Domains of a subgroup

A fundamental domain is the piece of the cell a group rebuilds the rest from. A group with more operations rebuilds more, so it needs less to start with — and the trade is exact rather than approximate.

If H sits inside G with index k, a fundamental domain for H is k copies of a fundamental domain for G. Not roughly k copies, and not k copies of something similar: the domain for H can be taken to be the union of the images of G’s domain under one representative of each coset, and the count of those cosets is the index.

That is the containment ordering of the classification restated as a statement about area, and it is worth having in that form because area is something a figure can show and an inclusion of matrix sets is not.

Why the copies are cosets #

The mechanism deserves a careful sentence, because “twice as many operations, half the area” is the kind of statement that sounds like a proportionality when it is really a partition.

Take a domain D for G: every point of the plane is equivalent under G to exactly one point of D. Now restrict attention to H. Two points of D that were equivalent under an operation of G outside H are no longer equivalent — H does not contain the operation that identified them — so D is no longer big enough to represent every H-orbit. What is missing is precisely one copy of D for each coset of H in G, since operations in the same coset identify the same pairs.

So the domain for H is D together with the images g·D for one g from each coset, and the number of pieces is [G : H]. The arithmetic is Lagrange’s theorem — the order of H divides the order of G — and the figures assert that division comes out whole before they draw anything.

Reading the ratio in the other direction #

The same relation is more useful backwards, and that is how a crystallographer actually meets it.

A structure that loses a symmetry — at a phase transition, on cooling, under strain — goes from a group G to a subgroup H. What was one Wyckoff position under G splits into several under H, because points that G identified are no longer identified, and each of them gains free parameters that G’s symmetry had fixed.

The number of pieces is the index. A position of multiplicity eight in G becomes, in an H of index two, either one position of multiplicity eight or two of multiplicity four, according to whether the lost operation acted within the orbit or between two halves of it. Both cases are common and telling them apart is the substance of the analysis.

This is the setting in which the index arithmetic earns its keep. A structure described in the higher group has fewer parameters to refine; a structure described in the lower one has more, fits better because it has more freedom, and may be fitting nothing but noise. The choice between them is the same threshold question as before, with the index supplying the count of extra parameters that a statistical test has to pay for.

Every group contains p1, and what that says #

The extreme case is the one every group shares, and it makes the arithmetic concrete in a way the index-two pairs do not.

p1 is a subgroup of all seventeen. Its only operations are the lattice translations, which every plane group contains by definition, so the index of p1 in a group is simply that group’s order — one for p1 itself, twelve for p6m.

The domain for p1 is therefore the whole cell, every time. That is not a special case needing separate treatment; it is the statement that a pattern with no symmetry beyond repetition has to be drawn in full before it repeats, and the more symmetry a group has, the less of the cell anybody has to draw.

The p1 comparison is worth doing once because it removes a possible misreading of all the others. A pair like cmm and cm looks as though the domain grows because a particular operation was lost, and that the shape of the larger domain records which one. It does record that; but the growth itself does not depend on which operation went, only on how many. Take away every operation but the translations and the domain grows by the whole order of the group, whatever those operations happened to be — and p4, whose four operations are rotations, and pmm, whose four are mirrors and a half turn, both go to the same whole cell.

Read as a practical statement, this is the amount of work symmetry saves. Drawing p6m means drawing a twelfth of a cell and letting the group do the rest; drawing p1 means drawing all of it. That ratio — twelve to one between the extremes of the classification — is the same number as the ratio of the group orders, and it is the most immediate reason ornament traditions favour the symmetric groups: the high-symmetry patterns are less work as well as easier to get right.

Where the measured ratio is not exactly the index #

The figures report a measured ratio and it is never quite the integer. The discrepancy is worth understanding, because it is the same one the domain essay has to explain and it comes from the special positions.

The domain is computed by sampling the cell and keeping one representative per orbit. A sample lying on a mirror or a rotation centre is its own whole orbit, so it is kept entire rather than shared between the images — and it is kept entire in both domains. The boundary is therefore counted once too often on each side, and since the two domains have different boundary lengths the ratio comes out slightly off.

It is a sampling artefact and it shrinks predictably. The boundary is one-dimensional and the interior is two-dimensional, so the excess falls as the grid is refined, and the assertion in the figure is an inequality with the slack stated in terms of the grid size rather than an equality with a fudge factor. That is the honest form of the claim and it is the form the site prefers: an assertion that has to be told how much room to leave is one whose author knows why the room is needed.

That census settles what the excess is, and it settles it by exhibiting the one case where it vanishes. If the discrepancy were a defect of the sampling it would appear everywhere, since every group is sampled the same way on the same grid. It appears in fifteen groups and not in the two whose operations hold no point of the plane still — which is exactly the condition under which no sample can be its own whole orbit. A quantity that disappears precisely when its stated cause is absent is not noise.

It also says why the four pairs drawn above miss the integer by such different amounts, and the amounts are worth putting side by side. p4m against p4 measures 1.51 where the index is two; cmm against cm measures 1.88; pmm against pm, 1.80; p6m against p3m1, 1.73. The worst of them is the pair that loses mirrors and keeps a large group — p4m’s four mirror lines are counted whole in a domain that is only an eighth of the cell, so the boundary is a large fraction of a small region. The best is cmm against cm, where the operation lost is a half turn and what it holds still is a handful of isolated points. The discrepancy is not a fixed property of the sampling; it is the boundary of the particular domain, divided by its area.

The pairs that cannot be compared #

Not every containment among the seventeen can be drawn this way, and the reason is a limitation this site has recorded before.

The comparison of two groups’ operation sets is legitimate only because both are written in one shared coordinate convention. Hexagonal and square lattices are described on bases that no integer matrix relates, so a containment between a hexagonal group and a square one is invisible to a test that compares matrices — even where a geometric relation exists.

So the pairs available to this figure are the pairs on a shared lattice: p4 inside p4m, cm inside cmm, pm inside pmm, p3m1 inside p6m. That is a large enough family to make the point and it is not the whole containment ordering, and saying so is more useful than quietly drawing only the cases that work.

Splitting a Wyckoff position, worked through #

The abstract statement — one position of G becomes several under H — is easier to believe with one case followed all the way.

Take p4m, of order eight, and its subgroup p4, of order four, of index two. A general point of p4m has an orbit of eight. Under p4 that same point has an orbit of four, so the eight points split into two p4 orbits — the direct images and the reflected ones, which p4m identified and p4 does not.

Now take a point on a mirror. Under p4m its orbit is four, because the mirror fixes it. Under p4 its orbit is also four, because p4 contains no operation fixing it and its four images are already all distinct. So this position does not split: it becomes an ordinary general position of p4, and what it loses is not multiplicity but its constraint — an atom there was forced to lie on the mirror, and in p4 it is free to move off.

The two cases are exactly the alternative the index arithmetic allows: an orbit of size m in G becomes either one orbit of m or k orbits of m/k in H, and which happens depends on whether the lost operations act inside the orbit or across it. Both are visible in the same pair of groups, which is why p4m and p4 is the standard first example.

What the round trip checked, and how #

Three things are asserted before either panel is drawn, and they are independent of one another.

Lagrange, as a division. The order of the subgroup divides the order of the group, checked as a remainder rather than assumed from the theory. A pair that failed it would mean the containment test had accepted something that is not a subgroup.

Containment, before comparison. The figure refuses outright to draw a pair where the smaller group is not contained in the larger. That refusal is what makes the index meaningful: without it the ratio of two domain areas is a ratio of two unrelated numbers.

The partition, on both sides. Each domain is produced by the construction that requires every sample’s orbit to meet the domain exactly once — no gaps, no overlaps — so the two areas being compared are both genuinely fundamental domains rather than regions that happen to look like them. That check is what caught a domain with a hole in it during an earlier phase, where the hole looked like an interesting feature of p3.

Normal, and not #

One distinction the area picture does not show, and which matters as soon as the subgroup is used for anything, is whether the subgroup is normal.

A subgroup is normal when conjugating it by any element of the group gives it back — when, in the language this site uses for conjugation, moving the subgroup’s operations to another part of the pattern leaves the same set of operations. Every subgroup of index two is normal, which is why all the pairs drawn here are, and it is the reason two-colourings work: the colour-preserving half must be normal for the colour rule to be consistent.

Subgroups of larger index need not be. p1 inside p4m has index eight and is normal — the translations are normal in every plane group by construction. But a group generated by one mirror inside p4m is not: conjugating it by the fourfold rotation gives the group generated by a different mirror, and the two are different subgroups of the same abstract type.

The consequence for domains is that the picture stays true and the interpretation changes. A non-normal subgroup still has a domain that is the index many copies, because the coset counting never used normality. What changes is that the copies are not interchangeable: there is no operation of the big group carrying the arrangement onto itself while permuting them, so the pieces are related to the original and not to each other in any uniform way.

That is the same distinction that makes conjugate subgroups “the same symmetry somewhere else” rather than the same subgroup, and it is the first thing that has to be settled before a subgroup relation can be used to describe a phase transition — a transition to a non-normal subgroup produces domains in the physical sense as well, regions of the material related by the lost operations.

What the ratio does not settle #

Two groups can stand in an index-two relation in more than one way, and the area picture cannot tell those ways apart.

p4m contains p4, and it also contains pmm, cmm and p4g — all of index two, all giving a domain of twice the area. Four different subgroups, four identical ratios, and a figure showing areas alone would give the same picture for each. What distinguishes them is which operations were kept, and that is a fact about the operation lists rather than about the region.

This is worth stating because it is the limit of the whole visual argument. A fundamental domain is a good picture of how much a group folds up and a poor picture of what it folds with. Two groups of the same order on the same lattice have domains of the same size, and the site’s other figures — the ones marking mirrors, glides and rotation centres — exist because the domain does not distinguish them.

Two things called a domain #

The word in this essay’s title is doing one job and the same word elsewhere in this collection does another, and the index counts both — which is convenient and is the reason the two get run together.

A fundamental domain is a region of the cell: the piece a group rebuilds the pattern from. Its area is the cell’s divided by the group’s order, and a subgroup of index k needs k times as much of it. That is this essay.

A domain state is a region of a crystal: one of the several arrangements a material can adopt when it loses a symmetry, counted by the same index. A descent of index k produces k of them.

The two are different objects and the coincidence of counts is not one. The k copies of a fundamental domain are the cosets applied to one region of space; the k domain states are the cosets applied to one arrangement of atoms. Both are indexed by the cosets, which is why both come out as the index, and neither is a picture of the other.

A useful way to keep them apart: the fundamental domains of a descent all sit inside one crystal at once, tiling its cell. The domain states do not — a given region of the crystal is in exactly one of them, and the others are elsewhere in the specimen or absent entirely.

The conjugates, and which one a crystal picks #

The normality remark has a physical form worth making explicit, because it is where the abstract distinction becomes something a microscope can see.

If H is not normal in G, its conjugates gHg⁻¹ are different subgroups of the same index, and each has its own fundamental domain of the same area in a different place. They are the same symmetry sitting differently, which is the essay’s own phrase for conjugation.

A crystal descending from G to H must choose one of them, and different regions choose differently — so the conjugates are precisely the orientation domain states, and how many there are is the number of conjugates rather than the index. Those two numbers agree when H is normal and not otherwise, which is the one place the coincidence above breaks.

Worked on the essay’s example: p4 is normal in p4m, so there is one conjugate and the descent produces domain states that differ by a translation rather than by an orientation. pm inside pmm is normal too. A non-normal case needs a larger group — and where one occurs, a descent of index four can produce two conjugate subgroups with two states each rather than four states of one.

The area picture cannot see any of that, since conjugate subgroups have domains of equal area, and it is the first thing to compute after the index rather than the last.

Where the exactness stops #

Area is measured in samples. Both domains are counted in grid cells, so every number here is a count with the grid’s resolution built into it. The index is exact; the measured ratio is not, and the two are reported separately for that reason.

The domain is one choice among infinitely many. Any region containing one point of each orbit is a fundamental domain, and the construction here picks a particular one — the representatives that come first in a fixed ordering. A different rule gives a differently shaped region of the same area, which is the invariant the whole essay is about.

Index two throughout. Every pair drawn here has index two, because those are the containments the shared-basis test finds among groups on one lattice. Larger indices exist — p1 sits inside p4m with index eight — and the same arithmetic applies, with a domain eight times the size and correspondingly more boundary to confuse the measurement.

Where the ladder goes next #

The construction underneath both panels, and the assertion that makes it a partition rather than a shading, is the fundamental domain.

The points that make the measured ratio miss the exact one, and the arithmetic of orbits and stabilisers behind them, is the points a group treats differently.

The ordering these indices belong to is the classification proof, and the notation that names a domain’s shape rather than its area is orbifold notation.

What the pictures here cannot show. Two shaded regions of different sizes are not evidence that one is a union of copies of the other: the copies are related by operations that no drawing displays, and the panels are drawn separately rather than superimposed. The claim that the larger domain is exactly the smaller one and its images is the output of the coset arithmetic, and what the figure shows is the consequence — the ratio of the two areas — rather than the reason.