Representations — the series · Crystallography & Symmetry

6mm: 6 irreducible characters on 6 classes. The character table of the plane point group 6mm, constructed rather than quoted. The columns are its 6 conjugacy classes, with the number of operations in each; the rows are its 6 irreducible representations, of dimensions 1, 1, 1, 1, 2, 2. The dimensions satisfy 1² + 1² + 1² + 1² + 2² + 2² = 12, which is the order of the group, and that identity together with the count of classes is what proves the list complete. Every entry is an exact element of the twelfth cyclotomic ring; the decimals shown are the numerical value of an integer vector, not a computed approximation.

What a group does to a function

A symmetry operation moves points about, and two hundred essays here have watched it do so. It also acts on everything defined over those points — a density, a displacement, a wave — and that action is linear, which turns a group of motions into a set of matrices and every question about it into arithmetic.

part 1 · point-groups

The ten plane classes, and the dimensions they permit. Every crystallographic point group of the plane, with one block per irreducible representation and each block as wide as its dimension. Nine of the ten have only one-dimensional representations; 4mm, 3m and 6mm carry a two-dimensional one, drawn in the measured colour. Nothing is wider than two, and the sum of the squares of the widths in each row is the order of that group — the identity that says the row is complete.

How large a degeneracy may be

Symmetry can force two things to have the same value, and in a crystal it can force three. It can never force five, and the reason is a sum of squares — the same kind of arithmetic that forbids a five-fold axis, arriving at a question about levels rather than about rotations.

part 2 · point-groups

4mm: a degeneracy tuned into existence, and gone at the next weight. Four invariant operators on one orbit of 8 points under 4mm, differing only in the weight given to a single class of pairs. The first column is not a choice: it is the value that weight has to take for two levels of different symmetry to arrive at the same number, found by sweeping the weight and closing on the crossing, and the two levels there agree to 1.0e-9. The character table predicts levels of sizes 1, 1, 1, 1, 2, 2; the tuned column shows 1, 1, 1, 2, 3 and every other column shows the predicted pattern again. That is the whole of what an accidental degeneracy is — a property of one choice of weights, not of the group — and it is why the weights have to be moved before a degeneracy is called forced. A degeneracy the group requires would be in all four columns, because nothing respecting the symmetry can lift it.

A coincidence the group did not ask for

Two levels sitting at the same value look identical whether symmetry required it or not. The difference is testable: move the numbers the symmetry does not decide and watch what survives, because a degeneracy the group forces cannot be shifted by anything the group leaves alone.

part 3 · point-groups

An orbit of 12 points under 6mm, split into 6 kinds. The functions defined on one orbit of 12 points under 6mm form a space of that dimension, and the group acts on it by permuting the points. The character of that action is the cheapest one in the subject — at each operation it is the number of points left where they are — and decomposing it against the table gives the multiplicities on the right. They are whole numbers, they weight the dimensions to 12 again, and the multiplicity of the trivial representation is the number of orbits, which is Burnside's lemma arriving as a special case.

The shell that splits into kinds

The neighbours of an atom carry a space of functions as large as the shell, and the group does not treat that space as one thing. It splits into pieces of a few kinds, in whole numbers, and the count is the cheapest character in the subject: how many neighbours each operation leaves where they are.

part 4 · point-groups

4mm: 17 of 25 transitions forbidden. Every pair of irreducible representations of 4mm, with the number of times the identity occurs in the product of the two with the vector operator. A zero is a prohibition: the integral that would give the transition rate vanishes for every choice of functions carrying those representations, whatever the material is made of. A positive number is a permission and nothing more. Rows are final states and columns initial ones; the labels are the dimensions of the representations, so the twos are the degenerate levels. Every one of the 17 prohibitions here was checked again against explicit polynomials.

What a group forbids to happen

Two levels and a thing that might carry a crystal from one to the other. Whether it can is one sum over the group — and a zero there is a prohibition that no material, no temperature and no intensity of light gets round.

part 5 · point-groups

A spin needs two full turns to come back. The number a rotation about a fixed axis multiplies a state by, against the angle turned through. A vector — anything of integer spin — is back where it started after one full turn; a spin-one-half state is multiplied by minus one and needs a second turn. So the operators acting on such a state do not form the rotation group: a full turn is an operation distinct from doing nothing, and the group is twice as large.

Two turns to come back

A rotation through a full turn does nothing to a crystal and multiplies a spin-one-half state by minus one, so the group acting on such a state is not the point group but a group twice its size. Building those eleven double groups from quaternions and averaging a random operator over each gives the degeneracies a spin may have — and shows that the doubling everybody calls Kramers' is time reversal's doing and not the double group's.

part 6 · point-groups