What the twenty lack is a product of mirrors

Twelve of the thirty-two are free sorted the crystal classes by the polynomials they leave alone. For twelve of them the invariant polynomials form a ring generated by three polynomials with no relation among them, so that counting the invariants of any degree is a partition problem in three fixed degrees. Those twelve are exactly the classes generated by their own mirror reflections, as the theorem of Chevalley, Shephard and Todd requires. Of the other twenty it said only that the count “is the partition number minus a correction, and the correction is where the errors live”.

This essay computes that correction, and it turns out to be simple. Every one of the twenty sits inside one of the twelve. Seventeen sit at index two and three at index four. The invariants of a smaller class are the invariants of the reflection class containing it, times one polynomial for each of a small set of characters. Each such polynomial is the product of the linear forms of some of the larger class’s mirror planes, the ones the smaller class does without. The count of invariants of each degree is then a sum of shifted partition counts, one for each of those polynomials, and nothing needs subtracting.

Primary and secondary invariants #

A theorem of Hochster and Eagon, for finite groups in characteristic zero, says that every ring of invariants has the same shape even when it is not free. Choose three invariant polynomials $\theta_1, \theta_2, \theta_3$ whose only common zero is the origin. Then every invariant can be written uniquely as $\sum_k \eta_k\, f_k(\theta_1, \theta_2, \theta_3)$ for a fixed finite list of invariants $\eta_k$ and arbitrary polynomials $f_k$. The $\theta$ are called primary invariants and the $\eta$ secondary ones. The ring is free over the primaries, and it is free over nothing at all exactly when one secondary, the constant 1, is not enough.

The number of secondaries is $d_1 d_2 d_3 / |G|$, where the $d_i$ are the primaries’ degrees, and their degrees can be read from the Molien series. Multiply the series by $(1 - t^{d_1})(1 - t^{d_2})(1 - t^{d_3})$, and what remains is a polynomial whose coefficient of $t^n$ counts the secondaries of degree $n$. The difficulty is choosing primaries, which the theorem says exist without saying which.

For a crystal class there is a natural choice. If $G$ is contained in a reflection class $W$, then the basic invariants of $W$ are invariants of $G$ too. They have only the origin as a common zero, since they are primaries for $W$ already. So they are primaries for $G$, and the number of secondaries is $|W| / |G|$, the index of $G$ in $W$. The smaller the reflection class that contains $G$, the fewer secondaries it needs.

Every class inside a reflection class #

The picture at the head of this essay lists every class with the smallest reflection class that contains it. The search tries each of the twelve reflection classes and each of the six ways of relabelling the three axes, since a monoclinic two-fold may be written along $y$ and the orthorhombic mirrors along $x$, $y$ and $z$. Twelve classes are reflection classes themselves, at index one, and their only secondary is 1. Seventeen sit at index two and need exactly one secondary besides 1. Three sit at index four: the centre of inversion $\bar{1}$ in $mmm$, the four-fold rotoinversion $\bar{4}$ in $4/mmm$, and the three-fold rotoinversion $\bar{3}$ in $6/mmm$. They need three.

None of the twenty needs a larger index. That is not forced by anything general, and it reflects how the crystal classes lie inside each other. Every one of the twenty can be completed to a reflection class by adding mirrors. The three that need index four are exactly the classes with an improper operation and no mirror at all: the inversion and the two rotoinversions whose powers include no reflection.

The secondaries are products of mirrors #

The secondary invariants can be written down, not merely counted. A reflection class’s characters of dimension one take only the values $+1$ and $-1$, because the class is generated by reflections, each of order two. For each such character $\chi$ that is $+1$ on all of $G$, let $J_\chi$ be the product of the linear forms of the mirrors on which $\chi$ is $-1$. Every operation of $W$ multiplies $J_\chi$ by $\chi$. So $J_\chi$ is invariant under $G$, where $\chi$ is $+1$, and changes sign under some operations of $W$. A theorem of Stanley, from 1977, says these are exactly the secondaries:

$$\mathbb{C}[x, y, z]^G = \bigoplus_{\chi} J_\chi \,\mathbb{C}[x, y, z]^W,$$

one for each character of $W$ that is trivial on $G$, including the trivial character with $J = 1$. So the degree of each secondary is the number of mirrors its character reverses.

That prediction can be tested against a computation that knows nothing about characters. The Molien series of each class is computed as an exact average over its matrices, as the earlier essay did, and multiplied by the reflection class’s factors. Separately, the characters of each reflection class are enumerated by assigning signs to its conjugacy classes of mirrors and keeping the assignments that extend to a homomorphism. The ones trivial on $G$ are kept, and the mirrors each reverses are counted. The two agree for all thirty-two classes: the twelve with nothing to add, the seventeen with one secondary, and the three with three.

The index-two cases read cleanly. $222$ sits in $mmm$, and the one character of $mmm$ trivial on $222$ reverses all three mirrors. Its relative invariant is $xyz$, of degree three. $23$ sits in $\bar{4}3m$, whose six mirrors are the diagonal planes $x = \pm y$ and so on, none of them in $23$, and its relative invariant is $(x^2 - y^2)(y^2 - z^2)(z^2 - x^2)$, of degree six. $3$ sits in $3m$ with a relative invariant of degree three, the product of the three mirror lines’ forms. The polynomials a textbook lists as the extra invariant of a subgroup are, in every case, a product of the mirrors the subgroup lacks.

The rotation classes carry a Jacobian #

Ten of the seventeen classes at index two have something in common. They are the classes of pure rotations, $2$, $3$, $4$, $6$, $222$, $32$, $422$, $622$, $23$ and $432$, each the subgroup of rotations inside a reflection class: $mm2$, $3m$, $4mm$, $6mm$, $mmm$, $\bar{6}2m$, $4/mmm$, $6/mmm$, $\bar{4}3m$ and $m\bar{3}m$. For each of them the character that picks out the rotations is the determinant, which is $-1$ on every mirror. So the relative invariant is the product of all of the reflection class’s mirror forms.

That product has a second name. The Jacobian determinant of a reflection class’s basic invariants, $\det(\partial\theta_i/\partial x_j)$, vanishes exactly on the mirror planes. By a theorem of Steinberg it equals the product of their linear forms up to a constant, and its degree is $\sum_i (d_i - 1)$. So the one extra invariant of a rotation class is the Jacobian of the reflection class around it, and its degree is fixed by the reflection class’s basic degrees before any mirror is counted. For $m\bar{3}m$, with basic degrees 2, 4 and 6, it is $1 + 3 + 5 = 9$. For $mmm$, with degrees 2, 2 and 2, it is 3, which is $xyz$. For $6/mmm$, with degrees 2, 2 and 6, it is 7, the degree the table gives for $622$. The census reaches each of these numbers twice, once from mirrors and once from the Molien series, and the Jacobian’s degree formula is a third route to the same number.

The other seven index-two classes, $2/m$, $4/m$, $\bar{4}2m$, $m\bar{3}$, $\bar{3}m$, $\bar{6}$ and $6/m$, each keep some of the reflection class’s mirrors and lack the rest. Their characters reverse only some conjugacy classes of mirrors, and their relative invariants are products of those alone. $m\bar{3}$ keeps the three axial mirrors of $m\bar{3}m$ and lacks the six diagonal ones, so its extra invariant is $(x^2 - y^2)(y^2 - z^2)(z^2 - x^2)$ again, of degree six. $4/m$ keeps the horizontal mirror of $4/mmm$ and lacks the four vertical ones, so its extra invariant has degree four, $xy(x^2 - y^2)$.

Nine mirrors, and the hand of a cube #

The most striking case is $432$, the twenty-four rotations of the cube. It sits in $m\bar{3}m$, the full cubic class with forty-eight operations, whose nine mirrors are drawn in the picture. The one character of $m\bar{3}m$ trivial on $432$ is the determinant, which is $-1$ on every mirror. So the relative invariant is the product of all nine mirror forms,

$$J = xyz\,(x^2 - y^2)(y^2 - z^2)(z^2 - x^2),$$

of degree nine. Every invariant polynomial of $432$ is an invariant of $m\bar{3}m$ plus $J$ times another.

That makes a physical statement exact. An invariant of $432$ that is not an invariant of $m\bar{3}m$ is one that changes sign under inversion. It distinguishes an object with the rotational symmetry of a cube from its mirror image, and it measures the hand of a cubic arrangement. The lowest degree of polynomial that can tell a 432-symmetric object from its mirror image is nine. No quadratic, cubic or sextic function of position will do it. How chiral, as a number measured handedness by comparing shapes. Here the same question for a cubic object reduces to one polynomial and its degree. The count of $432$’s invariants in each degree exceeds $m\bar{3}m$’s by the count of $m\bar{3}m$’s invariants nine degrees lower. $m\bar{3}m$ has one invariant each in degrees 0 and 2, two in degree 4 and three in degree 6, so $432$ has one more than $m\bar{3}m$ in degree 9, one more in degree 11, two more in 13 and three more in 15. The first extra one appears at degree nine, and there are more at eleven, thirteen and every odd degree above.

Three classes two steps away #

The three classes of index four are the mirror-free classes with an improper operation. The centre of inversion, $\bar{1}$, sits in $mmm$, and three characters of $mmm$ besides the trivial one are $+1$ on the inversion. Each reverses two of the three mirrors, so the secondaries are $1$, $xy$, $yz$ and $zx$. Every polynomial the inversion leaves alone, every even polynomial, is a polynomial in $x^2$, $y^2$ and $z^2$ plus one of those three products times another. The rotoinversion $\bar{4}$ in $4/mmm$ has secondaries of degrees three, three and four, among them $xyz$, $z(x^2 - y^2)$ and $xy(x^2 - y^2)$. $\bar{3}$ in $6/mmm$ has them in degrees four, four and six.

Relations follow directly. For a class of index two with one secondary $J$, the square $J^2$ is invariant under all of $W$, since $\chi^2 = 1$. So $J^2$ is a polynomial in $W$’s basic invariants, and that single equation is the only relation the ring needs. For $222$ it is $(xyz)^2 = x^2 \cdot y^2 \cdot z^2$, written in $mmm$’s basic invariants. For $432$ it is an identity of degree eighteen expressing $J^2$ in the cube’s invariants of degrees two, four and six. Three invariants and one relation found the plane’s version, a single identity forced on a class with three generators in two variables, and it has the same origin.

What the correction was #

The earlier essay feared that counting the invariants of a non-free class meant subtracting a correction found only by working out relations. The decomposition removes the subtraction. For a class $G$ of index two in $W$, with relative invariant of degree $\delta$, the number of independent invariants of degree $n$ is

$$\dim \mathbb{C}[x,y,z]^G_n = p_W(n) + p_W(n - \delta),$$

where $p_W$ counts the ways of reaching a degree from $W$’s three basic degrees. For index four there are four terms. Every count is a sum of partition counts, shifted by the degrees of mirror products, and the arithmetic is as easy as for the free classes once the containing reflection class is known.

This is also what Neumann’s principle uses when it counts the components of a symmetric property tensor. A symmetric tensor of rank $r$ is a polynomial of degree $r$ in disguise, as the parts a property splits into showed. So the number of independent components a class allows is $p_W(r)$ plus the shifted counts. For $222$, a rank-three symmetric tensor has $p_{mmm}(3) + p_{mmm}(0) = 0 + 1 = 1$ component, the $xyz$ one. That is the one independent coefficient of second-harmonic generation in a $222$ crystal when the susceptibility is fully symmetric, as Kleinman’s symmetry makes it far from any absorption, and it is the relative invariant itself.

The degree of the first secondary also says when a property can tell a class from its reflection class at all. A fully symmetric tensor of rank $r$ is a homogeneous polynomial of degree $r$, so its independent components in $G$ and in $W$ differ only when $r$ reaches $\delta$. For $2$ against $mm2$, $2/m$ against $mmm$, and the centre of inversion, the difference appears at rank two. For $222$, $3$, $\bar{4}2m$, $\bar{6}$ and $\bar{4}$ it appears at rank three, and for $4$, $4/m$, $32$, $\bar{3}m$ and $\bar{3}$ at rank four. For $422$ it waits until rank five, for $6$, $6/m$, $23$ and $m\bar{3}$ until rank six, and for $622$ until rank seven. For $432$ no fully symmetric property of rank below nine distinguishes it from $m\bar{3}m$. The two classes do differ in properties that tell a hand from its mirror image, such as optical activity, which fifteen may rotate light found $432$ permits and $m\bar{3}m$ forbids. Those properties are not fully symmetric tensors, and the ones that are cannot see the difference until the ninth rank.

What the account does and does not say #

The theorem is quoted and the agreement computed. The decomposition as a free module over the reflection class’s invariants is Stanley’s theorem, not derived here. What is computed is that its prediction, the degrees of the characters’ mirror products, matches the Molien series term by term for all thirty-two classes. The two computations share only the classes’ matrices.

The primaries are a choice. Using the containing reflection class’s invariants as primaries gives the fewest secondaries among choices of that kind. It is not a claim that no other primaries do better. For $23$ the choice gives degrees 2, 3 and 4 with two secondaries. Whether some class admits primaries of smaller product, and so fewer secondaries, than its smallest containing reflection class provides is a separate search, and it has not been made.

The polynomials are named where they are short. The mirror products for the classes with few mirrors are written out; for $\bar{3}$ in $6/mmm$ only their degrees are given.

Who worked out the pieces #

The free case is Chevalley’s theorem of 1955, with Shephard and Todd’s classification of the complex reflection groups the year before. That every invariant ring of a finite group is free over some choice of primary invariants, with a finite set of secondaries, follows from the Hochster–Eagon theorem of 1971, and its combinatorial reading through Molien’s series is Stanley’s. The decomposition by relative invariants, one for each linear character, is in Stanley’s 1977 paper on relative invariants of groups generated by pseudo-reflections.

What the account must refuse #

The second test carries the essay. If a class’s containing reflection class were misidentified, or a character mis-enumerated, the predicted degrees would miss the Molien series at some term, and the thirty-two comparisons would not all agree. The refusal asks the search not to report a mirror-free class as its own reflection class. Thirteen classes other than the trivial one have no mirrors, and every one of them is found at index two or four.

Still open: the magnetic classes #

The same decomposition applies wherever a group sits inside a reflection group with abelian quotient. The magnetic point groups of the operation that reverses time are crystal classes in which some operations are combined with the reversal of time. An invariant of a magnetic class is a polynomial the unprimed operations keep and the primed ones negate, which is a relative invariant with a character. So the counts for the fifty-eight magnetic classes that permit the magnetoelectric effect, and for every magnetic property tensor, should come from the same mirror products with different characters chosen. Whether every magnetic class’s relative invariants are mirror products of the smallest reflection class containing its unprimed part has not been checked, and it would reduce magnetic property counting to the same sums of shifted partitions.