Auslander's Theorem for permutation actions on noncommutative algebras

TL;DR

This paper generalizes Auslander's Theorem to various noncommutative algebras with permutation group actions, establishing isomorphisms of skew group algebras and fixed rings as graded algebras and singularities.

Contribution

It extends Auslander's Theorem to noncommutative settings like skew polynomial rings, quantum Weyl algebras, and Sklyanin algebras, revealing new algebraic isomorphisms and singularity properties.

Findings

Isomorphism of skew group algebra and endomorphism algebra for new classes

Identification of fixed rings as graded isolated singularities

Extension of Auslander's Theorem to noncommutative algebras

Abstract

When $A = \mathbb{k}[x_1, \ldots, x_n]$ and $G$ is a small subgroup of $\operatorname{GL}_n(\mathbb{k})$, Auslander's Theorem says that the skew group algebra $A \# G$ is isomorphic to $\operatorname{End}_{A^G}(A)$ as graded algebras. We prove a generalization of Auslander's Theorem for permutation actions on $(-1)$-skew polynomial rings, $(-1)$-quantum Weyl algebras, three-dimensional Sklyanin algebras, and a certain graded down-up algebra. We also show that certain fixed rings $A^G$ are graded isolated singularities in the sense of Ueyama.

Equations

$${\mathrm{Tr}}_A(g,t):=\sum _{n\ge 0}\mathrm{Tr}(g{|}_{A_n})t^n=\frac{1}{(1-t{)}^{n-1}q(t)}\text{ for }q(1)\ne 0,$$

$${\gamma }_{A,G}:A\#G$$

$$a\#g$$

$$\mathsf{p}(A,G)=\mathrm{GKdim}A-\mathrm{GKdim}(A\#G)/(f_G)$$

$$a{x}_1{x}_2+b{x}_2{x}_1+c{x}_3^2=a{x}_2{x}_3+b{x}_3{x}_2+c{x}_1^2=a{x}_3{x}_1+b{x}_1{x}_3+c{x}_2^2=0;$$

$$x^{2}y=\alpha xyx+\beta y{x}^2\text{mbox}andx{y}^2=\alpha yxy+\beta y^{2}x.$$

$${\mathrm{Tr}}_A(g,t)=(-1{)}^n\mathrm{hdet}(g{)}^{-1}{t}^{-\ell }+(\text{lower order terms}).$$

$$f_{i,j}=(x_i-x_j)\prod _{\begin{array}{c}(a,b)\ne (i,j)\\ a<b\end{array}}(x_a^2-x_b^2).$$

$$p_m=\sum _{\sigma \in G}\left((x_{a_1}^2-x_{\sigma (b_1)}^2)\cdots (x_{a_m}^2-x_{\tau (b_m)}^2)\#\sigma \right).$$

$${\hat{f}}_{i,j}=\{\begin{array}{ll}{y}_i\prod _{\begin{array}{c}(a,b)\ne (i,j)\\ a<b\end{array}}(y_a-y_b)& \text{ if }A=V_n\\ (2y_i-y_j)\prod _{\begin{array}{c}(a,b)\ne (i,j)\\ a<b\end{array}}(y_a-y_b)& \text{ if }A=S(1,1,-1).\end{array}$$

$$\mathsf{p}(A,G)=\mathrm{GKdim}A-\mathrm{GKdim}(A\#G)/(f)\ge n-(n-2)=2.$$

$$\mathsf{r}(A,g)=\mathrm{GKdim}A-k,\text{ where }{\mathrm{Tr}}_A(g,t)=\frac{1}{(1-t{)}^{k}q(t)}\text{ and }q(1)\ne 0.$$

$$\mathsf{r}(A,G)=\min \{\mathsf{r}(A,g):e\ne g\in G\}.$$

$$\alpha \#e=\sum _{i=1}^n(a_i\#{\sigma }_i)(f_G)(b_i\#{\sigma }_i^{\prime }).$$

$$\alpha \#e=\sum _{i=1}^n(a_i\#e)\left(\sum _{\sigma \in G}1\#\sigma \right)(b_i\#e)=\sum _{i=1}^n\sum _{\sigma \in G}{a}_i\sigma (b_i)\#\sigma .$$

$$\sum _{i=1}^n(a_i\#e)(f_H)(b_i\#e)=\sum _{i=1}^n(a_i\#e)\left(\sum _{\sigma \in H}1\#\sigma \right)(b_i\#e)=\sum _{i=1}^n\sum _{\sigma \in H}{a}_i\sigma (b_i)\#\sigma .$$

$$\mathsf{p}(A,G)\le \mathsf{p}(A,H).$$

$$q_{(12)(34)}$$

$$=(x_1^2+x_2^2+x_3^2+x_4^2)\#e+(x_1^2+x_2^2+x_3^2+x_4^2)\#(12)(34)\in (f).$$

$$r=(x_1^2+x_2^2+x_3^2+x_4^2)\in (f)\cap A.$$

$$H_{A^G}(t)=\frac{1-3t+5t^2-3t^3+t^4}{(1+t^2)(1-t{)}^4},$$

$$\tau ((h,g)(b\otimes a))=(g,h)\tau (a\otimes b).$$

$${\mu }_{\tau }((g,h)(a\otimes b)\otimes (g,h)(a^{\prime }\otimes b^{\prime }))$$

$$=({\mu }_A\otimes {\mu }_B)(g(a)\otimes \tau ((h,g)(b\otimes a^{\prime }))\otimes h(b^{\prime }))$$

$$=({\mu }_A\otimes {\mu }_B)(g(a)\otimes (g,h)\tau (a^{\prime }\otimes b)\otimes h(b^{\prime }))$$

$$=\sum g(a)g(a_i^{\prime \prime })\otimes h(b_i^{\prime \prime })h(b^{\prime })$$

$$=(g,h)(({\mu }_A\otimes {\mu }_B)(a\otimes \tau (b\otimes a)\otimes b))$$

$$=(g,h)({\mu }_{\tau }((a\otimes b)\otimes (a^{\prime }\otimes b^{\prime }))).\text{qed}$$

$$\max (\mathsf{p}(A,G),\mathsf{p}(B,H))\ge \mathsf{p}(A{\otimes }_{\tau }B,G\times H).$$

$$\left((A{\otimes }_{\tau }B)\#(G\times \{e_H\})\right)/(F)\cong \left((A\#G){\otimes }_{\tau }B\right)/(F)\cong (A\#G)/(f_G){\otimes }_{\tau }B.$$

$$\mathsf{p}(A\otimes B,G\otimes \{e_H\})$$

$$=(\mathrm{GKdim}(A)+\mathrm{GKdim}(B))-\mathrm{GKdim}((A\#G)/(f_G){\otimes }_{\tau }B)$$

$$=\mathsf{p}(A,G).\text{qed}$$

$$2\le \mathsf{p}(V_8,\{(1234),(5678)\})=\mathsf{p}(A{\otimes }_{\tau }B,G\times G)\le \mathsf{p}(A,G)=4.$$

$${\mathrm{Aut}}_{\mathrm{gr}}(A)=\left\{\left[\begin{array}{cc}{a}_{11}& 0\\ 0& a_{22}\end{array}\right],\left[\begin{array}{cc}0& a_{12}\\ a_{21}& 0\end{array}\right]:a_{11},a_{12},a_{21},a_{22}\in {\mathbb{k}}^{\times }\right\}.$$

Full text

1. Introduction

Throughout, $\Bbbk$ is an algebraically closed field of characteristic zero. All algebras are $\Bbbk$-algebras and all tensor products are over $\Bbbk$. For an algebra $A$ and $G$ a group of algebra automorphisms of $A$, the set of all elements of $A$ that are invariant under the action of $G$ (denoted by $A^{G}$) forms a subalgebra of $A$, called the ring of invariants of $G$ on $A$. Many notions in commutative algebra have their roots in the study of properties of rings of invariants of finite group actions on $\Bbbk[x_{1},\dots,x_{n}]$.

The homological properties of $A^{G}$ are often of particular interest. A classical result of Shephard-Todd and Chevalley [9, 23] states that if $A=\Bbbk[x_{1},\dots,x_{n}]$ and $G$ is a finite subgroup of $\operatorname{GL}_{n}(\Bbbk)$ acting as graded automorphisms, then $A^{G}$ has finite global dimension if and only if $G$ is generated by reflections; an element $g\in\operatorname{GL}_{n}(\Bbbk)$ is said to be a reflection if $g$ fixes a codimension-one subspace of $V=\bigoplus_{i=1}^{n}\Bbbk x_{i}$.

When $A$ is not the commutative polynomial ring the above definition of reflection is not appropriate, but there is a suitable generalization due to Kuzmanovich, Zhang and the second author [14, Definition 2.2]. For a graded algebra $A$ of Gelfand-Kirillov (GK) dimension $n$, a graded automorphism $g$ is a reflection if its trace series is of the form

[TABLE]

where $A_{n}$ denotes the $n^{\text{th}}$ graded component of $A$ and $\operatorname{Tr}(g|_{A_{n}})$ denotes the usual trace of the action of $g$ on $A_{n}$. When $A=\Bbbk[x_{1},\dots,x_{n}]$, $g$ is a reflection in this sense if and only if it is a reflection in the classical sense. In [15, Theorem 5.5], the same authors prove that if $A=\Bbbk_{p_{ij}}[x_{1},\dots,x_{n}]$ (a skew polynomial ring) and $G$ is a finite group of graded automorphisms of $A$, then $A^{G}$ has finite global dimension if and only if $G$ is generated by reflections in this more general sense.

Another classical theorem from invariant theory in which reflections play an important role is a theorem of Auslander. The skew group algebra $A\#G$ is $A\otimes\Bbbk G$ as a vector space over $\Bbbk$, with multiplication defined by $(a\otimes g)(b\otimes h)=a(g.b)\otimes gh$ for all $a,b\in A$, $g,h\in G$. A natural algebra homomorphism (due to Auslander) from $A\#G$ to $\operatorname{End}_{A^{G}}(A)$ (where we view $A$ as a right $A^{G}$-module) is given by

[TABLE]

Theorem 1**.**

(Auslander [1])* If $A=\Bbbk[x_{1},\dots,x_{n}]$, $|G|$ is invertible in $\Bbbk$, and $G$ does not contain any nontrivial reflections (i.e., $G$ is a “small group”), then $\gamma_{A,G}$ is an isomorphism.*

We call the map $\gamma_{A,G}$ the Auslander map of the $G$-action on $A$. In general, this map is neither injective nor surjective. The category of (left) modules over $A\#G$ is equivalent to the category of (left) $A$-modules with compatible left $G$-actions and their homomorphisms. Auslander’s Theorem identifies an $A^{G}$-module (namely $A$) whose endomorphism ring has finite global dimension.

Natural generalizations of Auslander’s result include replacing $A$ with a noncommutative ring and replacing the action of the group $G$ with the action of a Hopf algebra. Significant progress has been made in both directions, see [2, 3, 7, 8, 10, 21, 22]. Our interest is in the former. It is conjectured that the Auslander map is an isomorphism for a noetherian AS regular domain $A$ and a group $G$ that does not contain a reflection.

Bao, He, and Zhang [2, 3] (using some ideas that appeared in an earlier paper of Buchweitz [5]) prove that the Auslander map is related to an invariant of the $G$-action on $A$ known as the pertinency. Let $A$ be an affine algebra generated in degree 1 and $G$ a finite subgroup of $\operatorname{GL}_{n}(\Bbbk)$ acting on $A_{1}$. The pertinency of the $G$-action on $A$ [2, Definition 0.1] is defined to be

[TABLE]

where $(f_{G})$ is the two sided ideal of $A\#G$ generated by $f_{G}=\sum_{g\in G}1\#g$, and GKdim is the Gelfand-Kirillov (GK) dimension of the algebra. In what follows, we often drop the subscript on $f_{G}$ when the group $G$ is clear from the context.

A connected $\mathbb{N}$-graded algebra $A$ is Artin-Schelter (AS) regular provided it has finite GK dimension, finite global dimension $d$, and satisfies $\operatorname{\underline{Ext}}_{A}^{i}(_{A}\Bbbk,A)=\delta_{id}\Bbbk(\ell)$ for some $\ell\in\mathbb{Z}$. An algebra $A$ is Cohen-Macaulay (CM) if for every graded $A$ module $j(M)+\operatorname{GKdim}(M)=\operatorname{GKdim}(A)$ where $j(M)=\min\{i:\operatorname{\underline{Ext}}_{A}^{i}(M,A)\neq 0\}$.

Theorem 2** ([3, Theorem 0.3]).**

Let $A$ be a noetherian, connected graded, AS regular, CM $\Bbbk$-algebra of GK dimension at least $2$. Let $G$ be a group acting linearly on $A$. Then the Auslander map is an isomorphism if and only if ${\sf{p}}(A,G)\geq 2$.

In this paper we compute lower bounds on the pertinency of some group actions on several families of noetherian AS regular domains including

• •

the $(-1)$-skew polynomial algebras $V_{n}$, generated by $x_{1},\ldots,x_{n}$ subject to the relations $x_{i}x_{j}+x_{j}x_{i}=0$ for $i\neq j$;

• •

the $(-1)$-quantum Weyl algebras $W_{n}$, generated by $x_{1},\ldots,x_{n}$ with relations $x_{i}x_{j}+x_{j}x_{i}=1$ for $i\neq j$; taking the standard filtration, $\operatorname{gr}(W_{n})=V_{n}$;

• •

the three-dimensional Sklyanin algebras $S(a,b,c)$, generated by $x_{1},x_{2},x_{3}$ with parameters $(a:b:c)\in\mathbb{P}^{2}$ and relations

[TABLE]

• •

the graded noetherian down-up algebras $A(\alpha,\beta)$ with parameters $\alpha,\beta\in\Bbbk$ with $\beta\neq 0$ generated by $x$ and $y$ with relations

[TABLE]

The symmetric group $\mathcal{S}_{n}$ acts on $V_{n}$ by permutations; that is, if $\sigma\in\mathcal{S}_{n}$ then $\sigma(x_{i})=x_{\sigma(i)}$, extending linearly and multiplicatively. Throughout we assume that any subgroup of $\mathcal{S}_{n}$ acts on $V_{n}$ in this way unless otherwise stated. For this action on $V_{n}$, $\mathcal{S}_{n}$ does not contain any reflections [16, Lemma 1.7(4)] and our result (Theorem 2.4) proves that the Auslander map is an isomorphism for this action on $V_{n}$. This result is quite different than for commutative polynomial rings, where the Auslander map is not an isomorphism for subgroups of $\mathcal{S}_{n}$ that contain a transposition. In Section 3 we compute the pertinency of most subgroups of $\mathcal{S}_{3}$ and $\mathcal{S}_{4}$ acting on $V_{3}$ and $V_{4}$, respectively.

By [27, Theorem 0.2], [20, Corollary 6.2], and [18], $V_{n}$, $S(a,b,c)$, and $A(\alpha,\beta)$ are AS regular and CM and thus satisfy the hypotheses of Theorem 2. In the following theorem, for each $A$ and $G$ given, we verify that ${\sf{p}}(A,G)\geq 2$, and hence prove by Theorem 2 that the Auslander map is an isomorphism.

Theorem 3**.**

The Auslander map is an isomorphism for the following groups acting on the following algebras:

1. (1)

any subgroup of $\mathcal{S}_{n}$ acting on $V_{n}$ (Theorem 2.4), 2. (2)

any subgroup of $\mathcal{S}_{n}$ acting on $W_{n}$ (Theorem 2.4), 3. (3)

any subgroup of $\mathcal{S}_{3}$ acting on $S(1,1,-1)$ (Theorem 2.4), 4. (4)

any subgroup of weighted permutations acting on $A(-2,-1)$ (Theorem 4.3), 5. (5)

$\left\langle(1~{}2~{}3)\right\rangle$* acting on a generic Sklyanin algebra $S(a,b,c)$ (Theorem 5.2), and* 6. (6)

any subgroup of $\left\langle-I_{4},(1~{}3)(2~{}4)\right\rangle$ acting on $V_{4}$ (Theorem 5.3).

To bound ${\sf{p}}(A,G)$, we make frequent use of the following theorem.

Theorem 4** (Theorem 3.4).**

Let $G$ be a group acting on $A$ and let $H\leq G$ be a subgroup. Then ${\sf{p}}(A,G)\leq{\sf{p}}(A,H)$.

When $A^{\left\langle g\right\rangle}$ has finite global dimension, ${\sf{p}}(A,\left\langle g\right\rangle)<2$, and we obtain the following corollary.

Corollary 5** (Corollary 3.5).**

Let $A$ be a noetherian connected graded $\Bbbk$-algebra and let $G$ be a finite subgroup of $\operatorname{Aut}_{\operatorname{gr}}(A)$. Suppose that $g\in G$ is a reflection. If $A$ and $A^{\langle g\rangle}$ have finite global dimension, then the Auslander map $\gamma_{A,G}$ is not an isomorphism.

We remark that $A^{\left\langle g\right\rangle}$ has finite global dimension when $g$ is a reflection acting on an AS regular noetherian domain with Hilbert series $(1-t)^{-n}$ [14, Theorem 5.3], and hence this corollary applies to groups that contain a reflection acting on $V_{n}$.

Auslander’s Theorem is a component of the classical McKay correspondence. In [7, 8], Chan, Walton, Zhang, and the second-named author study a quantum version of the McKay correspondence. If $A$ is AS regular and $g\in\operatorname{Aut}_{\operatorname{gr}}(A)$, then the homological determinant of $g$, denoted $\operatorname{hdet}(g)$, may be computed using the following formula [12, Lemma 2.6]:

[TABLE]

With the exception of certain weighted actions on the down-up algebra, all actions of groups $G$ on $A$ in Theorem 3 have trivial homological determinant, that is, $\operatorname{hdet}(g)=1$ for all $g\in G$. Hence, as a consequence of our work, the following part of the McKay correspondence applies to the algebras of Theorem 3. Following the terminology in [7], call an $A$-module $M$ initial if $M$ is graded, generated in degree [math] and $M_{<0}=0$.

Theorem 6** ([7, Theorem A]).**

Let $A$ be a noetherian AS regular algebra and $G$ a finite group acting on $A$ as graded automorphisms with trivial homological determinant. If $A\#G\cong\operatorname{End}_{A^{G}}(A)$, then there are bijective correspondences between the isomorphism classes of

• •

simple left $G$-modules,

• •

indecomposable direct summands of $A$ as left $A^{G}$-modules, and

• •

indecomposable, finitely generated, projective, initial, left $A\#G$-modules.

We conclude in Section 5 by providing examples of graded isolated singularities in the sense of Ueyama [24]. For a graded algebra $A$, let $\operatorname{grmod}A$ denote the category of finitely-generated graded right $A$-modules. For a module $M\in\operatorname{grmod}A$, $x\in M$ is called torsion if there exists a positive integer $n$ such that $xA_{\geq n}=0$. The module $M$ is called a torsion module if every element of $M$ is torsion. Let $\operatorname{tors}A$ denote the full subcategory of $\operatorname{grmod}A$ consisting of torsion modules. We can then define the quotient category $\operatorname{tails}A=\operatorname{grmod}A/\operatorname{tors}A$. Following [24], we say that $A^{G}$ is a graded isolated singularity if $\operatorname{gl.dim}(\operatorname{tails}A^{G})<\infty$. Mori and Ueyama prove that if the Auslander map is an isomorphism, then $A^{G}$ is a graded isolated singularity if and only if $A\#G/(f_{G})$ is finite-dimensional [22, Theorem 3.10]. For several algebras $A$ and groups $G$, we are able to show that $A\#G/(f_{G})$ is finite-dimensional. These examples are of particular interest, since for a graded isolated singularity $A^{G}$, the category of graded CM $A^{G}$-modules has several nice properties (we refer the reader to [25] for undefined terminology).

Theorem 7** ([25, Theorem 3.10 and Example 3.13]).**

Let $A$ be a noetherian AS regular algebra of dimension $d\geq 2$, and let $G$ be a group acting linearly on $A$ with trivial homological determinant. If $A^{G}$ is a graded isolated singularity, then

• •

$A^{G}$* is an AS Gorenstein algebra of dimension $d\geq 2$,*

• •

$A\in\operatorname{CM}^{\operatorname{gr}}\left(A^{G}\right)$* is a $(d-1)$-cluster tilting module, and*

• •

for $M\in\operatorname{CM}^{\operatorname{gr}}\left(A^{G}\right)$ the $\Bbbk$-vector spaces $\operatorname{\underline{Ext}}_{A^{G}}^{1}(A,M)$ and $\operatorname{\underline{Ext}}_{A^{G}}^{1}(M,A)$ are finite-dimensional.

We show that $A^{G}$ is a graded isolated singularity in the following cases.

Theorem 8**.**

For the following groups $G$ acting on the following $\Bbbk$-algebras $A$, $A^{G}$ is a graded isolated singularity:

1. (1)

$\left\langle(1~{}2)(3~{}4),(1~{}3)(2~{}4)\right\rangle$* acting on $V_{4}$ (Proposition 3.7),* 2. (2)

$\left\langle(1~{}2)(3~{}4)\cdots(2n-1~{}2n)\right\rangle$* acting on $V_{2n}$ (Proposition 3.8),* 3. (3)

$\left\langle(1~{}2~{}\cdots~{}2^{n})\right\rangle$* acting on $V_{2^{n}}$ ([3, Theorem 5.7]),* 4. (4)

$\left\langle(1~{}2~{}3)\right\rangle$* acting on a generic Sklyanin algebra $S(a,b,c)$ (Theorem 5.2), and* 5. (5)

any subgroup of $\left\langle-I_{4},(1~{}3)(2~{}4)\right\rangle$ acting on $V_{4}$ (Theorem 5.3).

2. $(-1)$-skew polynomials and related algebras

In this section, we prove that if $G$ is a finite subgroup of $\mathcal{S}_{n}$ acting as permutations on the $(-1)$-skew polynomial ring $V_{n}$, then the pertinency ${\sf{p}}(V_{n},G)$ is bounded below by $2$, thus proving that, in this case, the Auslander map is an isomorphism by Theorem 2. We also consider related algebras including the Sklyanin algebra $S(1,1,-1)$ and the $(-1)$-quantum Weyl algebras $W_{n}$. The idea for the computation that follows was inspired by an argument of Brown and Lorenz [4, Lemma 2.2] for a commutative algebra $A$.

Let $A$ be a quotient of the free algebra $\Bbbk\langle x_{1},\ldots,x_{n}\rangle$ such that $\mathcal{S}_{n}$ acts on $A$. Denote by $C(A)$ the center of $A$. We assume $x_{\ell}^{2}\in C(A)$ for $1\leq\ell\leq n$, which clearly holds for $V_{n}$ and $W_{n}$, and is well-known for $S(1,1,-1)$ (see [26, Section 8.2]).

Let $G$ be a subgroup of $\mathcal{S}_{n}$ and set $f=\sum_{\sigma\in G}1\#\sigma$. For $i<j$, we define

[TABLE]

Lemma 2.1**.**

The elements $f_{i,j}\in(f)\cap A$ for $1\leq i<j\leq n$.

Proof.

Let $U=\{(a,b)\mid 1\leq a<b\leq n,(a,b)\neq(i,j)\}$ and let $m=|U|$. Enumerate the elements of $U$: $\{(a_{1},b_{1}),\ldots,(a_{m},b_{m})\}$. Set $p_{0}=f$. For $1<k\leq m$, we define $p_{k}$ recursively. Let $p_{k}=x_{a_{k}}^{2}p_{k-1}-p_{k-1}x_{b_{k}}^{2}$. Since $p_{0}=f$, it is clear that $p_{k}\in I$ for all $k$. Because the $x_{\ell}^{2}$ are in $C(A)$ by hypothesis, a direct computation shows

[TABLE]

Further, since $U$ was taken over all $a<b$ except $(a,b)=(i,j)$, if $\sigma\neq e$ and $\sigma\neq(i~{}j)$, then there exists some $a_{k},b_{k}$ such that $\sigma(b_{k})=a_{k}$. Hence, $p_{m}$ vanishes on all components except for the identity and possibly the transposition $(i~{}j)$, if $(i~{}j)\in G$. We now observe that $x_{i}p_{m}-p_{m}x_{j}=f_{i,j}\in(f)$, completing the proof. ∎

We define the Vandermonde determinant on the elements $y_{1},\dots,y_{n}$ in $T=\Bbbk[y_{1},\dots,y_{n}]$ by $\operatorname{Vdm}(T)=\prod_{i<j}(y_{i}-y_{j})$. If $A=V_{n}$ or $A=S(1,1,-1)$ (with $n=3$), we set $y_{i}=x_{i}^{2}\in C(A)$ and $T\subset C(A)$. By Lemma 2.1, $f_{1,2}\in(f)\cap A$. It follows that $\operatorname{Vdm}(T)=x_{1}f_{1,2}+f_{1,2}x_{2}\in(f)\cap A$. Moreover, for $1\leq i<j\leq n$ we define $\hat{f}_{i,j}=\frac{1}{2}(x_{i}f_{i,j}+f_{i,j}x_{i})\in(f)\cap A$. Then

[TABLE]

Let $J$ be the ideal generated by the $\binom{n}{2}+1$ elements $\hat{f}_{i,j}$ of degree $\binom{n}{2}$.

Proposition 2.3**.**

The GK dimension of the algebra $T/J$ is at most $n-2$.

Proof.

Set $\hat{f}=\sum_{1\leq i<j\leq n}\hat{f}_{i,j}$. We will show that $\hat{f}$ and $\operatorname{Vdm}(T)$ are relatively prime, and hence they form a regular sequence. It follows that the grade of $J$ (that is, the length of the longest regular sequence in $J$) is at least two, and hence the dimension of $T/J$ is at most $n-2$ by the depth inequality.

Let $1\leq a<b\leq n$. Consider the image of $\hat{f}$ in $T_{a,b}=k[y_{1},\dots,y_{n}]/(y_{a}-y_{b})$. Since each $\hat{f}_{i,j}$ for $(i,j)\neq(a,b)$ has $y_{a}-y_{b}$ as a factor, the image of $\hat{f}$ and $\hat{f}_{a,b}$ in $T_{a,b}$ agree. Since $T_{ab}$ is a domain and the image of all the irreducible factors of $\hat{f}_{a,b}$ are nonzero, the image of $\hat{f}_{a,b}$ is nonzero as well. Therefore $\hat{f}$ does not have $y_{a}-y_{b}$ as a factor for any such $a,b$, and hence $\hat{f}$ and $\operatorname{Vdm}(T)$ are relatively prime. As $T/J$ is commutative, its GK dimension is equal to its Krull dimension, so $\operatorname{GKdim}T/J\leq n-2$. ∎

Theorem 2.4**.**

The Auslander map $\gamma_{A,G}:A\#G\longrightarrow\operatorname{End}_{A^{G}}(A)$ is an isomorphism for $G$ a subgroup of $\mathcal{S}_{n}$ acting on $A=V_{n}$ or $W_{n}$, and for $G$ a subgroup of $\mathcal{S}_{3}$ acting on $S(1,1,-1)$.

Proof.

First, let $A=V_{n}$ or $A=S(1,1,-1)$. We assume $n=3$ in the second case. Then $A$ is finitely generated over the central subalgebra $T$. Let $A^{\prime}$ be the image of the map $A\hookrightarrow A\#G\longrightarrow(A\#G)/(f)$ and $T^{\prime}$ the image of $T$ in $A^{\prime}$. By [3, Lemma 5.2], $\operatorname{GKdim}T^{\prime}=\operatorname{GKdim}A^{\prime}=\operatorname{GKdim}(A\#G)/(f)$. Clearly, $\operatorname{GKdim}T^{\prime}\leq\operatorname{GKdim}T/J$ and $\operatorname{GKdim}T/J\leq n-2$ by the above argument. Hence

[TABLE]

Applying [3, Theorem 0.3] now completes the proof for $V_{n}$ and $S(1,1,-1)$.

By [2, Proposition 3.6 and Corollary 3.7], ${\sf{p}}(W_{n},G)\geq{\sf{p}}(V_{n},G)$ and so the Auslander map is an isomorphism for $W_{n}$ as well.∎

3. Pertinency computations

In this section we give more precise pertinency computations for subgroups of $\mathcal{S}_{3}$ and $\mathcal{S}_{4}$ acting on $V_{3}$ and $V_{4}$, respectively. In addition, we provide techniques for computing bounds on pertinency, distinct from the method in the previous section, including proving in Theorem 3.4 that the pertinency of any subgroup is an upper bound on the pertinency of the group. We also obtain in Theorem 3.12 an inequality that relates the pertinency of the actions of groups $G$ and $H$ (respectively) on algebras $A$ and $B$ (respectively) to the pertinency of $G\times H$ on the twisted tensor product $A\otimes_{\tau}B$ under certain conditions.

Pertinency values for $n$-cycles acting on $V_{n}$ were bounded, and for $n=2^{d}$ computed exactly, by Bao, He, and Zhang in a result we recall below.

Theorem 3.1** ([3, Theorem 5.7]).**

*Let $G=\left\langle(1~{}2~{}\cdots~{}n)\right\rangle$, $n\geq 2$ and let $\phi(n)$ be Euler’s phi function. If $n=2^{d}$ then ${\sf{p}}(V_{n},G)=n$. In general, ${\sf{p}}(V_{n},G)\geq\frac{\phi(n)}{2}$ when $n$ is even and ${\sf{p}}(V_{n},G)\geq\phi(n)$ when $n$ is odd. *

Remark 3.2**.**

Let $n\geq 2$ and suppose $\sigma$ is a $k$-cycle acting on $V_{n}$. Let $H=\left\langle\sigma\right\rangle$. If $k=n$, then ${\sf{p}}(V_{n},H)$ is as in Theorem 3.1. However, if $k<n$, then by reindexing we may assume $\sigma=(1~{}2~{}\cdots~{}k)$. We recognize $V_{n}$ as an Ore extension of $V_{k}$ with $H$ acting trivially on $x_{k+1},\ldots,x_{n}$. By [2, Lemma 6.4], ${\sf{p}}(V_{n},H)$ is still as above but with the $n$ on the right hand side replaced by $k$. We use this result frequently in what follows without mention.

Let $G$ be a finite subgroup of $\operatorname{Aut}_{\operatorname{gr}}(A)$. For a finite-order element $g\in G$, the reflection number of $g$, denoted ${\sf{r}}(A,g)$, is defined as

[TABLE]

The reflection number of the $G$-action on $A$ is defined to be

[TABLE]

Bao, He, and Zhang conjecture that ${\sf{p}}(A,G)\geq{\sf{r}}(A,G)$ [3, Conjecture 0.9]; if true this conjecture would imply that the Auslander map is an isomorphism for groups that contain no reflections. For all of the permutation actions considered in this paper, we show that the conjectured inequality holds.

Next we will bound the pertinency of the group action above by the pertinency of the action of a subgroup.

Lemma 3.3**.**

Suppose $G$ is a group acting on an algebra $A$ and $H\leq G$ is a subgroup. Let $f_{H}=\sum_{\sigma\in H}1\#\sigma$ and $f_{G}=\sum_{\sigma\in G}1\#\sigma$. If $\alpha\#e\in(f_{G})\cap A\subseteq A\#G$, then $\alpha\#e\in(f_{H})\cap A\subseteq A\#H$.

Proof.

Since $\alpha\#e\in(f_{G})$, there exist $a_{i},b_{i}\in A$ and $\sigma_{i},\sigma_{i}^{\prime}\in G$, $1\leq i\leq n$, such that

[TABLE]

Since, for all $i$, $(1\#\sigma_{i})f_{G}=f_{G}=f_{G}(1\#\sigma_{i}^{\prime})$, by replacing $b_{i}$ with $(\sigma_{i}^{\prime})^{-1}(b_{i})$, we may assume $\sigma_{i}=\sigma_{i}^{\prime}=e$ for all $i$. Therefore,

[TABLE]

That is, $\sum_{i=1}^{n}a_{i}\sigma(b_{i})=\alpha$ if $\sigma=e$ and [math] otherwise. Now consider

[TABLE]

By the computation above, this is equal to $\alpha\#e$, which proves our claim. ∎

The next theorem resolves a conjecture of Bao, He, and Zhang in the group case [3, Remark 5.6(2)].

Theorem 3.4**.**

Let $G$ be a group acting on $A$ and let $H\leq G$ be a subgroup. Then

[TABLE]

Proof.

By [3, Lemma 5.2], $\operatorname{GKdim}A\#G/(f_{G})=\operatorname{GKdim}A/((f_{G})\cap A)$. Lemma 3.3 implies that $\operatorname{GKdim}A/((f_{G})\cap A)\geq\operatorname{GKdim}A/((f_{H})\cap A)$ and the result follows. ∎

Corollary 3.5**.**

Let $A$ be a noetherian connected graded $\Bbbk$-algebra and let $G$ be a finite subgroup of $\operatorname{Aut}_{\operatorname{gr}}(A)$. Suppose that $g\in G$ is a reflection. If $A$ and $A^{\langle g\rangle}$ have finite global dimension, then the Auslander map $\gamma_{A,G}$ is not an isomorphism.

Proof.

Let $H=\langle g\rangle$ be the subgroup of $G$ generated by the reflection $g$. By [14, Lemmas 1.10 and 1.11], the graded $A^{H}$-module $A$ is finitely generated and free so $\operatorname{End}_{A^{H}}(A)$ is a matrix ring over $A^{H}$. This implies that $A\#H$ is not isomorphic to $\operatorname{End}_{A^{H}}(A)$ as $A\#H$ is concentrated in nonnegative degree and there must be a map of negative degree in $\operatorname{End}_{A^{H}}(A)$. One therefore has ${\sf{p}}(A,G)\leq{\sf{p}}(A,H)<2$ and so $\gamma_{A,G}$ is not an isomorphism. ∎

Corollary 3.6**.**

For $n\geq 2$, ${\sf{p}}(V_{n},\mathcal{S}_{n})=2$.

Proof.

By Theorem 2.4, ${\sf{p}}(V_{n},\mathcal{S}_{n})\geq 2$. The cyclic subgroup generated by a 2-cycle is a subgroup of $\mathcal{S}_{n}$ with pertinency exactly equal to $2$ (Theorem 3.1), so ${\sf{p}}(V_{n},\mathcal{S}_{n})\leq 2$ by Theorem 3.4. ∎

The above theorems, combined with prior results, are summarized in the table below as they apply to nontrivial subgroups of $\mathcal{S}_{3}$ acting on $V_{3}$. It is clear that pertinency is stable under conjugation, so we list the subgroups up to conjugacy.

[TABLE]

We need a few additional results to handle the subgroups of $\mathcal{S}_{4}$ acting on $V_{4}$.

Proposition 3.7**.**

Let $G$ be the Klein 4 subgroup $\langle(1~{}2)(3~{}4),(1~{}3)(2~{}4)\rangle$ acting on $A=V_{4}$. Then ${\sf{p}}(A,G)=4$.

Proof.

Let $f=\sum_{\sigma\in G}1\#\sigma$. Observe that

[TABLE]

Similarly, for each $\sigma\in G$, $q_{\sigma}=(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})\#e+(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})\#\sigma\in(f)$. Additionally, $(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})f\in(f)$, so by subtracting $\sum_{\sigma\in G}q_{\sigma}$, we have

[TABLE]

We also have $s=(x_{1}^{2}+x_{2}^{2}-x_{3}^{2}-x_{4}^{2})f+f(x_{1}^{2}+x_{2}^{2}-x_{3}^{2}-x_{4}^{2})\in(f)$ and therefore, $(x_{1}-x_{2})r+x_{1}s-sx_{2}=(x_{1}-x_{2})(x_{1}^{2}+x_{2}^{2})\in(f)\cap A$. Indeed, by the same argument, for any $1\leq i\neq j\leq 4$, $f_{i,j}=(x_{i}-x_{j})(x_{i}^{2}+x_{j}^{2})\in(f)\cap A.$ Now, we see that $x_{i}f_{i,j}+f_{i,j}x_{i}=2(x_{i}^{4}+x_{i}^{2}x_{j}^{2})\in(f)\cap A$ and $(x_{i}-x_{j})f_{i,j}=x_{i}^{4}+2x_{i}^{2}x_{j}^{2}+x_{j}^{4}\in(f)\cap A$. Subtracting, it follows that $x_{i}^{4}-x_{j}^{4}\in(f)\cap A$. Therefore, we have that $x_{i}^{2}x_{j}^{2}+x_{k}^{4}\in(f)\cap A$ for any $1\leq i\neq j\leq 4$, and $1\leq k\leq 4$.

Now $r^{2}$ is congruent to $-8x_{i}^{4}$ modulo the other six relations, and therefore $x_{i}^{4}\in(f)\cap A$ for all $i$, completing our proof. ∎

Using Molien’s Theorem, one can show that in lowest terms, the Hilbert series of $A^{G}$ as in Proposition 3.7 is

[TABLE]

and so $A^{G}$ is a rather complicated AS Gorenstein ring. Hence, Theorem 7 provides useful information for graded isolated singularities.

Proposition 3.8**.**

Let $\sigma=(1\ 2)(3\ 4)\cdots(2n-1\ 2n)$. Then ${\sf{p}}(V_{2n},\langle\sigma\rangle)=2n$.

Proof.

Let $f=1\#e+1\#\sigma$. Then for $i$ odd, one has $x_{i}f-fx_{i+1}=(x_{i}-x_{i+1})$. Now skew-commuting $x_{i}-x_{i+1}$ with either $x_{i}$ or $x_{i+1}$ shows that $x_{i}^{2}$ and $x_{i+1}^{2}$ are in $(f)\cap A$. Hence, $\dim_{\Bbbk}A/((f)\cap A)<\infty$ and ${\sf{p}}(A,\langle\sigma\rangle)=2n$. ∎

We list the pertinencies and reflection numbers of subgroups of $\mathcal{S}_{4}$ acting on $V_{4}$.

[TABLE]

We conjecture that ${\sf{p}}(V_{3},\left\langle(123)\right\rangle)=2$. If this conjecture holds, then ${\sf{p}}(V_{4},G)=2$ for $G=\left\langle(1~{}2~{}3)\right\rangle$ and $\left\langle(1~{}2~{}3),(1~{}2~{}4)\right\rangle$. We now give an alternate way to realize an upper bound on ${\sf{p}}(V_{4},\left\langle(1~{}2),(3~{}4)\right\rangle)$.

Let $A$ and $B$ be algebras with multiplication maps $\mu_{A}$ and $\mu_{B}$, respectively. A $\Bbbk$-linear homomorphism $\tau:B\otimes A\rightarrow A\otimes B$ is a twisting map provided $\tau(b\otimes 1_{A})=1_{A}\otimes b$ and $\tau(1_{B}\otimes a)=a\otimes 1_{B}$, $a\in A$, $b\in B$. A multiplication on $A\otimes B$ is then given by $\mu_{\tau}:=(\mu_{A}\otimes\mu_{B})\circ(\operatorname{id}_{A}\otimes\tau\otimes\operatorname{id}_{B})$. By [6, Proposition 2.3], $\mu_{\tau}$ is associative if and only if $\tau\circ(\mu_{B}\otimes\mu_{A})=\mu_{\tau}\circ(\tau\otimes\tau)\circ(\operatorname{id}_{B}\otimes\tau\otimes\operatorname{id}_{A})$ as maps $B\otimes B\otimes A\otimes A\rightarrow A\otimes B$. The triple $(A\otimes B,\mu_{\tau})$ is a twisted tensor product of $A$ and $B$, denoted by $A\otimes_{\tau}B$.

Let $G$ and $H$ be groups acting on $A$ and $B$, respectively. Then $G\times H$ acts naturally on $A\otimes B$ by $(g,h)(a\otimes b)=g(a)\otimes h(b)$.

Lemma 3.9**.**

Let $G$ and $H$ be finite groups of automorphisms acting on algebras $A$ and $B$. Let $\tau:B\otimes A\rightarrow A\otimes B$ be a twisting map. Then $G\times H$ acts on $A\otimes_{\tau}B$ provided

[TABLE]

Proof.

As observed above, $G\times H$ acts naturally on $A\otimes B$ and similarly $H\times G$ acts on $B\otimes A$. Let $\mu_{A}$ and $\mu_{B}$ be the multiplication maps on $A$ and $B$, respectively, and $\mu_{\tau}$ the (twisted) multiplication on $A\otimes_{\tau}B$. Recall that $A\otimes_{\tau}B$ has the same basis as $A\otimes B$.

Let $a\otimes b,a^{\prime}\otimes b^{\prime}\in A\otimes_{\tau}B$ and $\sum a_{i}^{\prime\prime}\otimes b_{i}^{\prime\prime}=\tau(b\otimes a^{\prime})$. Assuming (3.10),

[TABLE]

Example 3.11**.**

Suppose $A=\Bbbk_{-1}[x_{1},\ldots,x_{n}]$ and $B=\Bbbk_{-1}[y_{1},\ldots,y_{m}]$. Define a twisting map $\tau:B\otimes A\rightarrow A\otimes B$ by $b\otimes a\mapsto(-1)^{k\ell}(a\otimes b)$ for $a\in A_{k}$ and $b\in B_{\ell}$ and extend $\tau$ linearly. Then $A\otimes_{\tau}B\cong V_{n+m}$. If $G$ and $H$ are any groups acting linearly as automorphisms on $A$ and $B$ respectively then they preserve degree and hence (3.10) holds in this case.

The proof of the next theorem should be compared to [2, Lemma 6.4].

Theorem 3.12**.**

Let $A$ and $B$ be affine algebras generated in degree 1 and let $\tau:B\otimes A\longrightarrow A\otimes B$ be a twisting map. Assume $A\otimes_{\tau}B$ is Cohen-Macaulay. Let $G$ and $H$ be finite subgroups of automorphisms acting on $A$ and $B$, respectively. If (3.10) holds for $G$, $H$, and $\tau$, then one has the inequality

[TABLE]

Proof.

It suffices by Theorem 3.4 to prove that ${\sf{p}}(A\otimes B,G\times\{e_{H}\})={\sf{p}}(A,G)$ and ${\sf{p}}(A\otimes B,\{e_{G}\}\times H)={\sf{p}}(B,H)$. We will prove the first and the second follows similarly.

Since $G\times H$ satisfies (3.10), then clearly so does its subgroup $G\times\{e_{H}\}$. By a modification of [19, Proposition 3.11], $\operatorname{GKdim}(A\otimes_{\tau}B)=\operatorname{GKdim}(A)+\operatorname{GKdim}(B)$.

Set $F=\sum_{g\in G}1\#(g,e_{H})$. Then $F$ commutes with $B$ and so

[TABLE]

Now,

[TABLE]

Question 3.14**.**

Under what hypotheses does equality hold in (3.13)?

Example 3.15**.**

Let $A=\Bbbk_{-1}[y_{1},y_{2},y_{3},y_{4}]$ and $B=\Bbbk_{-1}[z_{1},z_{2},z_{3},z_{4}]$. Let $\tau$ be as in Example 3.11. Then $A\otimes_{\tau}B\cong V_{8}$. Let $G$ be the subgroup of $\mathcal{S}_{4}$ generated by $(1~{}2~{}3~{}4)$. Then $G$ acts on both $A$ and $B$ and so ${\sf{p}}(A,G)={\sf{p}}(B,G)=4$ by Theorem 3.1. By Theorems 2.4 and 3.12 we have

[TABLE]

4. The down-up algebra

$A(-2,-1)$

In this section, we demonstrate that our methods can be applied to certain groups of weighted permutations. Bao, He, and Zhang proved that the Auslander map is an isomorphism for noetherian $A(\alpha,\beta)$ and any finite group of graded automorphisms when $\beta\neq-1$, and also for the case $A(2,-1)$ [2, Theorem 0.6]. Here we prove that for any group acting on the down-up algebra $A(-2,-1)$ the Auslander map is an isomorphism (Theorem 4.3) solving one of the remaining open cases.

Recall that $A=A(-2,-1)$ is the $\Bbbk$-algebra generated by $x$ and $y$ subject to the two cubic relations $x^{2}y+yx^{2}+2xyx=xy^{2}+y^{2}x+2yxy=0$. The element $z=xy+yx$ is central in $A$.

By [13, Proposition 1.1],

[TABLE]

There exists a filtration $\mathcal{F}=\{F_{n}\}$ defined by $F_{n}A=(\Bbbk\oplus\Bbbk x\oplus\Bbbk y\oplus\Bbbk z)^{n}\subset A$ for all $n\geq 0$. The associated graded ring $\operatorname{gr}_{\mathcal{F}}(A)\cong V_{3}$ [17, Lemma 7.2(2)]. Note that $\mathcal{F}$ is stable with respect to the action of $\operatorname{Aut}_{\operatorname{gr}}(A)$ on $A$. Moreover, $V_{3}$ is a connected graded algebra and the group of automorphisms of $V_{3}$ induced by the filtration is

[TABLE]

As a $G$-module, $A\cong V_{3}$ so the $G$-action is inner faithful and homogeneous. Let $H$ be a finite subgroup of $\operatorname{Aut}_{\operatorname{gr}}(A)$. We identify $H$ with the corresponding subgroup of $G$, which, by an abuse of notation, we also call $H$. By [2, Proposition 3.6], ${\sf{p}}(A,H)\geq{\sf{p}}(V_{3},H)$. Hence, it suffices to prove ${\sf{p}}(V_{3},H)\geq 2$.

There are a few special cases of such groups. If $H$ is diagonal, then the Auslander map is an isomorphism by [2, Theorem 5.5]. In particular, $H$ is small in the commutative sense and the action of $H$ commutes with the graded twist sending $V_{3}$ to $\Bbbk[x_{1},x_{2},x_{3}]$. Also, if $H$ is small (in the commutative sense) when acting on $T=\Bbbk[y_{1},y_{2},y_{3}]\subset C(V_{3})$ where $y_{i}=x_{i}^{2}\in C(V_{3})$, then the Auslander map is an isomorphism by classical results.

Set $f=\sum_{h\in H}1\#h$ and write $H=H_{d}\cup H_{t}$ where the $H_{d}$ are diagonal and the $H_{t}$ are not. Denote elements of the first type as $M(a,b)$ and elements of the second type as $N(c,d)$.

Let $M=\operatorname{diag}(a,b,ab)\in H_{d}$. If two of $a,b,ab$ are equal to $1$ then so is the third. Hence, we conclude that for every nonidentity element of $H_{d}$, at least two of the entries are not $1$. Define $S_{t}=\{h_{2,1}:h\in H_{t}\}$, $S_{d}=\{h_{1,1}:h\in H_{d},h_{1,1}\neq 1\}$, and $S_{d}^{\prime}=\{h_{2,2}:h\in H_{d},h_{1,1}=1,h_{2,2}\neq 1\}$.

Lemma 4.1**.**

The element $\displaystyle V=x_{1}^{2|S_{d}|}x_{2}^{2|S_{d}^{\prime}|}\prod_{d\in S_{t}}(x_{1}^{2}-d^{-2}x_{2}^{2})\in(f)\cap T$.

Proof.

Set $p_{0}=f$. Enumerate the elements of $S_{t}$: $d_{1},\ldots,d_{n}$. Inductively define $p_{k}=x_{1}^{2}p_{k-1}-d_{k}^{-2}p_{k-1}x_{2}^{2}$. Set $\widehat{V}=p_{n}$. All components corresponding to $H_{t}$ are now zero in $\widehat{V}$ and the coefficient of the identity component is $\prod_{d\in S_{t}}(x_{1}^{2}-d^{-2}x_{2}^{2})$.

Enumerate the elements of $S_{d}$: $a_{1},\ldots,a_{\ell}$, and the elements of $S_{d}^{\prime}$: $b_{1},\ldots,b_{r}$. Set $q_{0}=\widehat{V}$ and inductively define

[TABLE]

Then $V=q_{r+\ell}$. ∎

Lemma 4.2**.**

There exists $\mu\in(f)\cap T$ such that $V$ and $\mu$ are relatively prime.

Proof.

We claim that for each factor $v\in V$, there exists $\mu_{v}\in(f)\cap T$ such that $v\nmid\mu_{v}$ and $u\mid\mu_{v}$ for all other factors $u$ of $V$ relatively prime to $v$. It then follows that $V$ and $\mu=\sum_{v}\mu_{v}$ are relatively prime.

Suppose $v=x_{1}^{2r}$ and let $\widehat{V}$ be as in Lemma 4.1. The proof follows almost identically to that lemma after replacing $S_{d},S_{d}^{\prime}$ with $\hat{S}_{d}=\{h_{2,2}:h\in H_{d},h_{2,2}\neq 1\}$ and $\hat{S}_{d}^{\prime}=\{h_{3,3}:h\in H_{d},h_{2,2}=1,h_{3,3}\neq 1\}$. Then $\gamma_{v}=q_{r+s}\in(f)\cap T$. The proof is similar when $v=x_{2}^{2(\ell-r)}$.

Suppose $v=(x_{1}^{2}-d^{-2}x_{2}^{2})$ for some $d$. Define $H_{t}^{\prime}$ as the set of $h\in H_{t}$ that are killed by $d^{-2}$ and not by any other constant. We may assume $H_{t}^{\prime}\neq\emptyset$. Set $p_{0}=f$. Enumerate the elements of $S_{t}$: $d=d_{1},\ldots,d_{n}$, and assume $d=d_{n}$. Inductively define $p_{k}=x_{1}^{2}p_{k-1}-d_{k}^{-2}p_{k-1}x_{2}^{2}$. Hence, all components in $p_{n-1}$ corresponding to $H_{t}$ are zero except those in $H_{t}^{\prime}$.

Let $\tilde{S}=\{h_{3,3}:h\in H_{d}\cup H_{t}^{\prime},h_{3,3}\neq 1\}$. Enumerate the elements of $\tilde{S}:\beta_{1},\ldots,\beta_{m}$. Set $q_{0}=p_{n-1}$. Inductively define $q_{k}=x_{3}(x_{3}q_{k-1}-\beta_{k}^{-1}q_{k-1}x_{3})$. Hence, the only remaining nonzero components in $q_{m}$ have one of the following forms ($d$ fixed, $a$ arbitrary): $M_{a}=M(a,a^{-1})$, $N_{1}=N(d^{-1},d)$, $N_{2}=N(-d^{-1},-d)$.

Consider $q=x_{1}q_{m}-d^{-1}q_{m}x_{2}$, which kills the $N_{1}$ component. The coefficient in the $M_{a}$ component is $(x_{1}-(ad)^{-1}x_{2})$, and the coefficient in the $N_{2}$ component is $2x_{1}$. Set $r_{0}=x_{2}q-dqx_{1}$. Since $(x_{2})(x_{1})-d(x_{1})(-d^{-1}x_{2})=x_{2}x_{1}+x_{1}x_{2}=0$, then the $N_{2}$ component is zero in $r_{0}$. What remains in the $M_{a}$ component is $2x_{2}x_{1}-ad(x_{1}^{2}+(ad)^{-2}x_{2}^{2})$.

Enumerate the distinct $(1,1)$-entries of the $M_{a}$ components whose coefficient is not zero: $a_{1},\ldots,a_{n}$, with $a_{1}=1$. Inductively define

[TABLE]

All coefficients in $r_{n}$ are central and it remains only to kill the remaining nonidentity $M_{a}$ components. For this we can proceed as in Lemma 4.1. ∎

Theorem 4.3**.**

Let $A=A(-2,-1)$. The Auslander map $\gamma_{A,G}:A\#G\rightarrow\operatorname{End}_{A^{G}}(A)$ is a graded isomorphism for any finite subgroup $G$ of $\operatorname{Aut}_{\operatorname{gr}}(A)$.

Proof.

As noted previously, it suffices by [2, Proposition 3.6] to prove this for $V_{3}$ and the corresponding group $H$ acting on $V_{3}$. By Lemmas 4.1 and 4.2, $(f)\cap T$ contains two relatively prime elements (see Proposition 2.3). We may now proceed as in Theorem 2.4. ∎

We observe that by [14, Proposition 6.4], the group $G$ in Theorem 4.3 contains no reflections.

5. Graded isolated singularities

Mori and Ueyama proved that if the Auslander map is an isomorphism, then $A^{G}$ is a graded isolated singularity if and only if $A\#G/(f_{G})$ is finite-dimensional [22, Theorem 3.10]. Thus, we have shown that $V_{4}^{G}$ is a graded isolated singularity when $G=\left\langle(1~{}2)(3~{}4)\right\rangle$ (Proposition 3.8) or $\left\langle(1~{}2)(3~{}4),(1~{}3)(2~{}4)\right\rangle$ (Proposition 3.7). In this section we present additional examples of graded isolated singularities using our methods. For these examples, we therefore obtain the conclusions of Theorem 7 as a corollary.

5.1. Generic three-dimensional Sklyanin algebras

Let $A=S(a,b,c)$ for a generic choice of $a,b,c\in\Bbbk$. Then $C_{3}=\langle\sigma\rangle$ acts on $A$ with $\sigma=(1~{}2~{}3)$ permuting the variables. We may diagonalize this action by choosing a different generating set of the algebra. Indeed, let $\xi$ be a primitive third root of unity, and set $X=x_{1}+\xi x_{2}+\xi^{2}x_{3}$, $Y=x_{1}+\xi^{2}x_{2}+\xi x_{3}$ and $Z=x_{1}+x_{2}+x_{3}$. Then $A$ is also generated by $X,Y$ and $Z$. To describe the relations that $X,Y$ and $Z$ satisfy, we have the following lemma:

Lemma 5.1**.**

In the notation of the previous paragraph, the generators $X,Y,Z$ satisfy Sklyanin relations with parameters

[TABLE]

Proof.

Let $f_{1}=axy+byx+cz^{2}$, $f_{2}=ayz+bzy+cx^{2}$, and $f_{3}=azx+bxz+cy^{2}$. Define $F_{1},F_{2}$ and $F_{3}$ similarly in terms of the generators $X,Y$ and $Z$ and the parameters $\alpha,\beta$ and $\gamma$. A direct verification shows that one has the following equalities which complete the proof of the lemma:

[TABLE]

Theorem 5.2**.**

Let $A$ be a Sklyanin algebra on generators $x,y,z$ with parameters $(a:b:c)$ and let $(\alpha:\beta:\gamma)$ be the coordinates of the Sklyanin algebra on generators $X,Y,Z$ as in the discussion above. If $\alpha,\beta,\gamma$, and $\alpha^{3}-\beta^{3}$ are nonzero and $\sigma$ is the permutation action of $(x~{}y~{}z)$ on $A$, then $A^{\left\langle\sigma\right\rangle}$ is a graded isolated singularity.

Proof.

It is easy to check that in our new coordinates, the action of $\sigma$ is given by $\sigma(X)=\xi^{2}X$, $\sigma(Y)=\xi(Y)$ and $\sigma(Z)=Z$. First, we claim that $Y^{2}\#e$ is in the ideal generated by $f=f_{G}$. Indeed, one has that

[TABLE]

hence $g=Y\#e+(1-\xi)Y\#\sigma$ is in $(f)$. The computation $Yg-\xi^{2}gY=(1-\xi^{2})Y^{2}\#e$ proves the desired claim. A similar calculation shows that $X^{2}\#e$ is in $(f)$.

Next, let $I$ be the ideal generated by the Sklyanin relations on $X,Y$ and $Z$, and $X^{2},Y^{2}$. Using the generators $X^{2}$ and $Y^{2}$ we may simplify these relations to:

[TABLE]

We use the diamond lemma with $X>Y>Z$ to produce a (partial) Gröbner basis of $I$. In the computations, we will assume that $\alpha,\beta,\gamma$, and $\alpha^{3}-\beta^{3}$ are nonzero

First, we compute the overlap between $\alpha XY+\beta YX+\gamma Z^{2}$ and $X^{2}$. In this calculation we use $=$ to denote equality in the tensor algebra and $\leadsto$ to indicate a reduction has taken place using the generators of the Gröbner basis of the defining ideal that have been found thus far. We have

[TABLE]

Therefore $Z^{2}X$ is in the ideal under our assumptions on the parameters. A similar calculation shows that $Z^{2}Y$ is in the ideal. Computing the overlap between $Z^{2}X$ and $\alpha XY+\beta YX+\gamma Z^{2}$ shows that $Z^{4}$ is in the ideal as well.

Hence, under the hypotheses on the parameters, the ideal $I$ contains elements with lead terms $X^{2}$, $XY$, $Y^{2}$, $YZ$, $XZ$, $Z^{2}X$, $Z^{2}Y$, $Z^{4}$. Therefore a spanning set of the quotient by the ideal $I$ consists of the monomials that do not contain those in the list above as subwords. A straightforward calculation shows that such a list of monomials is given by $1$, $X$, $Y$, $Z$, $YX$, $ZX$, $ZY$, $Z^{2}$, $ZYX$, $Z^{3}$. ∎

5.2. Weighted action on $V_{4}$

Consider the following action of the Klein-4 group on $V_{4}$. Let $K=\left\langle\alpha,\beta\right\rangle\subseteq\operatorname{Aut}_{\operatorname{gr}}(A)$ where $\alpha(x_{i})=-x_{i}$ for all $i$ and $\beta=(1~{}3)(2~{}4)$.

Theorem 5.3**.**

Let $K$ be the group above. Then ${\sf{p}}(V_{4},K)=4$ and thus $V_{4}^{K}$ is a graded isolated singularity, as is $V_{4}^{H}$ for any subgroup $H$ of $K$.

Proof.

Let $f=1\#e+1\#\alpha+1\#\beta+1\#\alpha\beta$ and set

[TABLE]

Then

[TABLE]

Combining these gives $x_{2}x_{3}-x_{4}x_{1},x_{2}x_{1}-x_{4}x_{3}\in(f)\cap V_{4}$.

Next we produce degree three elements in $(f)\cap V_{4}$. Set

[TABLE]

Thus, $x_{1}p_{3}-p_{3}x_{1}=4x_{1}^{2}x_{3}\#e\in(f)\cap V_{4}$. Conjugation and symmetry now give $x_{1}x_{3}^{2},x_{2}^{2}x_{4},x_{2}x_{4}^{2}\in(f)\cap V_{4}$. Finally we set

[TABLE]

Then $x_{1}p_{4}+p_{4}x_{1}=2x_{1}(x_{1}^{2}-x_{3}^{2})\#e$ and it follows that $x_{1}^{3}\in(f)\cap V_{4}$. Similarly, $x_{2}^{3},x_{3}^{3},x_{4}^{3}\in(f)\cap V_{4}$ hence $V_{4}/((f)\cap V_{4})$ is finite-dimensional. The remaining claim now follows from Theorem 3.4. ∎

Acknowledgments

We thank the referee for many helpful suggestions. Many computations for this paper were performed with the computer algebra system Macaulay2 [11]. The authors would also like to thank Andrew Conner and James Zhang for many stimulating conversations related to this project. E. Kirkman was partially supported by grant #208314 from the Simons Foundation.