Invariants — the series · Crystallography & Symmetry

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.

part 1 · point-groups

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.

part 2 · point-groups

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.

part 3 · point-groups

Which order parameters carry a cubic invariant, and therefore cannot grow from zero. Every order parameter of every plane class, with the number of independent cubic invariants it admits. The count is the degree-three coefficient of the Molien series of the representation's image — the same computation the invariant-ring figures make for a different reason — and Landau's condition is that it be zero. Where it is not, a free energy in that order parameter has a term of odd degree, which puts its minimum away from zero the moment the quadratic coefficient does anything at all, so the parameter jumps rather than growing. In the plane exactly two order parameters carry one, and both are the two-dimensional representation of a class with a threefold axis and no sixfold.

The cubic term that forbids a continuous change

A crystal may lose a symmetry gradually only if the quantity measuring the loss admits no cubic invariant. Whether it does is the third coefficient of a Molien series — so a question about how a material changes is answered by counting polynomials.

part 4 · point-groups

12 of the thirty-two classes have a free invariant ring. Every crystal class with its order, the number of its operations that are reflections, whether its ring of invariant polynomials is free, and the degrees of the generators when it is. A reflection here is an operation of determinant minus one whose fixed set is a plane; an inversion centre has determinant minus one and fixes only the origin and is not one. The classes with a free ring are exactly the classes generated by their reflections, which is Chevalley's theorem checked rather than quoted.

Twelve of the thirty-two are free

A crystal class leaves some polynomials alone, and the ones it leaves alone form a ring. For twelve of the thirty-two classes that ring is generated by three polynomials with no relation between them, and for the other twenty it is not — and the twelve are exactly the classes generated by their mirror planes. The two verdicts are computed by routes sharing no code, and an inversion centre is not a mirror.

part 5 · point-groups

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.

part 6 · point-groups

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.

part 7 · point-groups