Who coined the term “crystallographic root system”?

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.


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

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

Leave a Comment