Basic degrees, exponents, and the graded coinvariant algebra of a finite Coxeter system

Let $(W,S)$ be a Coxeter system of finite type with $n=\mathrm{\mid }S\mathrm{\mid }$, and let $V_{\mathbb{C}}$, ${\rho }_{\mathbb{C}}$, $S:=\mathbb{C}[V_{\mathbb{C}}]$, $R:=S^W$, $R_{+}={⨁}_{d>0}{R}_d$ and $I:=S{R}_{+}$ be as in Finite linear invariant and coinvariant polynomial algebras and Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers; fix coordinates $x_1,\dots ,x_n$ so that $S=\mathbb{C}[x_1,\dots ,x_n]$ as the iterated polynomial ring of Polynomial rings in finitely many commuting indeterminates by iteration, with the graded structure of Nonnegatively graded rings and modules, homogeneous elements, and twists.

(1) Existence of a basic family. Assume the Axiom of Choice. A minimal finite family $f_1,\dots ,f_r$ of homogeneous positive-degree invariants generating the ideal $I$ exists, generates $R$ as a $\mathbb{C}$-algebra, and is algebraically independent (Finite reflection invariant generators are algebraically independent); under the stated AC any such minimal family has exactly $n$ elements (Chevalley shephard todd for finite weyl groups, Reflection basic invariants form a regular sequence).

(2) Degrees and exponents. Fix one such family $f_1,\dots ,f_n$ and call $d_i:=\mathrm{deg}{f}_i\text{the }\mathbf{\text{basic}}\mathbf{\text{degrees}},e_i:=d_i-1\text{the }\mathbf{\text{exponents}},$ arranged in nondecreasing order $d_1\le \cdots \le d_n$. Each $d_i$ is a positive integer, and in fact $d_i\ge 2$: a degree-one invariant would be a nonzero $W$-invariant linear form, excluded by Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers (3).

(3) The graded coinvariant algebra. The coinvariant algebra of $(W,V_{\mathbb{C}})$ is the graded quotient $A:=S\mathrm{/}I={⨁}_{k\ge 0}{A}_k,A_k:=(S_k+I)\mathrm{/}I,$ with $A_0=\mathbb{C}$ (constants survive because $I$ has positive degree), as in Finite linear invariant and coinvariant polynomial algebras. Its Hilbert series is written $\mathrm{Hilb}(A,t)$.

(4) Well-definedness warning. This definition asserts nothing about the multiset $\{d_1,\dots ,d_n\}$ being independent of the chosen family, nothing about $\mathrm{Hilb}(A,t)$ or ${\prod }_i{d}_i$, and no relation between the $d_i$ and the reflection geometry; all of that is the content of the justifier The basic degrees are independent of the chosen family; Hilbert series of the invariants and of the coinvariant algebra; the order formula and the Molien identity ↗.

(5) Conventions. For $n=0$ the group is trivial, $S=R=\mathbb{C}$, $I=0$, $A=\mathbb{C}$ and the families and products are empty. For reducible $S$ the family in (2) is the concatenation of the families of the connected components, and $I$ is generated by all components' positive-degree invariants. No crystallographic or integrality assumption is made.