The coinvariant algebra of a finite Weyl group

Assume the Axiom of Choice inherited from the cited invariant-theory results (The Axiom of Choice). Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and Weyl group $W$, acting on ${\mathfrak{h}}^{\ast }$ by the root reflections of Root reflections and the Weyl group action (Finite Weyl root system, lattice and chamber conventions), hence on the symmetric algebra $S=S({\mathfrak{h}}^{\ast })$ by graded algebra automorphisms; a nondegenerate $W$-invariant bilinear form identifies $\mathfrak{h}$ with ${\mathfrak{h}}^{\ast }$ and thus identifies $S$ with $S(\mathfrak{h})$ as a graded $W$-algebra. Such a form is the complexification of a positive definite form on the real Cartan, obtained by averaging any positive definite form over finite $W$. Fix this identification; the page writes $S=S(\mathfrak{h})$ for either presentation. Write $S_{+}^W={⨁}_{d>0}(S^W{)}_d,I:=S{S}_{+}^W,$ so that $I$ is the homogeneous ideal of $S$ generated by the invariants of positive degree; this is the specialization to the reflection group case of the quotient construction of Finite linear invariant and coinvariant polynomial algebras, because $S_{+}^W$ is exactly the space of positive-degree elements of the invariant algebra $S^W$.

The coinvariant algebra of the pair $(\mathfrak{h},W)$ is the graded quotient $C=C(\mathfrak{h},W):=S\mathrm{/}I\mathrm{.}$ It is a commutative graded $\mathbb{C}$-algebra with $C_0=\mathbb{C}$: the ideal $I$ is homogeneous of positive degree, so constants survive and every homogeneous piece of $S$ maps onto its image. The quotient map $\rho :S⟶C,\rho (f)=f+I,$ is a surjective graded algebra homomorphism, and the action of $W$ on $S$ descends to a $\mathbb{C}$-linear action on $C$ by graded algebra automorphisms, because $I$ is $W$-stable. The algebra $C$ is finite dimensional: $W$ is generated by its root reflections, so Chevalley shephard todd for finite weyl groups writes $S$ as a free module of finite rank over the polynomial algebra $S^W$, and Weyl coinvariant hilbert series has order w dimension gives ${\dim }_{\mathbb{C}}C=\mathrm{\mid }W\mathrm{\mid }<\mathrm{\infty }$ with the graded Hilbert series recorded there.

Degree normalization. Throughout the Soergel material on this page the Cartan generators are given degree $2$, that is $\mathrm{deg}\mathfrak{h}=2$; the symmetric algebra $S$ is then graded with $\mathrm{deg}{S}_d=2d$ for homogeneous polynomials of ordinary degree $d$, and $C$ inherits this doubled grading. In particular a basic invariant of ordinary degree $d_i$ has page degree $2d_i$, and the top nonzero degree of $C$ is $2N$ with $N={\sum }_i(d_i-1)$. The separate degree-sum lemma on this page proves that this number equals the positive-root count.

Auxiliary permutation model for type $A$. For $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_n$ one may instead work with the place-permuting $S_n$-action on the auxiliary polynomial ring $S_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}}=\mathbb{C}[x_1,\dots ,x_n]$, where a permutation $\sigma$ sends $x_i$ to $x_{\sigma (i)}$, and with the quotient $C_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}}:=\mathbb{C}[x_1,\dots ,x_n]\mathrm{/}(e_1,\dots ,e_n),$ where $e_1,\dots ,e_n$ are the elementary symmetric polynomials. Splitting the permutation representation ${\mathbb{C}}^n=\mathfrak{h}\oplus \mathbb{C}\cdot (1,\dots ,1)$ into its sum-zero Cartan part and its fixed average coordinate exhibits $\mathbb{C}[x_1,\dots ,x_n]\cong S(\mathfrak{h})\otimes \mathbb{C}[e_1]$ with $e_1=x_1+\cdots +x_n$ the invariant coordinate, and restriction to the hyperplane $e_1=0$ has kernel $(e_1)$. By Fundamental theorem of symmetric polynomials: unique expression as a polynomial in $e_1,\dots ,e_n$, $e_1,\dots ,e_n$ generate the invariant algebra and its positive-degree ideal of the permutation action, while $e_2,\dots ,e_n$ restrict to basic invariants of $S(\mathfrak{h}{)}^W$, dividing out by the positive-degree invariants commutes with this presentation and yields a canonical isomorphism of graded algebras $C(\mathfrak{h},W)\cong C_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}}=\mathbb{C}[x_1,\dots ,x_n]\mathrm{/}(e_1,\dots ,e_n),$ in which $x_i$ has degree $2$ and $e_i$ has degree $2i$. Elements of $C$ are written as classes $\rho (f)$; the auxiliary model is used only for explicit rank computations and never replaces the intrinsic definition.

The quotient and finite coordinate constructions require no additional choice; the freeness and Hilbert-series conclusions inherit AC from their suppliers.