Auslander s Theorem for permutation actions on noncommutative algebras

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1 Auslander s Theorem for permutation actions on noncommutative algebras Robert Won Joint with Jason Gaddis, Ellen Kirkman, and Frank Moore AMS Western Sectional Meeting, Pullman, WA April 23, / 22

2 Invariant theory k = k, char k = 0. All rings are k-algebras. Classically: Let G be a group acting on A = k[x 1,..., x n ]. Study A G the subring of invariants. (Shephard-Todd-Chevalley, Molien, Auslander) A g G is a reflection if g fixes a codim 1 subspace of kx i. Noncommutative invariant theory: A noncommutative H a Hopf algebra April 23, 2017 Invariant theory 2 / 22

3 Reflections Definition (Kirkman, Kuzmanovich, and Zhang 2008) Let A be a graded algebra of GK dimension n. Then g G is a quasi-reflection if its trace series is of the form Example Tr A (g, t) := 1 (1 t) n 1, q(1) 0. q(t) The symmetric group S n acting naturally on the ( 1)-skew polynomial ring k 1 [x 1,..., x n ] = k x 1,..., x n x i x j = x j x i for i j contains no quasi-reflections. April 23, 2017 Invariant theory 3 / 22

4 Auslander s Theorem Theorem (Auslander 1962) Let G GL n (k) be a finite group acting on A = k[x 1,..., x n ]. Define the Auslander map: γ A,G : A#G End A G(A) a#g (b ag(b)). If G contains no reflections, then γ A,G is an isomorphism. Question: Noncommutative version? Conjecture If A a noetherian AS-regular algebra and G GrAut(A) contains no quasi-reflections, then the Auslander map is an isomorphism. April 23, 2017 Invariant theory 4 / 22

5 Invariant theory Some answers: Theorem (Chan, Kirkman, Walton, and Zhang 2016) If A a noetherian AS-regular algebra of global dimension 2 with trivial homological determinant, then the Auslander map is an isomorphism. Theorem (Mori and Ueyama 2016) A a noetherian AS-regular algebra of dimension d 2 and G GrAut(A) with trivial homological determinant. TFAE: 1 G is ample for A. 2 A#G = End A G(A) and gl. dim tails A <. A#G 3 dim k < where f = 1 1#g. (f ) G g G April 23, 2017 Invariant theory 5 / 22

6 This seems pertinent Definition (Bao, He, and Zhang 2016) The pertinency p(a, G) = GKdim A GKdim A#G (f G ) where (f G ) is the two-sided ideal generated by f G = g G 1#g. (Actually defined for Hopf algebras) Under hypotheses of Mori and Ueyama, G is ample if and only if p(a, G) = n. April 23, 2017 Pertinency 6 / 22

7 This seems pertinent In fact: Theorem (Bao, He, and Zhang 2016) The Auslander map A#G End A G(A) is an isomorphism if and only if p(a, G) 2. They also prove: Theorem (Bao, He, and Zhang 2016) Let A = k 1 [x 1,..., x n ] and G = (1 2 n). Then p(a, G) 2 so the Auslander map is an isomorphism. Can we prove something more general? April 23, 2017 Pertinency 7 / 22

8 This seems ideal Thanks to Bao, He, Zhang, we just need to understand the ideal (f G ) = 1#g. g G Theorem (Bao, He, and Zhang 2016) Let A be finitely generated over a central subalgebra T. Let A be the image of the map A A#G (A#G)/(f G ) and T A be the image of T. Then GKdim T = GKdim A = GKdim A#G (f G ). April 23, 2017 The ideal 8 / 22

9 This seems ideal So need only understand (f G ) A or even (f G ) T. A minus sign and a quotient: p(a, G) = GKdim A GKdim A#G (f G ) = GKdim A GKdim A (f G ) A (f G ) A A p(a, G). (f G ) A Idea: Construct enough elements in (f G ) so p(a, G) 2. April 23, 2017 The ideal 9 / 22

10 Constructing elements We borrow an idea from Brown and Lorenz. Assume A is commutative. For g G, define I(g) generated by {a g.a a A}. Lemma (Brown and Lorenz 1994) I(g) (f G ) A. g G,g e Issue 1: Proof requires commutativity. Issue 2: Produces elements of degree G 1 = n! 1 (often much higher than lowest degree element in (f G )). April 23, 2017 The ideal 10 / 22

11 Example: k 1 [x 1, x 2, x 3 ] and S 3 Let T = k[x 2 1, x2 2, x2 3 ] C(V 3) and f = σ S 3 1#σ. Define f 1 = x 2 1 f fx2 2 = (x 2 1 x2 2 )#(1) + (x2 1 x2 2 )#(13) + (x 2 1 x2 3 )#(23) + (x2 1 x2 3 )#(123) f 2 = x 2 1 f 1 f 1 x 2 3 = (x 2 1 x2 2 )(x2 1 x2 3 )#(1) + (x2 1 x2 3 )(x2 1 x2 2 )#(23) f 3 = x 2 2 f 2 f 2 x 2 3 = (x 2 1 x2 2 )(x2 1 x2 3 )(x2 2 x2 3 )#(1) (f ) C(A). This provides only one of the elements we need. We must use noncommutativity to obtain the second element. April 23, 2017 An example is worth a thousand theorems 11 / 22

12 Example: k 1 [x 1, x 2, x 3 ] and S 3 Recall Now f 2 = (x 2 1 x2 2 )(x2 1 x2 3 )#(1) + (x2 1 x2 3 )(x2 1 x2 2 )#(23). g 23 = (x 2 f 2 f 2 x 3 )(x 2 x 3 ) = (x 2 1 x2 2 )(x2 1 x2 3 )(x 2 x 3 ) 2 #(1) = (x 2 1 x2 2 )(x2 1 x2 3 )(x2 2 + x2 3 )#(1) (f ) C(A). We can similarly construct g 12 and g 13. Set g = g 12 + g 13 + g 23. April 23, 2017 An example is worth a thousand theorems 12 / 22

13 Example: k 1 [x 1, x 2, x 3 ] and S 3 The elements f 3 = (x 2 1 x2 2 )(x2 1 x2 3 )(x2 2 x2 3 ) g = (x 2 1 x2 2 )(x2 1 x2 3 )(x2 2 + x2 3 ) + (x 2 1 x2 2 )(x2 1 + x2 3 )(x2 2 x2 3 ) + (x x2 2 )(x2 1 x2 3 )(x2 2 x2 3 ) are relatively prime in T = k[x 2 1, x2 2, x2 3 ] and GKdim T/(f 3, g) 1. Theorem (Gaddis, Kirkman, Moore, and W) Let G be any subgroup of S n acting on A = k 1 [x 1,... x n ] as permutations. Then p(a, G) 2 so the Auslander map is an isomorphism. April 23, 2017 An example is worth a thousand theorems 13 / 22

14 Example: S(a, b, c) and (1 2 3) Let S(a, b, c) be the three-dimensional Sklyanin algebra S(a, b, c) = k x 1, x 2, x 3 ax 1 x 2 + bx 2 x 1 + cx 2 3 ax 2 x 3 + bx 3 x 2 + cx 2 1 ax 3 x 1 + bx 1 x 3 + cx 2 2. acted on by (1 2 3). Let f = 1#e + 1#(1 2 3) + 1#(1 3 2). Then f 1 = x 1 f fx 3 = (x 1 x 3 )#e + (x 1 x 2 )#(1 3 2). (x 1 x 2 )f 1 + f 1 (x 2 x 3 ) = (x 1 x 2 )(x 1 x 3 ) + (x 1 x 3 )(x 2 x 3 )#e = (x 2 1 x2 3 x 2x 1 x 1 x 2 )#e (f ). April 23, 2017 An example is worth a thousand theorems 14 / 22

15 Example: S(a, b, c) and (1 2 3) So x 2 1 x2 3 x 2x 1 x 1 x 2 (f ) S(a, b, c) x 2 2 x2 3 x 1x 2 x 2 x 1 (f ) S(a, b, c). Now a Gröbner basis argument implies dim k S(a, b, c) (f ) S(a, b, c) <. Theorem (Gaddis, Kirkman, Moore, and W) Let G = (1 2 3) acting on A = S(a, b, c) for generic (a : b : c) P 2. Then p(a, G) = 3 2 so the Auslander map is an isomorphism. April 23, 2017 An example is worth a thousand theorems 15 / 22

16 All the Auslanders! Theorem (Gaddis, Kirkman, Moore, and W) The Auslander map is an isomorphism for the following: 1 subgroups of S n acting on k 1 [x 1,..., x n ], 2 subgroups of S n acting on the ( 1)-quantum Weyl algebra, 3 subgroups of S 3 acting on the three-dimensional Sklyanin algebra S(1, 1, 1), 4 the cyclic group (1 2 3) acting on a generic three-dimensional Sklyanin algebra S(a, b, c), 5 subgroups of weighted permutations acting on the down-up algebra A(2, 1), 6 I n, (1 3)(2 4) acting on k 1 [x 1, x 2, x 3, x 4 ]. April 23, 2017 Results 16 / 22

17 Graded isolated singularities Definition (Ueyama 2013) A G is a graded isolated singularity if gl. dim tails A G <. Theorem (Ueyama 2016) If A G is a graded isolated singularity, then A G is an AS-Gorenstein algebra of dimension d 2, A CM gr (A G ) is a (d 1)-cluster tilting module, and Ext 1 A G (A, M) and Ext 1 A G (M, A) are f.d. for M CM gr (A G ). Theorem (Mori and Ueyama 2016) If GKdim A 2, A G is a graded isolated singularity if and only dim k A#G/(f G ) < if and only if p(a, G) = n. April 23, 2017 Results 17 / 22

18 Graded isolated singularities Theorem (Bao, He, and Zhang 2016) Let A = k 1 [x 1,..., x 2 n] and G = (1 2 2 n ). Then p(a, G) = 2 n so A G is a graded isolated singularity. Theorem (Gaddis, Kirkman, Moore, and W) For the following, A G is a graded isolated singularity: 1 (1 2)(3 4), (1 3)(2 4) acting on k 1 [x 1, x 2, x 3, x 4 ], 2 (1 2)(3 4) (2n 1 2n) acting on k 1 [x 1,..., x 2n ], 3 (1 2 3) acting on a generic Sklyanin algebra S(a, b, c), 4 I n, (1 3)(2 4) acting on k 1 [x 1, x 2, x 3, x 4 ]. April 23, 2017 Results 18 / 22

19 Whither the upper bounds? Constructing elements of (f G ) gives lower bounds for p(a, G). Upper bounds? Theorem (Gaddis, Kirkman, Moore, and W) If G G then p(a, G) p(a, G ). (This resolves a conjecture of Bao-He-Zhang for the group case.) Corollary (Gaddis, Kirkman, Moore, and W) Let A be a noetherian connected graded algebra and suppose G contains a reflection g. If A and A g have finite global dimension, then the Auslander map γ A,G is not an isomorphism. April 23, 2017 Results 19 / 22

20 Computing pertinency exactly Lower bounds: constructing elements of (f G ) Upper bounds: subgroup theorem Subgroups of S 3 acting on k 1 [x 1, x 2, x 3 ]: conjugacy class p(a, G) (12) 2 (123) 2 or 3 (12), (23) 2 April 23, 2017 Results 20 / 22

21 Computing pertinency exactly Subgroups of S 4 acting on k 1 [x 1, x 2, x 3, x 4 ]: conjugacy class p(a, G) (12) 2 (12)(34) 4 (123) 2 or 3 (1234) 4 (12), (34) 2 (12)(34), (13)(24) 4 (1234), (24) 2 (123), (124) 2 or 3 (123), (12) 2 (1234), (12) 2 April 23, 2017 Results 21 / 22

22 Thanks! April 23, 2017 Results 22 / 22

Noncommutative invariant theory and Auslander s Theorem

Noncommutative invariant theory and Auslander s Theorem Miami University Algebra Seminar Robert Won Wake Forest University Joint with Jason Gaddis, Ellen Kirkman, and Frank Moore arxiv:1707.02822 November