A row written as a product

Every question so far has been about one index at a time. At which indices does a group contain a copy of itself; which of those copies are maximal; what happens along an axis. The answers are rows of integers, and a row of integers is an object in its own right.

There is a standard way of holding a whole row at once. Attach to a sequence $a_1, a_2, a_3, \ldots$ the function

$$\sum_{n \ge 1} \frac{a_n}{n^s},$$

a Dirichlet series, and the arithmetic of the sequence becomes the algebra of the function. The useful thing that can happen to such a series is that it factors into a product over the primes, one factor for each, and that happens exactly when the coefficients are multiplicative — when the value at a product of coprime numbers is the product of the values.

Every one of the seventeen rows factors, and the content of the factorisation is the statement each of those answers has been making one index at a time: a copy at a composite index is built from copies at prime powers, one prime at a time, and nothing else.

What is being counted, and what it is not #

The coefficient at index n is the number of sublattices of index n that the group’s point group carries to itself. That is the right thing to count for this purpose and it is not the number of subgroups.

It is the right thing because maximality is about lattices. The criterion is that a copy is maximal exactly when no invariant lattice lies strictly between its lattice and the whole, and the argument for it needs nothing about which subgroup sits on a lattice. So the poset the whole question lives in is the poset of invariant sublattices, and its counting function is the series below.

It is not the number of subgroups, because one lattice can carry several copies — pm’s lattice doubled across its mirrors carries two, with the mirrors on the even lines or on the odd ones. How much the two counts differ is a separate census, and every number on this page is the smaller of the two.

The factors, read off the prime powers and multiplied back #

A series has an Euler product when its coefficients are multiplicative. Whether these are is not assumed.

The values at $1, p, p^2, \ldots$ are read off the row for each prime; those local sequences are multiplied together into a single series by convolution; and the result is compared with the row, index by index. For every one of the seventeen the two agree everywhere.

That agreement is a fact and not a definition, and what it means is worth spelling out. A sublattice of index 15 invariant under p4 corresponds to a pair — an invariant sublattice of index 3 and one of index 5 — and to exactly one such pair. The two primes do not interfere: the choices at three and the choices at five are made independently and every combination occurs. Put in the language the maximal copies use, every copy is a chain of maximal steps, the steps at different primes commute, and no chain is counted twice.

Where the coefficient is nought the factor is nought at that power and the product is nought at every multiple of it, which is why p4’s row has holes at 3, 6, 7, 11, 12, 14 and so on: three is inert in ℤ[i] and has no ideal of its norm, so nothing divisible by three to an odd power appears.

p1 has the series everybody knows #

The group with no operation but the identity preserves every sublattice, so its coefficients are the numbers of sublattices themselves.

How many ways there are to thin a lattice gives the count: a sublattice of index n in the plane corresponds to a Hermite normal form, and the number of them is the sum of the divisors of n. So p1’s row is 1, 3, 4, 7, 6, 12, 8, 15, 13, 18 — and a Dirichlet series with the divisor sums as coefficients is

$$\zeta(s)\,\zeta(s-1),$$

the Riemann zeta function times a shift of itself. The group with the least symmetry has the most famous series in mathematics, and it has it because having no symmetry means imposing no condition, so what is left to count is the sublattices of ℤ² and nothing more.

p2 has the same row, for the reason the primes a cell can grow by gives: a half-turn preserves every sublattice, since it is minus the identity and negation carries a lattice to itself. So the two groups with the largest rows are the two whose point groups impose nothing.

One row, read in the ring instead #

For the groups whose invariant lattices are ideals, the coefficients have a closed form that owes nothing to enumeration, and comparing the two is the strongest check the row admits.

p4’s invariant sublattices are the ideals of ℤ[i], and the number of ideals of norm n in that ring is a classical sum: add up, over the divisors d of n, plus one for each d one more than a multiple of four, minus one for each d one less, and nothing for the even ones. At 25 the divisors 1, 5 and 25 all count plus, giving three; at 10 only 1 and 5 count, giving two; at 9 the divisors 1 and 9 count plus and 3 counts minus, giving one; at 3 the count is nought.

The row computed by enumerating sublattices and testing H⁻¹MH for integrality gives exactly those numbers. Two counts with nothing in common but their answer — one a search over Hermite normal forms, the other a divisor sum with a character in it — and the agreement is what licenses reading the row as a statement about the ring rather than about a search.

The same sum with residues modulo three counts p3’s and p6’s ideals in ℤ[ω]. And the groups with mirrors have no such closed form here, because a mirror keeps only the ideals that are their own mirror images and that is a condition on a generator rather than on a norm.

cm’s row is worth seeing for what a single mirror leaves. Its factor at two is 1, 1, 3, 5, 7 — one invariant lattice at index two, three at four, five at eight — where the factors at three and five are the plain 1, 2, 3, 4. A local factor is exactly the right object to hold that asymmetry: it is a statement about the powers of one prime, and the awkwardness of two for a centred lattice is a property of two alone.

And this is where counting lattices and counting copies come apart most visibly. The maximal indices record that cm has no copy of itself at index two, although the row above shows an invariant lattice there. The lattice exists and the subgroup on it is a different group, which is the same distinction the axial case makes with the enantiomorphic partner — and it is the reason the census of copies counts subgroups rather than lattices.

Every operation is a condition, and conditions subtract #

Past those two the rows thin, and they thin in a way that is nearly monotone in the order of the point group.

p1 and p2, of order one and two, have seven hundred and sixty-two invariant sublattices between index two and index thirty. pm, pg, pmm, pmg and pgg have a hundred and fifty-six; cm and cmm have ninety-eight and a hundred; p4 has twenty-four; p3 and p6 have eighteen; and p4m, p4g, p3m1, p31m and p6m have eight each.

An operation of a point group is a condition a sublattice must satisfy, and the conditions are close to independent, so each one cuts the count by roughly a constant factor. That is why the order and the room run in opposite directions, and why the relationship is a trend rather than a law: cm has fewer lattices than pm at the same order, because a mirror along a diagonal of a centred cell is a different condition from a mirror along an edge.

The rows with the fewest entries are the most informative about what the series is for. p6m has eight invariant sublattices to index thirty, at indices 3, 4, 9, 12, 16, 25, 27 and 36 — a very sparse set — and its Euler product has a factor at each prime that is almost entirely zeros. A row that sparse would be tedious to describe one entry at a time and is two lines as a product.

The holes are where the primes do not split #

A row’s zeros are as informative as its entries, and for the rotation groups they have a one-word cause.

p4’s row is nought at 3, 6, 7, 11, 12, 14, 15, 19, 21, 22, 23, 24, 28 and 30 up to thirty. Every one of those is divisible by a prime three more than a multiple of four to an odd power. Such a prime stays prime in ℤ[i], so its ideal has norm $p^2$ rather than $p$, and no ideal of norm p exists at all — which is the same statement as a sum of two squares failing to represent it.

p3’s and p6’s rows are nought wherever a prime two more than a multiple of three appears to an odd power, for the identical reason in ℤ[ω]. And p1’s row has no zeros at all, because there is no condition to fail.

So the shape of a row is a statement about which primes split in a ring the group never mentions. That is the sentence every answer so far has been converging on, and the series is where it is shortest: the local factor at a split prime is the geometric-looking sequence 1, 2, 3, 4 …, the factor at an inert prime is 1, 0, 1, 0, 1 …, and the factor at a ramified prime is 1, 1, 1, 1 …. Three patterns, one per way a prime can behave, and the whole row is their product.

What the series is not #

Three things a Dirichlet series invites a reader to expect, and which this one does not supply.

It does not converge to anything useful here. A Dirichlet series is an analytic object and the interesting questions about it are about poles and growth. p1’s is ζ(s)ζ(s−1), which has poles at s = 1 and s = 2 and whose behaviour there says the number of sublattices of index at most $N$ grows as $N^2$, and that is a genuine statement. For the other sixteen the analytic side is not pursued on this page at all: the series is used as a bookkeeping device for a row, which is the smaller half of what such an object is for.

It does not count subgroups. The Tables list subgroups, and the correction is the census of copies on one lattice.

And it does not distinguish the copies from the other invariant lattices. Every invariant sublattice carries some subgroup with the parent’s point group; whether that subgroup is the parent’s own type is a further question, and in the plane the answer is almost always yes — which is why the distinction did not arise until an axis in space made the along-axis steps land on the enantiomorphic partner. The rows above are rows of lattices, and for the plane they are also rows of types.

Where the exactness stops #

Computed here. For each of the seventeen and each index to thirty, the sublattices of that index and which of them every operation of the point group carries to itself, by the integrality test on H⁻¹MH. From those coefficients, the local factor at each prime, the product of those factors, and a comparison with the coefficients at every index. And, for p1, the sums of the divisors computed separately and compared.

Thirty is where it stops, and multiplicativity is checked only there. That the coefficients are multiplicative is verified at every index to thirty and is not proved. It would follow from the sublattices of coprime indices being in bijection with pairs, which is the Chinese remainder theorem applied to lattices and is a short argument that is not made here.

The series is written and not summed. No analytic property of any of the seventeen series is computed. Only p1’s is identified with a known function, and that identification rests on the divisor sums.

The zeros are zeros of the lattice count, not of the subgroup count. A row entry of nought means no invariant lattice at that index, and then certainly no copy; a row entry of one means a lattice, and whether a copy sits on it is a further question the row does not answer. Every statement above about “a copy at index n” is therefore a statement about the possibility of one.

And the point group is taken as acting on a fixed lattice. Which sublattices are invariant depends on the lattice type as well as the point group, which is why cm and pm differ; the rows are properties of an arithmetic class rather than of a geometric one, in the sense the thirteen classes draw.

The refusal is a cubic sublattice of index three, which is the cubic arithmetic arriving early: thirteen sublattices of index three exist in space and not one of them keeps the cubic symmetry, so the cubic row’s coefficient there is nought and the series is sparser than anything the plane offers.

Who counts lattices this way #

Counting sublattices by a Dirichlet series is standard in the arithmetic of groups, where the object is called a subgroup zeta function and the general theory is due to Grunewald, Segal and Smith from 1988. Their subject is much larger — the zeta function of a finitely generated nilpotent group, of which a lattice is the easiest case — and the fact that ℤ²’s is ζ(s)ζ(s−1) is the first example in every account of it.

What is added here is the point group. A zeta function counting only the sublattices a given finite group of matrices preserves is not a standard object under a standard name, and the reason is that for most purposes the group is what is being counted rather than what is doing the constraining. The seventeen rows above are a small, completely computable instance of the general construction, with the finite group supplying the conditions and the ring supplying the answer.

The connection to the International Tables is the practical end. The Tables’ series of maximal isomorphic subgroups is the Euler product’s prime factors, written out as text; the composite entries the Tables leave implicit are what the product generates.

Still open: the series as a function #

Everything above treats the series as a way of writing a row. The other half of what a Dirichlet series is for is untouched, and two questions are worth naming.

Where are the poles? p1’s series has a pole at s = 2, which says the number of invariant sublattices of index at most $N$ grows as $N^2$. Every other row is sparser, so every other series has its rightmost pole further left, and the position measures how much the point group costs. For p4 the count should grow as N times a constant, since the Gaussian ideals of norm at most N number about N times π over four; for p6m it should grow as the square root of N. Those are predictions from the arithmetic, and nothing above computes the growth to test them.

And does the axial case factor too? The axial case finds the invariant sublattices of an axial group to be an ideal across times a multiple along, which is a product of two posets — so its series ought to be a product of two series, one for the plane’s cross-section and one for ℤ. Whether that survives the screw congruence, which forbids some along-axis steps outright and sends others to the enantiomorph, is a question about a series with two kinds of coefficient and is not asked here.