# Uniform proof that a finite (irreducible real) reflection group is determined by its degrees?

Given a finite (irreducible real) group $G$ generated by reflections acting on euclidean $n$-space, it was shown by Chevalley in the 1950s that the algebra of invariants of $G$ in the associated polynomial algebra over $\mathbb{R}$ is itself a polynomial algebra with homogeneous generators of uniquely determined degrees $d_1, \dots, d_n$. (The converse is true for groups generated by "quasi-reflections", which applies in the complex setting.) Moreover, $|G|= \prod d_i$. Determination of the degrees is usually done via the classification of the possible groups. These are in fact the finite (irreducible) Coxeter groups, characterized by their Coxeter graphs. The "crystallographic" ones are the Weyl groups familiar in Lie theory.

Textbook references include Bourbaki Groupes et algebres de Lie, Chap. V, $\S5$ and my book Reflection Groups and Coxeter Groups (Cambridge, 1990), Chapter 3. In my section 3.7 is a table giving the degrees for each irreducible type. Once one verifies this table, it's clear that the group $G$ is uniquely determined (up to isomorphism) by its degrees. When the degrees are listed in non-decreasing order, we have $d_1 =2$ and $d_n =h$, the Coxeter number of $G$. (Coxeter showed that the eigenvalues of a Coxeter element are the $m_i$th powers of a primitive $h$th root of 1, where it turns out that $m_i+1 = d_i$.)

Is there a uniform way to prove (without using the classification) that $G$ is determined by its degrees?

[EDIT: As the comment by Noam Elkies indicates, the following paragraph should be ignored.]

Though I'm less familiar with the behavior of complex (=unitary) groups generated by quasi-reflections, the same question seems to arise there. For a modern treatment, including the Shephard-Todd classification and a list of degrees in Appendix D, see the book by Lehrer and Taylor Unitary Refelction Groups (Cambridge, 2009).

• For complex reflection groups there are counterexamples: exceptional groups #8 and #13 both have degrees $8,12$, and there are several examples of exceptional groups with the same invariant degrees as a group in the second infinite family (generalized hyperoctahedral group), e.g. #5 ($6,12$), #10 ($12,24$), #18 ($30,60$), and #25 and #26 ($6,9,12$ and $6,12,18$). Jun 29, 2014 at 2:40
• @Noam: Yes, the complex reflection groups were only a superficial afterthought that I meant to look at more closely but obviously didn't. (In that case should one be considering the "codegrees" as well as the degrees?) Jun 29, 2014 at 14:15

I dont know a proof (other than by inspecting the list of degrees) in the irreducible case. However in the reducible case, a finite real reflection group is not determined up to isomorphism by its degrees: The Weyl groups $W(A_2B_2)$ and $W(A_1D_3)$ both have degrees 2,2,3,4 but they are not isomorphic as abstract groups.