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The coinvariant algebra is a complete intersection with one-dimensional socle

Statement

Assume the Axiom of Choice, inherited from the published invariant-theory items. Let $W$ act on the Cartan space $h$ by reflections and on $S = S (h)$ by graded algebra automorphisms, with coinvariant algebra $C = S / I$ as in The coinvariant algebra of a finite Weyl group. Let $w_{1}, \dots, w_{r}$ be homogeneous basic invariants of page degrees $2 d_{1}, \dots, 2 d_{r}$ , where $d_{1}, \dots, d_{r}$ are the classical degrees ( $\deg h = 1$ ; for the Cartan of ${s l}_{n}$ they are $2, 3, \dots, n$ ), so that $S^{W} = C [w_{1}, \dots, w_{r}]$ (Chevalley shephard todd for finite weyl groups), and let $ρ : S \to C$ be the quotient by the ideal $I = (w_{1}, \dots, w_{r})$ . Put $N : = \sum_{i = 1}^{r} (d_{i} - 1)$ . Then:

The sequence $w_{1}, \dots, w_{r}$ is a homogeneous $S$ -regular sequence and $C$ is a complete intersection: $C = S / (w_{1}, \dots, w_{r})$ with $Hilb (C, t) = \frac{\prod_{i} (1 - t^{2 d_{i}})}{(1 - t^{2})^{r}} = \prod_{i} (1 + t^{2} + \dots + t^{2 (d_{i} - 1)}),$ so that $\dim_{C} C_{2 N} = 1$ and $C_{j} = 0$ for $j > 2 N$ .
The multiplication pairing $C_{i} \times C_{2 N - i} \to C_{2 N} ≅ C$ is nondegenerate for every $i$ , and consequently $soc (C) = {c \in C : C_{+} c = 0} = C_{2 N}$ is one dimensional.
Equivalently, $C$ is a graded Gorenstein algebra with one-dimensional socle in the top degree $2 N$ .

Facts & Assumptions

Given: A finite Weyl reflection group $W$ acting on the Cartan space $h$ of rank $r$ , the symmetric algebra $S = S (h)$ with page grading $\deg h = 2$ , homogeneous basic invariants $w_{1}, \dots, w_{r}$ of page degrees $2 d_{i}$ , the ideal $I = (w_{1}, \dots, w_{r})$ , the coinvariant algebra $C = S / I$ , and a basis $x_{1}, \dots, x_{r}$ of $h^{*}$ viewed as degree-two elements of $S$ .

[F1]

Under AC, $S^{W} = C [w_{1}, \dots, w_{r}]$ with the $w_{i}$ algebraically independent and homogeneous of positive page degree $2 d_{i}$ , $I = S S_{+}^{W}$ , $C$ is a finite-dimensional commutative graded $C$ -algebra with $C_{0} = C$ generated in degree two, and $ρ : S \to C$ is the quotient map (Chevalley shephard todd for finite weyl groups, The coinvariant algebra of a finite Weyl group, The Axiom of Choice).

[F2]

Minimal homogeneous invariant generators $f_{1}, \dots, f_{m}$ of $I = S S_{+}^{W}$ have common zero set ${0}$ , satisfy $m = r$ , and form an $S$ -regular sequence; every homogeneous lift of a homogeneous basis of $C$ is a graded free $S^{W}$ -basis of $S$ (Reflection basic invariants form a regular sequence).

[F3]

For a finite sequence $x = (x_{1}, \dots, x_{n})$ in a commutative ring and a module $M$ , the Koszul complex $K (x; M)$ has $H_{i} = 0$ for $i > n$ and $H_{n} (K (x; M)) = {m \in M : x_{i} m = 0 for all i}$ ; if $M$ is finite free and $x$ is $M$ -regular, then $K (x; M)$ is a finite free resolution of $M / (x) M$ . The differential preserves internal degree when the $x_{i}$ are given their degrees, the top exterior generator $e_{1} \land \dots \land e_{n}$ carrying the sum of the degrees of the $x_{i}$ (Koszul Complex Of A Sequence With Coefficients, Basic Koszul Homology, Koszul Complex Resolves A Regular Quotient).

[F4]

Under AC, $Hilb (S / I^{o r d}, t) = \prod_{i} (1 + t + \dots + t^{d_{i} - 1})$ and $\prod_{i} d_{i} = ∣ W ∣$ in the classical normalization $\deg h = 1$ (Weyl coinvariant hilbert series has order w dimension).

[F5]

In the polynomial ring $S = C [x_{1}, \dots, x_{r}]$ , the variables form a regular sequence with $S / (x_{1}, \dots, x_{r}) = C$ , and $S / (x_{1}, \dots, x_{i}) ≅ C [x_{i + 1}, \dots, x_{r}]$ (Polynomial rings in finitely many commuting indeterminates by iteration).

[F6]

Under AC (hence Dependent Choice) the two constructions of ${Tor}_{n}^{S} (-, -)$ by resolving either variable are naturally isomorphic (Tor from a projective resolution of the left module, Tor from a projective resolution of the right module, The left and right projective constructions of Tor are naturally isomorphic).

Proof

technique · direct

1.1F1

The invariant ideal is generated by the basic invariants: $I = S S_{+}^{W} = (w_{1}, \dots, w_{r})$ , because $S^{W} = C [w_{1}, \dots, w_{r}]$ and every element of $S_{+}^{W}$ is a polynomial in the $w_{i}$ with zero constant term. Hence $C = S / (w_{1}, \dots, w_{r})$ ; since $C$ is generated by the images of the degree-two elements, its augmentation ideal satisfies $C_{+} = (x_{1}, \dots, x_{r}) C$ .

1.2F1F2F3

The $r$ basic invariants form an $S$ -regular sequence: they are minimal homogeneous invariant generators of $I$ (they generate $I$ , and no $w_{i}$ lies in the ideal generated by the others, since applying the $S^{W}$ -linear Reynolds projection to a relation in $S$ would give a relation in $C [w_{1}, \dots, w_{r}]$ , where no variable belongs to the ideal generated by the others), and by [F2] the minimal homogeneous invariant generators of $I$ are $r$ in number with common zero set ${0}$ and form an $S$ -regular sequence; consequently, by [F3], $K (w; S)$ is a finite free resolution of $C$ .

1.3F3F5

The variables $x_{1}, \dots, x_{r}$ form an $S$ -regular sequence with $S / (x_{1}, \dots, x_{r}) = C = : k$ , by [F5]; consequently, by [F3], $K (x; S)$ is a finite free resolution of $k$ .

1.4F1F3algebra

Koszul differentials vanish after base change to $k$ : tensoring $K (w; S)$ with $k$ multiplies by the images of the $w_{i}$ in $k$ , which lie in the positive-degree part and hence act as zero; so $H_{r} (K (w; S) \otimes_{S} k) = K_{r} (w; S) \otimes_{S} k$ , a one-dimensional space in internal degree $\sum_{i} 2 d_{i}$ .

2.1F3F6step 1.2step 1.4

First computation: by resolution 1.2, resolving the $C$ variable, and the balance isomorphism [F6], ${Tor}_{r}^{S} (C, k) ≅ H_{r} (K (w; S) \otimes_{S} k)$ , which by step 1.4 is one dimensional and concentrated in internal degree $\sum_{i} 2 d_{i}$ .

2.2F3F6step 1.3

Second computation: by resolution 1.3, resolving the $k$ variable, and [F6], ${Tor}_{r}^{S} (C, k) ≅ H_{r} (C \otimes_{S} K (x; S)) = H_{r} (K (x; C))$ ; by [F3] the top homology is ${c \in C : x_{i} c = 0 for all i}$ , placed in internal degree $2 r$ above the degrees of its elements, since $e_{1} \land \dots \land e_{r}$ carries degree $2 r$ .

2.3F3F4step 1.2algebra

Hilbert series: applying the Euler characteristic to the finite free resolution $K (w; S)$ of $C$ gives $Hilb (C, t) = \prod_{i} (1 - t^{2 d_{i}}) \cdot Hilb (S, t) = \frac{\prod_{i} (1 - t^{2 d_{i}})}{(1 - t^{2})^{r}} = \prod_{i} (1 + t^{2} + \dots + t^{2 (d_{i} - 1)}),$ the last equality by cancelling one factor $1 - t^{2}$ per invariant. Hence $Hilb (C, t)$ is a polynomial of degree $2 N$ with leading coefficient $1$ , so $\dim_{C} C_{2 N} = 1$ and $C_{j} = 0$ for $j > 2 N$ ; the classical formula [F4] gives the same series after substituting $t^{2}$ for $t$ .

3.1F6step 2.1step 2.2

Comparing steps 2.1 and 2.2 through the natural isomorphism [F6] identifies the top Koszul homology of step 2.2 with a one-dimensional space in internal degree $\sum_{i} 2 d_{i}$ ; shifting back by the degree $2 r$ of the top exterior generator, the space ${c \in C : x_{i} c = 0 for all i}$ is one dimensional and concentrated in internal degree $\sum_{i} 2 d_{i} - 2 r = 2 \sum_{i} (d_{i} - 1) = 2 N$ .

4.1step 1.1step 2.3step 3.1

Since $C_{+} = (x_{1}, \dots, x_{r}) C$ by step 1.1, the space of step 3.1 is exactly the socle $soc (C) = {c \in C : C_{+} c = 0}$ ; it is one dimensional in degree $2 N$ . The top graded piece $C_{2 N}$ is one dimensional by step 2.3, and every element of $C_{2 N}$ is annihilated by $C_{+}$ because $C$ has no degrees above $2 N$ ; hence $0 \neq C_{2 N} \subseteq soc (C)$ and both are one dimensional in degree $2 N$ , so $soc (C) = C_{2 N}$ .

5.1step 2.3step 4.1algebra∎

Nondegeneracy of the top pairing: let $0 \neq c \in C_{i}$ be homogeneous and consider the nonzero graded ideal $c C$ of the finite-dimensional graded algebra $C$ ; let $m$ be the largest degree with $(c C)_{m} \neq 0$ , so $m \leq 2 N$ by step 2.3. Choose $0 \neq y \in (c C)_{m}$ ; then $C_{+} y \subseteq (c C)_{> m} = 0$ , so $y \in soc (C) = C_{2 N}$ by step 4.1, forcing $m = 2 N$ . Writing $y = c b$ with $b \in C_{2 N - i}$ gives a nonzero product $c b$ , so the pairing $C_{i} \times C_{2 N - i} \to C_{2 N} ≅ C$ has no nonzero left radical; by commutativity it has no nonzero right radical either, and finite-dimensionality of the two sides makes it a perfect pairing for every $i$ . Hence $C$ is a graded Gorenstein algebra with one-dimensional socle in top degree $2 N$ .

Depends on

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Sources

Wolfgang Soergel, Kategorie O, perverse Garben und Moduln ueber den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990) 421-445 (MPIM preprint 1989/46 full text) (standard reference, not scraped)
Arthur Garnier, The classical theory of Soergel bimodules and the link with Lusztig's conjecture, Oberwolfach Arbeitsgemeinschaft note (2022) (standard reference, not scraped)