Crystallography & Symmetry
patterns derived from their groups
Concept

Reflection group — where it appears

A group generated by its own reflections, which in the plane means the classes 1, m, 2mm, 3m, 4mm and 6mm. Such a group has a free invariant ring, and no group without reflections does.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

restriction · Restriction 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.

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.

point-groups · Invariants 4: three generators in two variables, and the one relation between them. The invariant ring of 4 needs 3 generators, of degrees 2, 4, 4, and three functions of two variables cannot be algebraically independent. The relation between them is found rather than quoted: every monomial in the generators of the degree at which they can first be dependent is written out, the map back to polynomials in x and y is formed, and its kernel is the relation. It is then evaluated at points of the lattice, where all three generators take integer values and the combination comes to exactly zero. A group with a reflection has no such relation, which is the same statement as its ring being free.

Three invariants and one relation

Four of the ten plane classes need three invariants where two variables can only support two, so exactly one polynomial identity ties them together. The identity is not recognised or recalled: it is the kernel of a linear map, computed and then checked at lattice points where every term is an integer.

point-groups · Invariants Every crystal class, as a reflection class and its relative invariants. The thirty-two crystal classes, each with the smallest reflection class containing it after a relabelling of the axes, the index, and the degrees of the secondary invariants that make its invariant ring a free module over the reflection class's polynomial ring. Twelve classes are reflection classes themselves; seventeen have index two and one secondary invariant besides 1; three — 1̄, 4̄ and 3̄ — have index four and three.

What the twenty lack is a product of mirrors

Twelve crystal classes have invariant rings generated by three independent polynomials, and they are the twelve generated by their own mirrors. The other twenty were left as the classes whose counts need a correction. The correction has a simple shape. Each of the twenty sits inside one of the twelve, at index two or four, and its invariants are the larger class's invariants times one or three extra polynomials. Each extra polynomial is a product of the mirror planes the smaller class lacks. For the rotations of the cube it is a single polynomial of degree nine, the product of all nine mirrors of the full cubic class.

point-groups · Invariants The fifty-eight black-and-white classes, sorted by reflection class. Every black-and-white magnetic point group, grouped by the smallest reflection class containing its underlying crystal class, with the degrees of the mirror products that generate its relative invariants for a polar and for an axial time-odd quantity. Every class's two series are exactly those sums of mirror products, and the number of generators is the class's index in its reflection class: one, two or four.

Every magnetic class counts its properties by mirrors

The invariants of every crystal class turned out to be products of mirrors of a reflection class, times that reflection class's own free invariants. A magnetic class asks for polynomials that some of its operations negate, which is harder in principle. It is not harder in fact: all fifty-eight black-and-white classes, for a time-odd quantity that is polar and for one that is axial, are counted by mirror products — provided the reflection class contains the whole class and not only its unprimed part, which for thirty of them it does not.

point-groups · Invariants The square net drawn as the Cayley graph of four different groups. The same square net four times. In each panel the edges are coloured by the group operation that spells them, read from one regular subgroup of the net's symmetry group: the lattice of translations, a p2 whose half-turn centres sit at the midpoints of edges, a pmm whose four mirrors run between rows of vertices, and a p4 whose quarter-turn centres sit in the middle of the squares. No symmetry element drawn passes through a vertex, which is what makes each group act without fixing any, and the large dot is the vertex named by the identity.

The square net draws eleven groups

The Cayley graph of a wallpaper group is a periodic net. Asked the other way, the question is which nets are Cayley graphs, and of what. A net is one exactly when some group of its symmetries carries every vertex to every other and fixes none. All eleven uniform nets pass. The square net is the Cayley graph of eleven of the seventeen groups, the kagome net of exactly one, and between them the eleven nets use all seventeen.

operations · Presentations

Named alongside it

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

Invariant ringMolien seriesInvariant polynomialAlgebraic independenceChevalley theoremHypersurfaceInvariant degreesSyzygyArchimedean tilingCayley graphCharacterChirality

All concepts