Crystallography & Symmetry
patterns derived from their groups
Concept

Character table — where it appears

The array of characters of a finite group's irreducible representations, one row per representation and one column per conjugacy class. It is complete exactly when the squares of the dimensions sum to the group's order and the rows number the classes.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

point-groups · Representations 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.

point-groups · Representations p3m1 at (0, 0): the levels the little group requires, and the ones measured. The levels of the p3m1 model at (0, 0), with degenerate ones drawn thick. The little group there has order 6, and its characters predict levels of dimensions 1, 1, 2, 2. The measurement is 1, 2, 2, 1, and the account is "unitary". The values are numerical and the multiplicities are read at a stated gap; the prediction they are compared against is exact.

Where two levels must meet

At most wavevectors nothing of a crystal's symmetry survives, and its levels are as unconstrained as any operator's. At a handful of them a whole point group survives, and where that group has a two-dimensional representation, two levels are obliged to coincide — before anything about the material is known.

space-groups · K symmetry 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.

point-groups · Representations

Named alongside it

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

Irreducible representationDegeneracyConjugacy classRepresentationAbelian groupAccidental degeneracyBloch stateCharacterCrystallographic restrictionCyclotomic integerFrobenius schur indicatorInvariant operator

All concepts