is the convention for a pair of simple roots, the ones a fundamental wedge is built from — gives ninety degrees, a hundred and twenty, a hundred and thirty-five and a hundred and fifty.

Fixing the angle does not yet give the collection of roots. A root system has to be closed under the reflections in its own members, and reflecting α in β produces a third vector which must also be a root, which must then be reflected in turn. Whether that process stops is the question, and it is the one the enumeration here actually runs.

The computation is done in integer coordinates, which is the part worth stating carefully. Every root of a rank-two system is an integer combination of the two simple roots — that is a consequence of the axioms rather than an assumption — so a root can be stored as a pair of integers and the reflection becomes an integer matrix acting on that pair. Both reflections and the closure are then exact, with no angles, no cosines and no floating point anywhere in the loop.

The four closures stop at four, six, eight and twelve roots. Those numbers are not put in; they come out of a breadth-first search that reflects until nothing new appears, and the search is what the figure above is a picture of.

The four systems, drawn #

Having the roots as integer pairs is enough to count them and not enough to draw them, because a picture needs positions. The positions come from the Cartan matrix too, and this is the second place where nothing is chosen by hand.

The Cartan matrix determines the Gram matrix of the two simple roots up to one overall scale: the diagonal entries are the roots’ lengths squared, the off-diagonal entry is their inner product, and the ratio of the two lengths is the ratio of the two Cartan integers. Fixing the shorter root at length squared two makes every entry an integer. From the Gram matrix, placing the first simple root along an axis and the second at the inner product the Gram demands puts every root somewhere definite, and the drawing is then a consequence of the integers rather than an illustration of them.

The four have names, and the names are worth knowing because they belong to a much larger classification this essay is a corner of. A₁×A₁ is two perpendicular pairs — two independent mirrors, and the group is the one a rectangle has. A₂ is six roots at sixty degrees, all the same length, which is the hexagonal arrangement. B₂ is eight roots in two squares of four, one square longer than the other by a factor of root two. G₂ is twelve roots in two hexagons, one longer than the other by root three.

Two of them have all roots the same length and two have two lengths. That is not a decoration: a system with two lengths is one whose group does not act transitively on its roots, and the long and short roots behave differently under everything that follows from the system. It is the same distinction that separates a lattice whose point group is transitive on its shortest vectors from one whose is not.

The rotations, which are the point #

Each system generates a group — its Weyl group — as the group generated by the reflections in all of its roots. Closing that group is another breadth-first search, this time over integer matrices, and it terminates at four, six, eight and twelve elements.

Each of those groups is dihedral, and each contains a rotation subgroup of half the order. The rotation is the product of the two simple reflections, and its order can be read off directly by multiplying the two integer matrices together until the identity comes back.

Two, three, four and six. With the identity that is one, two, three, four and six — the crystallographic restriction, in the same five numbers, from an argument in which no lattice appears at any point.

That deserves a moment rather than a note, because there are two ways to read it and only one of them is right.

The wrong reading is that this is a second proof of the same theorem, in the sense of an independent confirmation. It is not independent. Both arguments come down to 2cos θ being an integer: the trace argument says the trace of an integer matrix is an integer and the trace is 2cos(2π/n), and this one says a product of two Cartan integers is an integer and that product is 4cos²θ, which is 2 + 2cos2θ. The two expressions are the same quantity looked at through different halves of the angle. Discovering that two arguments share a step is worth more than discovering that two arguments exist.

The right reading is about where the integrality comes from, and there the two differ completely. In the lattice argument it comes from the lattice — the matrix is integral because there is a discrete set of translations for it to permute. Here it comes from a compatibility condition imposed on a set of vectors with no periodicity in sight, and the periodicity is a consequence: the integer span of a root system is a lattice, and the Weyl group preserves it because every Cartan integer is a whole number. The root system builds the lattice rather than living in one, and the restriction is what makes that possible.

The lattice the roots build #

The claim that periodicity is downstream can be made to do some work, because the lattice a root system generates is a computable object and it is worth looking at.

Take the integer span of the roots — every whole-number combination of them. That is a lattice by construction, and the Weyl group carries it to itself, because reflecting a root gives an integer combination of roots and integer combinations are what the span consists of. This is where the periodicity comes from in this account. The lattice was not there to begin with and no rotation was tested against it; it is what the integrality condition manufactures.

Which lattice is it? The answer needs a reduction rather than a look, because a Gram matrix written on the simple roots does not announce which of the plane’s lattice types it is. Reducing each one — the ordinary Lagrange reduction, the same one that decides the shortest basis — gives two answers between the four systems.

A₁×A₁ and B₂ both build the square lattice. A₂ and G₂ both build the hexagonal one. So the four root systems produce two lattices, and each lattice arrives twice.

The pairs are not duplicates, and the difference between the members is exactly the one this collection calls the holohedry is the ceiling. A lattice has a point group of its own — the largest group of rotations and reflections carrying it to itself — and any group acting on that lattice is a subgroup of it. The square lattice’s is of order eight and the hexagonal lattice’s of order twelve. B₂’s Weyl group is of order eight and G₂’s of order twelve, so those two systems reach the ceiling. A₁×A₁’s is of order four and A₂’s of order six, so those two sit at exactly half.

A root system therefore carries strictly more information than the lattice it generates. Two different systems can produce the same lattice and then act on it with different groups, and there is no way to recover the system from its lattice alone. That is worth holding beside the fact that a crystal’s lattice and its point group are separate pieces of data — a hexagonal lattice hosts a crystal of class 3 as readily as one of class 6/mmm, and what a class pins down about the axes is a smaller thing than what the lattice does.

The two halves also explain the two root lengths. A system whose Weyl group is the whole point group of its lattice must move every shortest vector to every other, and it does — but a system at half the ceiling cannot, and the roots divide into two orbits of different length. Two lengths and half a point group are the same phenomenon counted twice, which is why the length column in the census and the fraction column in the lattice figure agree case for case.

What the enumeration must refuse #

A count of four is only a result if the enumeration could have produced five. The products the argument rejects are therefore run rather than argued about: a product of four, which is the parallel case the axioms exclude, and products of five and six, which correspond to no real angle at all.

Each of the rejected Cartan matrices produces a set that grows past four hundred roots and is still growing when the search gives up. The reason is easy to see once the closure has been watched: reflecting one root in another at an angle that is not a rational part of a turn produces a new direction every time, and the orbit is dense in the circle rather than finite. A root system’s finiteness and the integrality of its Cartan integers are the same condition, and the enumeration is the demonstration.

There is a second refusal worth stating because it is about scope. Nothing above says anything about rank three or higher. The rank-two classification is small enough to run to completion by brute force and the general one is not: it is the theorem that the connected root systems are the four infinite families and five exceptional ones, and it is a classification this collection does not carry. What the plane case supplies is the mechanism — a product of two integers bounded by four — and that mechanism is exactly the one the general proof uses, applied one pair of simple roots at a time.

The same list, from three directions #

Three arguments in this collection now produce the same five orders.

The trace argument is the shortest. A lattice automorphism has an integer trace and the trace of a rotation is 2cos(2π/n), so n is one, two, three, four or six. It needs a lattice and nothing else.

The degree argument — the degrees that name the restriction — says the same thing in the language of algebraic numbers: 2cos(2π/n) generates a field of degree φ(n)/2 over the rationals, and being an integer means that degree is one, which happens for exactly those n. It replaces a computation by a fact about cyclotomic fields, and it is the version that generalises cleanly: in three dimensions the same test asks whether φ(n) divides the dimension, and in four dimensions it lets five-fold symmetry in, which is where five-fold becomes legal.

The root argument here is the one that needs no ambient object. It says: two reflections whose relative measurement is integral meet at one of four angles. Everything else — that the group is finite, that its rotation has one of four orders, that the integer span of the roots is a lattice the group preserves — follows.

Having three routes matters for a practical reason as well as a pleasing one. Each of them extends differently. The trace route extends to any dimension and gets hard, because the integer to be constrained is a sum of many cosines. The degree route extends to any dimension and stays easy, which is why it is the one the site uses for the four-dimensional case. The root route does not extend by dimension at all — it extends by rank, and what it produces there is the classification of finite reflection groups, of which the crystallographic ones are the Weyl groups and the rest are the two families and three exceptional groups that are not.

That last extension is the one with the largest consequences elsewhere in this collection. The finite reflection groups are exactly the groups whose ring of invariant polynomials is free, which is Chevalley’s theorem, and running it on the crystal classes says twelve of the thirty-two are free. The two-dimensional Weyl groups above are four of the smallest members of that family, and the reason their invariant rings are free is the reason any reflection group’s is.

What a crystallographer should take from it #

Very little, operationally — no structure is solved differently because the restriction has a root-system proof. What changes is what the restriction is understood to be.

Read through the trace, the restriction is a fact about lattices: a constraint that periodicity places on rotation. Read through roots, it is a fact about reflections measuring each other, and periodicity is downstream of it. A crystal is periodic and has a point group; the root-system view says that a group of reflections satisfying an integrality condition produces a periodic thing to act on, whether or not anybody was looking for one.

That reversal is the same one the seventeen makes for wallpaper groups when they are derived from their generators rather than found in patterns, and the same one every wall names a generator makes for a fundamental domain: the group is the primary object and the pattern is what it leaves behind. Root systems are the version of that reversal in which the group comes first by construction, because a root system is nothing but the data needed to write down a group of reflections and be sure it is finite.

And there is one number in it that is worth carrying out. The bound is four — 4cos²θ < 4 — and it is the whole of the argument. Everything else is bookkeeping: which integers multiply to nought, one, two or three, and what happens when the corresponding reflections are turned loose on each other. A restriction that governs every crystal that has ever been measured is, at bottom, the observation that a cosine is at most one.