The shell that splits into kinds

An atom in a crystal has neighbours, and the neighbours at one distance form an orbit under the site’s own symmetry. Anything defined on that shell — a displacement, a charge, a bond order, a weight in a sum — is a list of numbers, one per neighbour, and the group permutes the list.

That is a representation, of dimension equal to the size of the shell, and it is the one a crystallographer meets before any other. It is also the easiest one in the subject to decompose, because its character requires no matrices at all.

The cheapest character there is #

The character of a representation is the trace of the matrix at each group element. For a permutation representation the matrix is a permutation matrix — ones and zeros — and its trace is the number of ones on the diagonal, which is the number of points the operation leaves where they are.

So the character is a count of fixed points. No matrix has to be written down: apply each operation to each neighbour, count the ones that do not move, and the list of counts is the character.

That is worth pausing on, because it makes the whole decomposition available with nothing but arithmetic on a set of points. The identity fixes all twelve. A mirror fixes the neighbours lying on it. A six-fold rotation fixes none, unless a neighbour sits at the centre, which it does not.

Decomposing, and what the multiplicities mean #

With the character in hand, the multiplicity of each irreducible representation is an inner product against the character table — a sum over the group, divided by its order, exact in the cyclotomic ring.

The answer is a list of non-negative integers, one per irreducible representation, and it says how many independent combinations of the shell’s values transform in each way. Those combinations are the symmetry-adapted basis, and every calculation over the shell is easier in it.

Two properties make the list trustworthy without any further checking. The multiplicities are whole numbers, and they weight the dimensions back to the size of the shell. A computation producing a fraction, or producing dimensions that do not add up, has failed rather than found something.

A worked shell, count by count #

6mm on twelve neighbours is worth walking through, because every number in it can be checked by hand.

The identity fixes twelve. The six-fold rotations fix none — a neighbour would have to sit at the centre. The three-fold rotations fix none. The two-fold fixes none. The six mirrors each fix two neighbours, the pair lying on that mirror line, provided the shell is oriented so that they do; a generic shell has its neighbours off the mirrors and the character there is zero.

For the generic shell the character is therefore twelve at the identity and zero everywhere else — which is the character of the regular representation, the one every group has on itself. Its decomposition is known in advance: each irreducible representation appears as many times as its dimension. So 6mm gives one copy of each of the four one-dimensional pieces and two copies of each two-dimensional one, weighting back to 4 × 1 + 2 × 2 × 2 = 12.

That is the case the figures show, and it is the case worth learning first: a shell in general position carries the regular representation, always, and its decomposition needs no computation at all.

Burnside’s lemma, as one line of this #

The multiplicity of the trivial representation in a permutation representation is the number of orbits, and it is worth seeing that this recovers a result the collection already has.

The trivial representation’s character is one at every element, so its multiplicity is the average number of fixed points over the group. That is exactly Burnside’s lemma: the number of orbits is the average number of things each element leaves alone.

So the counting essays’ central tool is a special case of the decomposition here, obtained by asking about one row of the character table rather than all of them. That is a pleasant kind of unification, and it is the sort this collection keeps finding: the same computation appearing in two fields under two names, with one of them a restriction of the other.

The general version says more. Burnside counts how many quantities are invariant; the full decomposition counts how many transform in each of the other ways as well, which is what a calculation needs when it is not looking only for averages.

What changes when the shell is special #

A shell whose neighbours sit on symmetry elements is not in general position, and its character is not the regular one.

Take four neighbours on the mirror lines of 4mm rather than eight in general position. The identity fixes four; each of the two mirrors containing them fixes two; the four-fold fixes none. That character is not the regular representation’s, and its decomposition contains fewer pieces — four neighbours can carry only four dimensions’ worth.

Which pieces survive is decided by the stabiliser. A general result covers it: the representation carried by an orbit is the one induced from the trivial representation of the stabiliser, and its decomposition contains each irreducible representation as often as the trivial representation appears in the restriction of that irreducible to the stabiliser.

That sentence is a mouthful and its content is simple: a shell on a mirror carries only the combinations that are even under that mirror, because the neighbours cannot tell the two sides apart.

What the split is for #

Three uses, in increasing order of how much work the symmetry saves.

Independent parameters. A model with one value per neighbour has as many parameters as neighbours, and the symmetry says how many are independent: the number of pieces, not the number of neighbours. Twelve neighbours under 6mm give six independent quantities, and fitting twelve would be fitting six numbers and six redundancies.

Block-diagonalising an operator. An operator commuting with the group has no matrix elements between pieces of different kinds, so a twelve-by-twelve problem becomes several small ones. That is the practical content of a symmetry-adapted basis, and it is why the decomposition is the first step of every serious calculation on a symmetric structure.

Selection rules. Whether a quantity of one kind can couple to a quantity of another is decided by whether the trivial representation appears in their product, which is a character sum. That is Neumann’s principle in its general form, and the property essays’ counts of independent tensor components are instances of it.

The special shell, where the count changes #

A shell whose neighbours sit on symmetry elements behaves differently, and this is the same phenomenon as the special positions of the direct lattice and the special reflections of the reciprocal one.

If a neighbour lies on a mirror, that mirror fixes it, so the character at the mirror is larger and the decomposition shifts: fewer pieces of the kinds that are odd under the mirror, more of the even ones. If the shell is smaller than the group’s order — an orbit of four under a group of eight, say — the shell is a special orbit and its decomposition is correspondingly restricted.

The rule underneath is the orbit–stabiliser theorem again, which this collection has now used for forms, Wyckoff positions, stars of wavevectors and reflection multiplicities: the orbit’s size is the group’s order divided by the stabiliser’s, and a point with a stabiliser has a short orbit.

Three shells at once, and why the count is additive #

A real neighbourhood has several shells — nearest neighbours, second nearest, third — and the decomposition of the whole neighbourhood is the sum of the decompositions of the shells.

That is immediate from the definition: the space of functions on a union of orbits is the direct sum of the spaces on each, and characters add. So a neighbourhood of eight nearest and four next-nearest neighbours carries the sum of the two characters, and its multiplicities are the sums of the multiplicities.

The consequence is that a large calculation’s bookkeeping never gets harder than the shells it is made of. A hundred neighbours in twelve orbits is twelve small decompositions added, rather than one decomposition of a hundred-dimensional space — and each of the twelve is a count of fixed points on a handful of points.

That additivity is the practical reason the method scales, and it is also why the special shells matter disproportionately: a neighbourhood is mostly general-position shells whose decomposition is known in advance, plus a few special ones that have to be computed.

An instrument, and what it exhibits #

The decomposition is a prediction, and predictions in this collection get exhibited rather than asserted.

Build an operator on the shell that commutes with the group — a weighted adjacency whose weight between two neighbours depends only on the orbit of the pair — and diagonalise it. Its levels come in multiplicities equal to the dimensions of the pieces, with as many levels of each dimension as the multiplicity says.

Then change the weights and do it again. The levels move; the pattern of multiplicities does not, because nothing respecting the symmetry can separate the states of an irreducible piece. That is the same experiment the accidental-degeneracy essay runs, applied here as a check on a count rather than as a test of a coincidence.

The vibrational reading, and its boundary #

The decomposition of a shell of neighbours is the first step of a normal-mode analysis, and it is worth saying what this collection does and does not claim about that.

What transfers. The counting. How many independent displacements a shell has, which combinations transform together, and which of them can couple to which — all of that is the arithmetic here, and it is the same whether the object is a crystal’s neighbour shell, a molecule’s atoms or a framework’s joints.

What does not. The frequencies. Which combination is stiff and which is soft is a question with a force constant in it, and this collection has no force constants anywhere. The instrument above assigns weights to orbits of pairs precisely because it needs some numbers and refuses to pretend they are physical.

The division is the same as everywhere else in this ladder, and the reason for stating it again is that the counting half is so often presented with the physical half attached — a table of modes with frequencies beside them — that the separation stops being visible.

The projection that produces the combinations #

The multiplicities say how many combinations of each kind there are. Producing the combinations themselves takes one more step, and it is short enough to describe.

For each irreducible representation there is a projection operator: a weighted sum over the group of its operations, with the weight at each element being the conjugate of that representation’s character there, scaled by the dimension over the group’s order. Applying it to any function on the shell gives a function transforming in that way, or zero.

So the symmetry-adapted basis is produced by applying each projection to a few starting functions and keeping what comes out. Nothing about the procedure needs the matrices of the representation — only its character — which is the same economy that made the decomposition cheap.

What the projection cannot do is choose which combination of a multi-dimensional piece to call the first one. That is a basis choice inside the piece, it is arbitrary, and any calculation using the basis has to carry the arbitrariness rather than pretend it away.

Why the trivial multiplicity is the interesting one so often #

A pattern runs through the applications, and it is worth naming.

The number of independent components of a tensor a class permits is the multiplicity of the trivial representation in the tensor’s representation. The number of orbits of points is the multiplicity of the trivial representation in the permutation representation. The number of symmetry-allowed terms in an expansion is again a count of trivial pieces.

The reason is that “allowed by symmetry” means “unchanged by every operation”, and the unchanged things are exactly the trivial pieces. So a great many crystallographic counts are one row of one character table, dotted with whatever the object’s own character happens to be.

The other rows are not idle — they say what happens to the quantities that are not invariant, which is what a mode, a displacement or a wave is — but the trivial row is the one an answer is usually read from, and recognising it saves re-deriving the same sum in each field.

What this does not decide #

Two limits, both familiar from the rest of the ladder.

The values. A decomposition says how many independent quantities there are and which combinations they are; it says nothing about their sizes, which depend on everything the symmetry does not fix.

The ordering. Which piece has the largest value, which the smallest, is not a group-theoretic question. The figures above show it changing as the weights change, which is the demonstration that it was never fixed.

Both are the standing division of this whole phase: the structure is exact and comes from the group, the values are measurements and come from everything else.

Where this goes #

This rung closes the representations anchor, which now has the machinery, the bound on what it can force, the test that separates a forced degeneracy from a coincidence, and the everyday computation the machinery is actually used for.

Its natural continuation is in the reciprocal-space ladder, where the same decomposition is performed at each wavevector against a different little group — and where the shell being decomposed is not a set of neighbours but the finite space of Bloch amplitudes, which is the same kind of object with a phase attached.

What a chemist does with the same decomposition #

The split of a shell into kinds is the first step of a construction used constantly outside crystallography, and naming it says what the combinations are for rather than only how many there are.

Take a central atom with a shell of neighbours around it. The neighbours’ functions decompose as above, giving a set of symmetry-adapted combinations — one for each copy of each irreducible representation. The central atom’s own functions decompose too, under the same group, and the two lists can be compared.

A combination on the shell can mix with a central function only when the two transform in the same way. That is the whole of the selection rule, and it turns an interaction problem of dimension shell plus centre into a set of small problems, one per irreducible representation, each involving only the pieces of that kind.

For an octahedral shell of six neighbours, the decomposition gives one totally symmetric combination, one two-dimensional piece and one three-dimensional piece. The central atom’s s function is totally symmetric, its p functions are the three-dimensional piece, and of its d functions two are the two-dimensional piece and three are of a kind the shell does not supply at all. So two of the five d functions can mix with the neighbours and three cannot, and that split is the crystal-field splitting arriving as a consequence of the shell’s decomposition rather than as a fact about d orbitals.

None of that needs an energy. Which combinations exist, and which can interact with which, is decided by two characters and one inner product.

Why the shells’ decompositions add and their levels do not #

The additivity of the counts is a convenience and it can be over-read, so the limit deserves stating.

The multiplicities add. A neighbourhood of three shells has, for each irreducible representation, the sum of the three shells’ multiplicities of it — which is what makes the bookkeeping of a large neighbourhood no harder than the bookkeeping of its parts.

The levels do not. An operator commuting with the group has no matrix elements between pieces of different kinds, which is the block-diagonalisation the essay describes. It has every freedom to connect pieces of the same kind coming from different shells, and it generally does — a nearest neighbour and a second-nearest one whose combinations transform alike are mixed by any operator that reaches both.

So the block structure is by symmetry kind and not by shell, and a decomposition that produced two copies of one representation from two shells produces a two-by-two block rather than two separate levels. The counting is additive and the diagonalisation is not, and a reader taking the first for the second would expect a spectrum with one level per shell per kind.

That is the same distinction the essay draws about frequencies, made one step earlier: symmetry says which blocks exist and how large they are; what is inside a block is a matrix nobody has been told the entries of.