Who coined the term "crystallographic root system"?

Question

Who coined the term "crystallographic root system" and when? In particular is there a connection to applied 3D crystallography?

It does not seem to be Killing or Cartan's terms (so presumably after 1900), and before Humphrey in 1990.

Answer 1

I contacted Dr. Humphreys, his take:

No, I definitely didn't invent this usage. The notion of "crystallographic Coxeter group" was apparently first discussed by Bourbaki in their infuential Chapters IV-VI of "Groupes et algebres de Lie" (1968), at the end of Section 2 in Chapter VI. This unfortunately doesn't agree with usage in fields like chemistry. But it's motivated at first by the notion of Weyl group of a root system (as defined by Bourbaki or similarly in my 1972 textbook on Lie algebras in characteristic 0).

Coxeter studied in a geometric way the finite groups generated by orthogonal reflections, which include all of these Weyl groups. But the classification of irreducible finite Coxeter groups includes some other examples: dihedral groups which aren't Weyl groups, along with H3 in rank 3 and H4 in rank 4. (The latter groups are nowadays associated with quasi-crystals, but they involve rotations with 5-fold symmetry.) The Bourbaki notion "crystallographic" selects precisely the Weyl groups in this setting, as those which leave some lattice invariant in the natural representation. From the classification one finds that these are the finite Coxeter groups with products of two generators having order 1, 2, 3, 4, or 6.

Similarly, there are "root systems" associated with infinite dimensional Lie algebras studied independently by Kac and Moody in the late 1960s. Here too there is a "Weyl group", and the natural definition of "crystallographic" group in this case also leads to the numbers above as orders of products of two generators. By now this has become a general definition in the study of arbitrary Coxeter groups, though like the notion of "hyperbolic Coxeter group" there is some conflict with other language used.

In principle, all these root systems are "crystallographic" in the sense that their Weyl groups are, but a Coxeter group (a sort of generalized reflection group) sometimes is and sometimes isn't crystallographic in the Bourbaki sense.

As usual, mathematicians are concerned with precision of arguments, but definitions can be made however one wants. In physics, for example, we are often frustrated by the absence of any definition of terms such as "state" which are in common use. So communication across discipline lines remains quite tricky, as I've learned from trying to communicate with physicists who use radically different terminology than I use.

I should add, to be more precise, that for infinite Coxeter groups the mathematical definition of "crystallographic group" allows products of two generators (which generalize "reflections" of order 2) to have infinite order, not just the numbers I listed. This applies for example when the group is an "affine Weyl group", the product of a finite Weyl group with a suitable translation group on which it acts

Answer 2

Wikipedia gives Humphreys as a reference for the term "crystallographic root system" in 1992. He refers to a book of Vinay Deodhar on reflection groups in 1981, see this MO-question. Weyl groups are "finite crystallographic Coxeter groups". In Bourbaki this term was not used, instead "integrality" is used. So it seems to me that this term came up around 1980. Perhaps this is coincidence, but this is exactly the time where crystallographic groups were studied intensively again, after a longer break. And after all, you are talking about them. The classification of crystallographic groups in dimension $2$ and $3$ goes back to Barlow (1894), Fedorov and Schönflies (1891), and is mainly of geometric nature. Bieberbach solved Hilbert's question on crystallographic groups in $1910$, and showed that there are only finitely many crystallographic groups in each dimension $n\ge 1$. The proof used Jordan's result on finitely many conjugacy classes of finite subgroups in $GL_n(\mathbb{Z})$. Then Zassenhaus devised an algorithm; and in in $1978$ crystallographic groups were classified by this algorithm in dimension $4$, by H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus. Around this time, it seems, root systems of semisimple Lie algebras were also called crystallographic, because of the similarity of the "crystallographic" argument for the classification of conjugacy classes of finite subgroups of $GL_n(\mathbb{Z})$ and the classification of root systems of semisimple Lie algebras.