Every magnetic class counts its properties by mirrors

What the twenty lack is a product of mirrors found the shape of every crystal class’s invariant polynomials. Twelve of the thirty-two classes are generated by their own reflections, and their invariants are free: three basic invariants, and every invariant is a polynomial in them. The other twenty each sit at index two or four inside one of the twelve, and their invariants are that reflection class’s free invariants multiplied by one, two or four products of mirror planes. The count of invariants of each degree, the Molien series, then comes out as a sum of shifted partition counts, and the essay checked that for all thirty-two.

It ended on the magnetic classes. The operation that reverses time introduced them: crystal classes in which some operations are combined with the reversal of time, marked with a prime. A property that time reversal changes the sign of, such as a magnetisation, is then constrained by polynomials the unprimed operations keep and the primed ones negate. Those are relative invariants with a character, and the question the essay left was whether they also come as mirror products. This essay answers it for all fifty-eight black-and-white classes. The answer is yes, but not quite in the form the question proposed.

Two characters for one class #

A black-and-white class is a crystal class $G$ with half of its operations primed. The unprimed half is a subgroup $H$ of index two. A property odd under time reversal picks up a factor $-1$ from every primed operation, on top of whatever the operation does in space. So its components are counted by polynomials $f$ with $g\cdot f = \chi(g)\,f$, where $\chi$ is $+1$ on $H$ and $-1$ on the rest.

That is the character for a polar time-odd quantity: something that transforms like a position under the spatial operations and changes sign under time reversal. A toroidal moment is the standard example, a polar vector made from circulating magnetisation. An axial time-odd quantity, such as the magnetisation itself, picks up the determinant of each operation as well, because an axial vector does not change sign under inversion. Its character is $\chi \cdot \det$. The two characters give two relative Molien series:

$$\frac{1}{|G|}\sum_{g \in G} \frac{\psi(g)}{\det(1 - t\,g)}, \qquad \psi = \chi \text{ or } \chi\cdot\det .$$

The coefficient of $t^d$ counts the independent components of a totally symmetric rank-$d$ quantity of that kind which the class permits. At degree one that is the number of components of a toroidal moment or of a magnetisation. Which magnetism a class permits computed those numbers by averaging a character, and they are the first test the series has to pass.

The prediction #

Stanley’s theorem, which the earlier essay used, says more than it was asked to there. For a reflection group $W$ and any sign character $\varphi$ of it, the polynomials that $W$ multiplies by $\varphi$ are exactly $J_\varphi$ times $W$’s invariants, where $J_\varphi$ is the product of the linear forms of the mirrors that $\varphi$ reverses. If $G$ sits inside $W$, and the character $\psi$ of $G$ is the restriction of such sign characters, then $G$’s relative invariants decompose over them:

$$\mathbb{C}[x,y,z]^{G,\psi} = \bigoplus_{\varphi|_G = \psi} J_\varphi \cdot \mathbb{C}[x,y,z]^W .$$

The series is then $\sum_\varphi t^{\deg J_\varphi} / \prod(1 - t^{d_i})$, where the $d_i$ are $W$’s degrees and $\deg J_\varphi$ is the number of mirrors $\varphi$ reverses. Both sides are computed separately, the left from Molien’s average over $G$’s matrices, the right by listing $W$’s sign characters and counting mirrors. They are compared term by term to degree twenty-nine. If $\psi$ extended to no sign character of $W$, the right side would be zero and the comparison would fail at its first nonzero term. That failure was possible for every class, and it does not happen.

Why a sign character is a choice of mirrors #

A reflection group is generated by its mirrors, so anything it does is decided by what it does to them. A sign character assigns $+1$ or $-1$ to each operation and multiplies along products. On a mirror it can only be $\pm 1$, and two mirrors that the group carries onto each other must get the same sign, because a character takes one value on a whole conjugacy class. So a sign character is exactly a choice of sign for each class of mirrors, and every such choice is one.

4/mmm makes this concrete. Its five mirrors fall into three classes: the mirror across the four-fold axis, the pair of mirrors containing the axis along the cell edges, and the pair along the diagonals. Three classes give eight sign characters. The trivial one is the ordinary invariant, and the other seven have seven kernels, the seven subgroups of index two. Priming the operations outside one of those subgroups is what makes a black-and-white class. Two pairs of the seven are exchanged by a rotation through forty-five degrees about the axis, which carries the edge mirrors onto the diagonal ones. That leaves the five classes the figure below lists. Each chooses which classes of mirrors are reversed, and the degree of its generator is simply how many mirrors those classes hold.

That is why the prediction is so cheap to make. The degrees of a reflection class’s invariants and the sizes of its mirror classes are both small, fixed lists. Once the character of a magnetic class is read off as a set of mirror classes, its whole series follows without averaging anything.

Fifty-eight of fifty-eight #

Every one of the fifty-eight classes passes for both characters, 116 series in all. The reflection class $W$ is the smallest one containing the underlying crystal class, the same $W$ the earlier essay found for that class. Twenty-eight magnetic classes live on crystal classes that are themselves reflection classes, and their relative invariants are a single mirror product times the free invariants. Twenty-seven sit at index two and have two mirror products, and three sit at index four and have four. The number of generators is the index, as it was for the ordinary classes. Nothing about time reversal changes the structure. It only chooses which mirrors are reversed.

4/mmm has five ways of priming half its operations, and each gives a different pair of series. 4/mmm has five mirror planes: one across the four-fold axis and two pairs containing it. A sign character of 4/mmm reverses some set of those mirrors, and each magnetic class picks the one that matches its primes. In 4/m′m′m′ every mirror is primed, so the polar character reverses all five and its generator has degree five. In 4/m′mm only the horizontal mirror is primed, and the generator is the single linear form $z$. The axial character reverses the complementary set, because the determinant reverses every mirror. Every axial degree is the number of mirrors less a polar one, in all fifty-eight classes. For 4/m′m′m′ the axial generator therefore has degree zero: the constant one. That class permits a time-odd pseudoscalar, a single number that time reversal and inversion both negate.

Two cases, worked through #

The two ends of the census show what the generators mean.

4/m′m′m′ primes every mirror. The polar character therefore reverses all five, and the generator is the product of all five mirror forms: $z$ for the horizontal mirror, $x$ and $y$ for the edge mirrors, and $x+y$ and $x-y$ for the diagonal ones. That product is $xyz(x^2-y^2)$, of degree five. Multiplied by 4/mmm’s invariants it gives every polar time-odd polynomial the class permits. So a polar time-odd quantity first appears in 4/m′m′m′ at rank five: there is no toroidal moment and no symmetric polar time-odd tensor of rank two, three or four, and the series reads zero until $t^5$. The axial character reverses no mirror at all, its generator is the constant one, and the class permits a time-odd pseudoscalar.

2′ is the opposite case, with almost nothing primed. Its crystal class is a single two-fold axis, which is not a reflection group, and the smallest reflection class containing it is mm2. Its two mirrors both contain the axis and are not conjugate, so mm2 has four sign characters. The polar character of 2′ is $-1$ on the two-fold rotation, and two of mm2’s characters restrict to it: each reverses exactly one mirror. The generators are therefore the two mirror forms, $x$ and $y$, the directions across the axis. The same holds for the axial character, which reverses the complementary single mirror in each case.

So in 2′ a magnetisation and a toroidal moment are both allowed, each with two components, both perpendicular to the primed axis. In the ordinary class 2, with no prime, a magnetisation lies along the axis instead. The series say this without being told what a magnetisation is. The prime moves the allowed direction from the axis into the plane across it, because it moves the character from trivial to one that reverses a mirror.

$\bar 1'$ is the third kind of case, and one of only three at index four. Its crystal class is the inversion alone, and priming it means the polar character is $-1$ on the inversion. A polynomial the inversion negates is an odd polynomial, so the relative invariants are simply every odd polynomial in $x$, $y$ and $z$. The census writes them through mmm, the smallest reflection class containing the inversion, whose invariants are $x^2$, $y^2$ and $z^2$. Four of mmm’s eight sign characters restrict to the polar character, and their mirror products are $x$, $y$, $z$ and $xyz$, of degrees one, one, one and three. That is the familiar statement that every odd polynomial is one of those four times a polynomial in the squares, recovered as a case of the general rule. The other two classes at index four are $\bar 4'$ and $\bar 3'$, whose crystal classes $\bar 4$ and $\bar 3$ were the two besides $\bar 1$ that sat at index four in the earlier essay. Their four generators have degrees one, two, two and five inside 4/mmm, and one, three, three and seven inside 6/mmm. In each case the lowest is a single mirror form, so both permit a toroidal moment along their axis.

The reflection class of the whole class, not of its half #

The question as it was left asked about the reflection class containing the unprimed part $H$. That is the natural guess, since $H$ is an ordinary crystal class and the earlier essay had already found its reflection class. For twenty-eight of the fifty-eight it would work, because the primed operations happen to lie in that reflection class too. For the other thirty they do not.

The simplest case is 2′, a two-fold axis combined with time reversal. Its unprimed part is the identity alone, whose smallest reflection class is the trivial one, and a trivial group cannot act on anything. The two-fold rotation that makes 2′ what it is lies outside it. The reflection class that works is mm2, the smallest containing the two-fold axis. There the polar character extends to the two sign characters that reverse one mirror each, and the relative invariants of 2′ are $x$ and $y$ times the invariants of mm2. That is the right answer: a toroidal moment perpendicular to a primed two-fold axis is allowed, with two components.

The decomposition always has to be made inside a group that acts on every operation of the class, and the reflection class of the unprimed part need not. What the computation shows is that the reflection class of the whole class always suffices, and that the character always extends to it. Neither is guaranteed in advance. A character of an index-two subgroup need not extend to a sign character of every reflection group containing it, and if one had failed to extend the series would have disagreed at its first term.

The first terms, read as properties #

The axial series’ degree-one coefficient is the number of magnetisation components a class permits. It is nonzero for eighteen of the fifty-eight. With the thirteen ordinary crystal classes that permit a magnetisation, that gives the thirty-one ferromagnetic magnetic classes the literature records, and the site’s census of magnetic properties found by averaging the magnetisation’s own character. Here the eighteen come out of the series with the same component counts class by class: two for 2′ and m′ and 2′/m′, one for the other fifteen. The series knows nothing about magnetisation. It counts polynomials, and a magnetisation is a degree-one relative invariant of the axial character.

The polar series’ degree-one coefficient counts components of a toroidal moment, and it is nonzero for twenty-one classes. Seven permit both a magnetisation and a toroidal moment, among them 2′ and m′. The degree-zero axial term, a time-odd pseudoscalar, is nonzero for twenty-one classes, including $\bar 1'$, $\bar 3'm'$ and 4/m′m′m′. A pseudoscalar of that kind is the part of a magnetoelectric response that is the same along every direction, and those twenty-one are the classes that allow it.

None of these counts is new. What is new is where they come from. The earlier approach to magnetic properties was to average a character over the class, one property at a time, which is Neumann’s principle with a sign attached. Here every property of every rank comes out of one series per class and character. That series is a mirror product over a denominator the reflection class already had, and the same sums of shifted partitions count them all.

What this does not settle #

The decomposition counts totally symmetric tensors, because those are what polynomials are. A property like the magnetoelectric tensor, which pairs a polar index with an axial one and is not symmetric in them, is counted by a different average and is outside this series. The degree-zero pseudoscalar is only its trace. Nor does the computation cover the thirty-two grey classes, in which time reversal alone is an operation. There every time-odd polynomial is negated by an operation that does nothing in space, so both series vanish identically and there is nothing to decompose.

The result also does not explain itself. The character extends to the reflection class in all fifty-eight cases, and the reason is not a general theorem: an index-two character of a subgroup can fail to extend. That it never fails here is a fact about the thirty-two crystal classes and their eleven reflection classes, checked rather than derived. Twelve of the thirty-two are free and the groups whose invariants are free set up the reflection classes, and how many invariants of each degree the series. The magnetic case adds a sign and changes nothing about the structure.

What the check has to refuse #

The tests that matter most are the two that reach outside the series. The ferromagnetic count is compared with the number the literature records. The component counts are compared with an independent average of the magnetisation’s own character, which never forms a series. The complement test is a check on the sign characters: if the determinant were read wrongly as a character of the reflection class, the axial generators would not be the polar ones reflected in the mirror count. The refusal records the correction to the question: the unprimed part’s reflection class is where the essay proposed to look, and for thirty classes it cannot see the primed operations at all.

Still open: the magnetic space groups #

The same question one level down is about the magnetic space groups, 1,651 of them, of which 1,191 are black-and-white in one of two senses. In some of them the primes fall on translations, which is why magnetic reflections land where nuclear ones cannot. The invariants a magnetic space group imposes at a wavevector are relative invariants of its little co-group with a character that now also depends on how the primes fall on translations. Whether those characters always extend to a reflection class of the co-group, as they do for the point groups, is a finite computation, and it would reduce every magnetic order parameter’s allowed terms to the same sums of shifted partitions.