Crystallography & Symmetry
patterns derived from their groups
Concept

Weyl group — where it appears

The finite group generated by the reflections in the roots of a root system. Its elements permute the roots, and the orders of its rotations are among the few a lattice allows, which is how the restriction appears with no lattice in the argument.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Four angles, and the integer that picks them. Two roots at angle θ have Cartan integers whose product is 4cos²θ. Both are whole numbers and the product is below four, so it is nought, one, two or three — and each value fixes the angle between the two roots, and with it the angle between the mirrors perpendicular to them. The shaded wedge is the region the pair of mirrors folds the plane onto; the smaller it is, the larger the group they generate.

Four root systems, and the same four rotations

Two mirrors meeting at an angle generate a group. Ask that the group be finite and that a certain pairing between the mirrors come out a whole number, and the angle has only four possible values — from which the rotations that survive are of order two, three, four and six. The crystallographic restriction arrives with no lattice anywhere in the argument.

restriction · Restriction Which four-dimensional lattices carry which rotations of order twelve. Six four-dimensional lattices, the order of each one's symmetry group, and how many of its symmetries are of each of the three kinds of order twelve: the single block Φ₁₂, the third-turn with a quarter-turn Φ₃Φ₄, and the quarter-turn with a sixth-turn Φ₄Φ₆. D₄ has 96 of the first among 1,152 symmetries, the two hexagonal planes 24 among 288, and the hexagonal-and-square sum 4 of each of the other two among 96. No lattice has all three, and none has the first with either other.

No lattice carries all three twelves

Four-dimensional lattices admit three different operations of order twelve: one that turns a single irreducible block by a twelfth, and two that turn two planes, by a third and a quarter or by a quarter and a sixth. Computing the symmetry group of every kind of four-dimensional lattice that can have a maximal one shows that no lattice carries all three, and none carries the first with either of the others. D₄, the most symmetric of them, has only the first.

restriction · Restriction Twelve kinds of operation six dimensions adds, and the lattices that carry them. The twelve rational kinds of operation whose orders first appear in five or six dimensions — 7, 9, 14, 15, 18, 20 (two kinds), 24 (two) and 30 (three) — against six lattices whose symmetry groups were computed element by element. Every kind is carried by at least one. All three kinds of order thirty are carried by A₄ ⊕ A₂, forty-eight symmetries each; the single-block kinds Φ₇ and Φ₁₄ only by A₆, and Φ₉ and Φ₁₈ by E₆ and by three hexagonal planes.

The thirties share a lattice

Four dimensions has three kinds of operation of order twelve and no lattice carries all three. Six dimensions has three kinds of order thirty, and one lattice carries every one of them, forty-eight times each. The difference is not the order. One of the twelves is a single block that cannot be split, and no thirty is: thirty needs eight dimensions to be irreducible, so in six it is always a five-fold piece beside a three-fold one, and the lattice that is both pieces holds every combination.

restriction · Restriction Seventeen ways to have order thirty in eight dimensions, and four on E₈. Every product of cyclotomic polynomials of total degree eight whose order is thirty: 17 of them. E₈'s symmetry group holds 4: the single block Φ₃₀ and three split kinds, each with a one-dimensional piece of each sign or a doubled minus one. The other thirteen fit the dimension and have no element on this lattice.

Thirty as one block lives on two lattices

Eight dimensions is where an operation of order thirty can finally be a single irreducible block. It was predicted to sit alone on E₈ with the split kinds elsewhere, as the twelves do in four dimensions. It does not: E₈ holds three split thirties beside it, and exactly three of the sixteen that fit, because a five-fold block leaves nothing but a permutation of five things. The block also lives on a second lattice, A₂ ⊗ A₄, and there it really is alone.

restriction · Restriction

Named alongside it

The objects these essays reach for when they reach for this one.

Crystallographic restrictionCyclotomic polynomialsLatticeCartan integerCentraliserCharacteristic polynomialCoxeter elementDihedral groupMirrorPoint groupReflectionRoot lattice

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