The copies a group keeps of itself grow at four speeds

A plane group contains copies of itself: subgroups of the same type with a larger cell, the patterns a crystal makes when it keeps every kind of symmetry it had and repeats less often. The richest group has the poorest arithmetic counted where such copies can sit. A copy needs a sublattice that every operation of the point group carries to itself, and the more operations there are, the fewer sublattices survive: p1 keeps every sublattice there is, p6m only its own scaled and turned copies. In the cubic case the count collapsed to one lattice at each cube, twice a cube and four times a cube, and nothing else.

That essay ended on a complication it could not resolve. A sublattice is not a subgroup. A lattice is not a subgroup found that one invariant sublattice can carry several copies of the group, differing in how the translations attached to its operations are shifted, and that the number of them is a cohomology group which grows with the point group. So two effects run in opposite directions as symmetry increases: fewer lattices, more copies on each. Over the indices to six, the falling count won. Whether it goes on winning, or whether the rising multiplicity catches it somewhere further out, was left as a measurement to make, “the copy count taken to the indices where the lattice count has thinned”.

This essay makes it, to index ninety-six for all seventeen groups. The answer is that the two effects never cancel in the long run, and the reason is that they are not the same kind of quantity. One is a growth rate and the other is a bounded factor.

What is counted #

For each plane group and each index $n$ from two to ninety-six, every sublattice of index $n$ is enumerated by its Hermite form, and kept if every operation of the point group carries it to itself. On each kept lattice $L'$ the copies of the group are counted as the first cohomology of the point group with coefficients in the finite quotient $L/L'$, the count a lattice is not a subgroup set up: the ways of shifting each operation’s translation by an element of the quotient that close into a group, modulo the shifts that a change of origin produces. Two cumulative numbers come out for each group: $L(N)$, the invariant lattices of index at most $N$, and $C(N)$, the copies on them.

The lattice counts have a second route wherever the arithmetic names them. For p1 and p2 they are the divisor sums $\sum_{n \le N} \sigma(n)$, since every sublattice is kept. For pm they are a divisor sum in which each divisor counts twice when it is even. For p4 they are the ideals of the Gaussian integers of norm at most $N$, for p3 and p6 the ideals of the Eisenstein integers, and for p4m and p6m the scaled copies of the lattice and the copies turned through forty-five or thirty degrees, at indices $k^2$ and $2k^2$ or $3k^2$. All eight formulas agree with the enumeration at every index to ninety-six, seven hundred and sixty comparisons rather than a handful.

Four speeds #

The figure at the head of this essay is the result, and it has a cleaner structure than the lead suggested. The groups fall into four tiers, and within a tier they climb at the same rate.

$N^2$. p1 and p2 keep every sublattice, because the half-turn reverses every vector and so carries every lattice to itself. There are 7,656 sublattices of index at most ninety-six, and the count grows as the square of the index, with slope 1.99 between forty-eight and ninety-six on the logarithmic plot.

$N \log N$. pm, pg, cm, pmm, pmg, pgg and cmm keep only the sublattices a mirror carries to itself, which in a rectangular lattice means those whose Hermite form is nearly diagonal: 656 of them for the rectangular groups and 428 for the centred ones, growing with measured slope 1.22, the signature of $N \log N$ over this range.

N. p3, p4 and p6 keep the sublattices a rotation of order three, four or six carries to itself, which are the ideals of a ring of quadratic integers, 57 for the hexagonal groups and 72 for p4. Their count grows linearly, as the number of lattice points in a growing disc does, because an ideal is named by a lattice point up to the ring’s units.

$\sqrt N$. p3m1, p31m, p4m, p4g and p6m keep only the sublattices that are the whole lattice scaled, or turned through the one angle the mirrors allow, so there are one or two at each square or twice or thrice a square and none in between: 13 or 14 to ninety-six. That is the plane’s version of the cubic collapse.

The tiers are set by the point group’s arithmetic and not by its size. A half-turn changes no lattice count at all: p1 and p2 keep the same 7,656 lattices, pm and pmm the same 656, p3 and p6 the same 57, because the half-turn is minus the identity on the plane and minus the identity keeps every lattice. What moves a group down a tier is an operation that is not $\pm 1$: a mirror, or a rotation of order three or more, or both.

Why each tier has its exponent #

The exponents are not fitted numbers; each one is a count of how many free choices a kept sublattice has.

A sublattice of the plane of index $n$ is two choices: a first row, along any direction, and a second, in any of a number of positions that grows with $n$. How many sublattices gave the count of them as the divisor sum $\sigma(n)$, which is about $n$ on average, so the total to $N$ is about $N^2$. A group whose only operation is the half-turn leaves both choices free.

A mirror pins one of them. A sublattice a mirror keeps must have a row along the mirror and a row across it, or be the centred version of such a rectangle, so it is fixed by two divisors whose product is the index. The number of ways to split $n$ into two such factors is the number of its divisors, about $\log n$ on average, and the total to $N$ is about $N \log N$. The direction is gone; only the lengths along two fixed axes remain.

A rotation of order three or more pins the direction too, but not the length or the angle within the rotation’s own symmetry. A sublattice it keeps is the lattice multiplied by a complex number that respects the rotation, an element of the Gaussian or Eisenstein integers, and the sublattices that are the same shape found exactly these as the similar sublattices. Such a multiplier is a point of the lattice itself, and the multipliers giving index at most $N$ are the lattice points in a disc of area proportional to $N$. So the count is linear.

Mirrors and a rotation of order three or more together pin the angle as well, to the parent’s own directions or the one turned direction the mirrors allow. Only the length is free, and a length whose square is the index gives about $\sqrt N$ choices to $N$. Each step down the tiers removes one degree of freedom from a kept lattice, and the exponent drops with it.

The multiplicity is a factor #

The copies per lattice do grow with the point group, as the earlier essays found. p1 has exactly one copy on every lattice, since a group of translations has nothing to shift. p2 has 1.76 on average and up to four; pmm, pmg and pgg 4.87 on average and up to sixteen on a single lattice; p4m 3.14 and up to four. But the averages stop there. Across all ninety-five indices, the largest number of copies on any one lattice in any plane group is sixteen.

That ceiling has a reason. The cohomology group counting the copies is killed by the order of the point group, and a cocycle is fixed by its values on the point group’s generators, of which a plane group needs at most three, so the group has a bounded number of generators each of bounded order, whatever the index. Its size therefore has a ceiling set by the point group alone, and for pmm, pmg and pgg the census reaches sixteen. However large the index, the multiplicity is drawn from a fixed finite menu.

So the copy count is the lattice count multiplied by something between one and sixteen. A bounded factor cannot change a growth rate. $C(N)$ grows at the speed of $L(N)$ in every group, with the same measured slopes to two decimal places in the first three tiers, and the factor only decides by how much a group leads others in its own tier.

Where the lead ends #

The two effects do compete at small indices, and that is where the earlier essays saw them.

To index six, pmm has seventy-two copies of itself, more than p2’s fifty-six and more than twice p1’s thirty-two. Its sixteen copies on a single lattice outweigh the few extra lattices the lower groups have at small indices. p4m likewise leads p4, eight to six. Then the lattice counts pull ahead. p4 passes p4m from index ten, p2 passes pmm from index eleven, and p1 passes pmm from index twenty-seven, and from each of those indices the order is never reversed again to ninety-six. It cannot be reversed later either: the poorer group’s lattices are growing faster by a power of the index or by a logarithm, and the richer group’s advantage is a constant factor.

That is the cancellation the lead asked about, and it happens exactly once for each pair, at a small index. Inside a tier there is no cancellation at all. pmm and pm keep the same lattices and pmm carries at least as many copies on each, so pmm leads pm at every index, and p2 leads p1 at every index for the same reason. The only contests are between tiers, and each is settled within the first thirty indices.

What the crossover index means for a crystal #

The crossover indices are small, and that is the useful part. A structural search for an ordered superstructure, or a calculation that tries every supercell up to some size for a lower-energy arrangement, usually works within a few tens of cells. At ten or twenty cells, which is where the counts above cross, the richer groups still carry their multiplicity: pmm has more ways to stay pmm in a cell of ten than p2 has to stay p2. Above about thirty they no longer do. A search that budgets its time by the number of candidate subgroups per size will find the rectangular groups expensive at small sizes and cheap at large ones, and the change happens inside the range such searches use.

It also says something about which superstructures a crystal can adopt without losing symmetry. The superstructures the reflections a superlattice adds are seen in a diffraction pattern; those that keep the full plane group are the copies counted here, and a crystal of high symmetry has very few of them at any size. A mirror-symmetric hexagonal layer that orders without losing a single operation can only do so on a cell enlarged by a square, three times a square, or, for the square groups, twice a square. Anything else is a symmetry loss as well as a cell change.

The ranking settles #

Ranking the seventeen by their copies shows the same thing for the whole set. At index six the rectangular groups with four operations lead, and the correlation between a group’s copies and the order of its point group is −0.55: negative already, since the six-fold and four-fold groups with mirrors are at the bottom, but loose. By twelve p2 leads; by thirty p1 has passed the rectangular groups; from thirty on the order barely changes and the correlation sits at −0.76 and −0.77. The richest groups, p6m, p31m and p3m1, have thirteen or twenty-nine copies to ninety-six. p2 has 13,446.

In a crystal this is the difference between a family of superstructures that grows with every new composition and one that is nearly closed. A structure of a low-symmetry group can order into a copy of its own group in thousands of ways within a modest supercell; a hexagonal structure with mirrors can do it at a handful of sizes, and a row written as a product already showed which sizes, prime by prime.

Counted twice, refused once #

The enumeration and the closed forms meet at every index for the eight groups that have one; a missed sublattice or a wrongly kept one would show as a count one off at a single index. The cohomology counts are the module the earlier essays already checked against direct construction at small indices, and here they are only carried further; seventeen, without a picture used a different cohomology of the same point groups to count the seventeen themselves. And the claim the lead left open is refused in its strongest form: p4m’s extra copies do not let it catch p1 at any index, since by ninety-six it has forty-four copies against p1’s 7,656.

Where the count stops #

The plane. In space the same tiers exist with different exponents, and the earlier essay’s cubic row is the extreme one: the full cubic group keeps a lattice only at $k^3$, $2k^3$ and $4k^3$, a growth like the cube root of the index. Groups with a single axis of order three or more keep the sublattices an ideal across the axis and any period along it allow, as an ideal across and a prime along found, and they grow faster. The bounded-multiplicity argument carries over unchanged, since the cohomology of a finite point group is still killed by its order.

Copies up to conjugacy in the group. Two copies that a change of origin outside the group, a normaliser element, would exchange are counted twice here. Counting them up to the normaliser would divide some counts by small factors and change no growth rate.

And copies of the same type only. A group has many subgroups of lower symmetry at each index, and those are counted by other rows of the tables. The question here was the narrow one: how often a crystal can keep its whole symmetry and grow its cell.

Who counted what #

That the number of subgroups of index $n$ in a group grows at a rate fixed by the group is the subject of subgroup growth, developed by Lubotzky, Segal and others from the 1980s on, mostly for infinite groups far wilder than these. For crystallographic groups the copies of a group in itself are the isomorphic subgroups of the International Tables, listed there by series indexed by primes or by values of a quadratic form, with the counting of the series made explicit by Bertaut, Billiet and Müller among others. What a census adds is the comparison across the seventeen at the same indices, which a list of series per group does not make.

Still open: the space groups’ tiers #

The plane’s four tiers are $N^2$, $N \log N$, N and $\sqrt N$. Space has more room: a monoclinic group keeps the sublattices a two-fold axis keeps, an orthorhombic one those three perpendicular axes keep, a tetragonal or hexagonal one an ideal across times anything along, a cubic one almost nothing. Each should grow as a definite power of the index times perhaps a logarithm, and the crossovers between a richer group’s early lead and a poorer group’s faster growth should again fall at small indices. Which exponents occur, whether any two of the seven crystal systems share one, and where the crossovers fall would be the same enumeration over the sublattices of $\mathbb{Z}^3$, much larger at each index, with the same cohomology on each. It has not been run here.