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**

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

**Attribution***Source : Link , Question Author : Matthew Dougherty , Answer Author : Matthew Dougherty*