What a group does to a function

A symmetry operation moves points. That is what the word has meant everywhere in this collection so far: a rotation carries an atom onto another atom, a mirror carries a tile onto its reflection, and the group of a pattern is the set of motions under which the point set is unchanged. Everything the round trip checks is a statement about where points go.

There is a second thing the same operation does, and almost every calculation anybody performs on a crystal lives inside it. Alongside the points there are functions defined over them — an electron density, a displacement of the atoms from their ideal sites, a temperature, a wave. An operation acts on those too. If g carries the point x to g·x, then it carries the function f to the function which at g·x has the value f had at x, and that new function is written f ∘ g⁻¹.

The action on functions has a property the action on points does not. It is linear: the sum of two functions goes to the sum of their images, a multiple goes to the same multiple. So the group is acting on a vector space, and a group acting linearly on a vector space is a group of matrices. That is the whole idea, and the consequences run through diffraction, elasticity, vibration, and the physical properties a class permits.

A representation, and why it is not just a copy of the group #

A representation of a group is a rule assigning a matrix to each element such that the matrices multiply the way the group does. The identity goes to the identity matrix, and the product of two elements goes to the product of their matrices, in that order.

The group’s own operations are already such a rule — in the lattice basis they are integer matrices, and they multiply correctly by construction — and that one is called the natural representation. But it is one representation among many, and the useful ones are usually not it.

The examples that matter arrive by themselves. Take any orbit of points under a group and consider all functions on that orbit: the group permutes the points, hence permutes the values, and the result is a representation of dimension equal to the size of the orbit. Take the components of a stress tensor: the group acts on them, linearly, and that is a six-dimensional representation. Take the displacements of the atoms in a cell: three per atom, and the group acts on all of them at once.

Each of those spaces is large, and each of them falls apart.

The falling apart, which is the whole subject #

A representation is reducible when the space it acts on contains a smaller subspace that the group carries onto itself. When that happens the matrices can all be written, in a suitable basis, with a block of zeros in the same corner, and the representation is really two smaller ones sitting side by side.

A representation with no such subspace is irreducible. Every representation of a finite group is a sum of irreducible ones, in exactly one way, and the irreducible ones are few: a finite group has as many of them as it has conjugacy classes, which for a crystallographic point group is a handful.

That is why the subject is worth the machinery. A twelve-dimensional space of atomic displacements is not twelve unrelated numbers; it is a small number of pieces, each of which the symmetry treats as a unit, and a physical quantity that respects the symmetry cannot mix one piece with another. The whole of what symmetry decides about a crystal’s properties, its vibrations and its transitions is decided by which pieces are present and how many times.

The character, which is a trace and forgets everything else #

Working with matrices directly is unpleasant, because a representation is only defined up to a change of basis: two rules that differ by conjugating every matrix by the same fixed matrix are the same representation described twice. What is needed is something that survives that freedom.

The trace does. The trace of P M P⁻¹ is the trace of M, so the function assigning to each group element the trace of its matrix does not depend on the basis at all. That function is the representation’s character, and it is a far smaller object than the representation: a handful of numbers rather than a handful of matrices.

Two facts make the character sufficient rather than merely convenient. It is constant on conjugacy classes — conjugate operations are the same operation seen from two coordinate systems — so it is a list with one entry per class. And two representations with the same character are the same representation. Nothing is lost by throwing the matrices away.

The characters of the irreducible representations are orthonormal under a natural inner product, and that single fact does all the work. It says how to find the multiplicity of an irreducible representation inside any representation: take the inner product of the two characters, and the answer is a non-negative integer. It says when a representation is irreducible: exactly when its character has inner product one with itself. And it is what makes the whole business a computation rather than a search.

There is a second orthogonality relation, over the columns rather than the rows, and it is the one that closes the table. Read a character table as a matrix with one row per representation and one column per class: the rows are orthonormal, which is the relation above, and so — after weighting each column by the size of its class — are the columns. A matrix cannot have more orthogonal rows than it has columns, nor more orthogonal columns than rows, so the number of representations and the number of classes are forced to be equal. That is where the second half of the construction’s stopping condition comes from, and it is why a table is square rather than merely happening to be. It also gives the practical test a reader can apply to any table in any book: pick two different columns, multiply them entry by entry with one conjugated, weight by the class sizes, and the answer must be zero.

The column relation has a use the row one does not. It says how many operations are in each class without counting them — the class of an element g has |G| divided by the sum of |χᵢ(g)|² members — so a table with a mistyped entry usually announces itself as a class size that is not a whole number. Every table this collection draws is checked both ways for exactly that reason: the row relation would not catch a duplicated column and the column relation would not catch a duplicated row.

Constructed here, not looked up #

Character tables for the crystallographic point groups are in every textbook of the subject, and this collection has a standing rule about numbers that arrive that way: every count is enumerated rather than quoted. A table is a count of a more elaborate kind, and it gets the same treatment.

The construction has four steps and each of them is short.

Every one-dimensional representation, by search. A one-dimensional representation is a homomorphism into the non-zero complex numbers, and on a finite group its values are roots of unity. Which roots? The order of any element divides the group’s exponent, which for a crystallographic point group divides twelve — because the orders a lattice permits are one, two, three, four and six, and twelve is their least common multiple. So the search assigns a twelfth root of unity to each generator, propagates the assignment through the group, and keeps the assignments that never arrive at the same element with two different values. At most a hundred and forty-four tries, and no table consulted.

The natural representation. The group’s own matrices, whose character is the trace: an integer, because in the lattice basis the matrices are integral. That integrality is not decoration. It is what makes the whole construction decidable rather than approximate — the trace of a symmetry operation of a lattice is a whole number, which is the fact the crystallographic restriction rests on, and it arrives here as the first row of the table needing no arithmetic at all.

Products, and the symmetric and exterior squares. Multiplying a representation by a one-dimensional one gives another representation; the symmetric and exterior squares of the natural one give further ones. Each character obtained this way is reduced against the irreducibles already found, and whatever is left with inner product one with itself is new.

A proof that the list is complete. This is the step that turns a collection into a table. The dimensions must satisfy the sum rule, and the number of irreducibles must equal the number of classes. Both are checked, and the construction throws rather than returning what it has if either fails.

The last step is worth dwelling on because of what its absence looks like. A table one row short passes every orthogonality test that remains in it. The rows it still has are still orthonormal; nothing about them is wrong. What goes wrong is every decomposition computed against it afterwards — each multiplicity comes out too small, and the dimensions no longer add up, and by then the error is several arguments downstream of its cause. The dimension sum catches it immediately, and it is the reason this collection’s machinery refuses rather than reports when a construction has not closed.

The arithmetic is exact, which is not obvious #

A character’s values are traces of matrices whose entries are complex numbers, so it is fair to expect the computation to be numerical, with tolerances to choose and residuals to interpret. It is not, and the reason is the crystallographic restriction arriving in a new place.

Every character value here lies in the ring generated by the twelfth roots of unity. A three-fold rotation contributes a cube root, a four-fold rotation contributes i, a six-fold rotation a sixth root — and twelve, once more, is the number that holds all of them at once. So a character value is stored here as four integers against the basis 1, ζ, ζ², ζ³, with the reduction ζ⁴ = ζ² − 1, and every operation on it — addition, multiplication, complex conjugation — is integer arithmetic. An orthogonality relation is an integer identity. A multiplicity is an integer or it is a bug.

That places character theory alongside the rest of this collection’s habits rather than beside them. Symmetry here is decidable because the operations are integer matrices in the lattice basis; the representations are decidable for the same reason, one level up.

What the multiplicities are for #

A representation decomposed is a question answered. Three examples, all of which this collection either has already or reaches shortly.

How many independent components a physical property has. A tensor of a given rank spans a representation of the point group; the number of independent components a class permits is the multiplicity of the trivial representation inside it, because a quantity that must be unchanged by every operation is one that transforms trivially. That is Neumann’s principle counted — and it is the same character sum the property essays already perform, arriving here from the general theory rather than as a special construction.

How a shell of neighbours splits. The functions on an orbit of points form a representation whose character is the cheapest one in the subject: at each operation it is simply the number of points left where they are. Decomposing it says which symmetry-adapted combinations exist, and its trivial multiplicity is the number of orbits — which is Burnside’s lemma arriving as a special case of something more general.

Which degeneracies are forced. An operator commuting with the group cannot separate the members of an irreducible piece, so a two-dimensional irreducible representation is a pair of levels that nothing respecting the symmetry can split. That is the argument the rest of this ladder is about, and it is the reason the plane’s largest degeneracy is two.

Reading a table, and what the letters are for #

A character table in a textbook has letters down its left-hand side — A₁, B₂, E — and they are a naming convention rather than a result. A and B are one-dimensional, E two-dimensional, T three-dimensional; the subscripts record behaviour under a chosen secondary axis. The convention is Mulliken’s and it is worth knowing, because every table a reader meets elsewhere uses it.

It is deliberately not used here. The tables on this page are labelled Γ₁, Γ₂ and so on in the order the construction produced them, because a name assigned by convention is a name that has to be checked against a convention, and this collection’s habit is to carry the object rather than its label. What identifies a representation on these pages is its row of character values, which no convention can disagree with.

The columns are worth as much attention as the rows. Each is a conjugacy class with the number of operations in it, and the classes are not the operations: the two mirrors of 2mm are in different classes, because nothing in the group carries one onto the other, while the four mirrors of 4mm fall into two classes of two. Which operations a group cannot tell apart is a fact about the group, computed rather than eyeballed, and it decides the shape of the table before any character is found.

What the machinery does not decide #

Two boundaries, and both of them matter later.

A representation says which degeneracies are forced, not which occur. Two levels of different symmetry may coincide for reasons having nothing to do with the group — the weights conspire, or a quantity nobody wrote down is conserved — and no character table forbids it. Separating the two cases takes an experiment rather than a calculation, and the third rung of this ladder performs one.

And a character says how many, never which. A multiplicity is a count of pieces of a given symmetry; it does not hand over the pieces, and it says nothing at all about the numbers attached to them. Two crystals in the same class have identical character tables and different elastic constants, different vibration frequencies, different everything a measurement returns — because the table decides the shape of the answer and the material decides its values. That division runs through every use of this machinery in the collection, and it is the reason a symmetry argument is cheap: it is answered once for a class and never again for a substance.

A real operator has a symmetry no group of motions contains. Complex conjugation is not a motion of the plane, and yet an operator with real matrix elements commutes with it; where a representation is not equivalent to its own conjugate, that extra symmetry joins the two into a single level. The indicator figure above says exactly where this happens — in 4, 3 and 6 — and it is why a group with no mirror has doubled levels that its character table, read naively, does not predict.

The one representation that contains all of them #

The completeness step is described above as the one that turns a collection into a table. There is a single object that supplies it, and it is worth naming because everything numerical about character theory falls out of the same source.

Take the functions on the group itself — one coordinate per element — and let the group act by relabelling. That is the regular representation, its dimension is the order of the group, and its character is easy: the identity fixes every element, so its character is |G|, and every other operation fixes none, so its character is zero.

Decompose it, and the orthogonality relations give the multiplicity of each irreducible representation as its own dimension. So the regular representation contains every irreducible piece, each as many times as its dimension — and taking dimensions on both sides gives

$$|G| = \sum_i d_i^2,$$

which is the sum the next rung is built on, arriving as a bookkeeping identity rather than as a fact to be quoted.

It also supplies the completeness proof. Every irreducible representation occurs in the regular one, so a list that accounts for all of |G| in that sum has accounted for all of them, and the search can stop. That is exactly the fourth step of the construction, and it is why a table can be proved complete rather than merely believed to be.

Getting the basis, not just the count #

Characters answer how many of each piece a space contains. Producing the pieces takes one more object, and it is one this collection already uses under another name.

For an irreducible representation i, form

$$P_i = \frac{d_i}{|G|}\sum_{g} \overline{\chi_i(g)}\, D(g),$$

the average of the operators weighted by the conjugate character. That operator is a projector onto the part of the space transforming as representation i, and applying it to any vector gives either zero or a genuine symmetry-adapted function.

The reason it works is orthogonality again: the average kills every component except the one whose character it was weighted by. And the construction is not new here — setting χᵢ to the trivial character, whose value is one everywhere, gives exactly the averaging projector of Neumann’s principle, whose trace counts invariants. The general projector is the same operator with the trivial representation replaced by any other.

So the whole apparatus is one average, three times over. Averaged with the trivial character it counts invariants and produces them. Averaged with any other character it produces the piece of that symmetry. And its trace, in each case, is a dimension — which is why every count in this field is a character sum and none of them requires a matrix to be written out.

Where this ladder goes #

The next rung takes the dimension sum seriously and asks what it forbids: why no crystal can force a five-fold degeneracy, and why the answer in space is three rather than two. The one after separates a degeneracy the group requires from one that merely happens, by an experiment that moves what has no business deciding it.

And then the whole apparatus moves into reciprocal space, where the group acts not on the crystal but on the waves that can travel through it, and where the translations — dropped as uninteresting here, since a point group is what is left after forgetting them — come back as phases that decide whether two levels may cross.