# Uniform proof that a finite reflection group is determined by its degrees?

Given a finite (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 done without loss of generality in the irreducible case, based on the classificaiton of the possible groups. These are in fact the finite 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?

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).

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