Degree bounds and rationality of Hilbert series in noncommutative - PowerPoint PPT Presentation

Degree bounds and rationality of Hilbert series in noncommutative invariant theory M´ aty´ as Domokos ——– (based on joint work with Vesselin Drensky) ——– MTA R´ enyi Institute of Mathematics Budapest, Hungary June, 2017

Basic setup ◮ G a group, V a K -vector space (char( K ) = 0) with basis x 1 , . . . , x n ρ : G → GL ( V ) a representation ◮ Induced action of G on –the tensor algebra T ( V ) = K � x 1 , . . . , x n � (free associative K -algebra) –symmetric tensor algebra S ( V ) = K [ x 1 , . . . , x n ] (commutative polynomial algebra) ◮ The algebra of G -invariants : for R = T ( V ) or S ( V ) R G := { f ∈ R | g · f = f ∀ g ∈ G }

Relatively free algebras ◮ Let A be a K -algebra, f ∈ K � x 1 , . . . , x n � . f = 0 is a polynomial identity (PI) on A if f ( a 1 , . . . , a n ) = 0 ∈ A ∀ a 1 , . . . , a n ∈ A. ◮ The T-ideal of identities of the variety R of associative algebras: I ( R , V ) = { f ∈ T ( V ) | f = 0 is a PI ∀ A ∈ R } ◮ Relatively free algebra of the variety R : F ( R , V ) = T ( V ) /I ( R )

Invariant theory T ( V ) ։ F ( R , V ) ։ S ( V ) If G acts completely reducibly on T ( V ), then we get surjections T ( V ) G ։ F ( R , V ) G ։ S ( V ) G Theorem [E. Noether]: For a finite group G the algebra S ( V ) G is finitely generated. In fact S ( V ) G is generated by elements of degree ≤ | G | .

Theorem of Kharchenko The following conditions on a variety R are equivalent: (i) For every finitely generated A ∈ R and finite group G acting on A via K -algebra automorphisms the algebra A G is finitely generated. (ii) Every finitely generated algebra in R is weakly noetherian. ——————————————————————————- β ( G, R , V ) = min { m | F ( R , V ) G is generated in degree ≤ m } , β ( G, R ) = sup V { β ( G, R , V ) | V is a G -module } .

Questions β ( G ) := min { m | S ( V ) G is generated in degree ≤ m ∀ G -module V } Noether’s bound: β ( G ) ≤ | G | . ———————————————————————————— Let R be a weakly noetherian variety of unitary associative K -algebras. 1. Is β ( G, R ) finite for all finite groups G ? 2. If the answer to 1. is yes, find an upper bound for β ( G, R ) in terms of | G | and some numerical invariants of R .

Answer 1 Let R be a weakly noetherian variety properly containing the variety of commutative algebras and G a finite group. Then β ( G, R ) ≤ ( ν ( n ( R )) − 1) · ν (2 β ( G ) ℓ ( R , | G | )) − 1 where ◮ R satisfies a multihomogeneous identity x 2 x n ( R )+1 x 3 + x 1 h 1 ( x 1 , x 2 , x 3 ) + h 2 ( x 1 , x 2 , x 3 ) x 1 = 0 (see [L’vov]) 1 ◮ ν ( n ) = min { d ∈ N | x 1 · · · x d ∈ ( x n ) T-id } (see [Nagata-Higman]) ◮ ℓ ( R , dim( V )) = index of nilpotency of the commutator ideal of F ( R , V ) (see [Latyshev]) 1 [M. Domokos and V. Drensky, J. Alg. 463 (2016), 152-167.]

Hilbert series F ( R , V ) = � ∞ d =0 F ( R , V ) d is graded ∞ d ) q d ∈ Z [[ q ]] . � H ( F ( R , V ) G , q ) = dim( F ( R , V ) G d =0 ———————————————————————————— Theorem. 2 Suppose that G is a reductive subgroup of GL ( V ) or G is the unipotent radical of a Borel subgroup in a reductive subgroup of GL ( V ). Assume I ( R ) � = 0. Then P ( q ) H ( F ( R , V ) G , q ) = m � (1 − q d j ) j =1 for some m, d 1 , . . . , d m ∈ N and P ∈ Z [ q ]. 2 M. Domokos and V. Drensky, arXiv:1512.06411v2

Example 1 1 Let dim( V ) = 2, so H ( T ( V ); t 1 , t 2 ) = 1 − t 1 − t 2 . Then ∞ 1 � 2 n � 1 q n = � � H ( T ( V ) SL ( V ) , q ) = 1 − 4 q 2 ) 2 q 2 (1 − n + 1 n n =0 by [Almkvist-Dicks-Formanek].

Theorem of Belov and Berele F ( R , V ) is Z n -graded ( n = dim( V )), � dim( F ( R , V ) γ ) t γ 1 1 · · · t γ n H ( F ( R , V ) , t 1 , . . . , t n ) = n . α ∈ N n 0 ————————————————————————— Supose I ( R ) � = 0. Then P ( t 1 , . . . , t n ) H ( F ( R , V ); t 1 , . . . , t n ) = � (1 − t α 1 1 · · · t α n n ) α where α ranges over a finite subset of N n 0 and P ∈ Z [ t 1 , . . . , t n ].

Corollary. P ( t 1 , . . . , t n , q ) H ( F ( R , V ); qt 1 , . . . , qt n ) = � (1 − t α 1 1 · · · t α n n q k ) ( α,k ) where P ∈ Z [ t 1 , . . . , t n ][ q ] is a polynomial in q and the product ( α, k ) range over a finite multiset of pairs with α ∈ Z n , k ∈ N 0 . Def. An element of Z [ t 1 , . . . , t n ] S n [[ q ]] of the above form is called a nice rational function .

Characterization of nice rational functions The formal character of a polynomial GL n -representation on Y is ch Y ( t 1 , . . . , t n ) = Tr(diag( t 1 , . . . , t n ) | Y ) When Y = � ∞ d =0 Y d is graded: � ch Y d ∈ Z [ t 1 , . . . , t n ] S n [[ q ]] . ch Y ( t 1 , . . . , t n , q ) = d

C := graded polynomial GL n -representations Y = � ∞ d =0 Y d , such that – Y is a finite module over S ( W ) for some finite dimensional graded polynomial GL n -representation W ; – for g ∈ GL n , f ∈ S ( W ), and m ∈ Y , we have g · ( fm ) = ( g · f )( g · m ). Proposition. f ∈ Z [ t 1 , . . . , t n ][[ q ]] is a nice rational function if and only if it belongs to the Z -submodule generated by { ch Z | Z ∈ C} .

Corollary. H ( F ( R , V ); qt 1 , . . . , qt n ) can be expressed as an alternating sum of formal characters ch Y with Y ∈ C . For any polynomial GL n -module Z and subgroup G ≤ GL n , dim( Z G ) depends only on ch Z . Therefore H ( F ( R , V ) G , q ) can be expressed as an alternating sum of Hilbert series of the form H ( Y G , q ) with Y ∈ C . Y G is a finite module over S ( W ) G , which is a finitely generated algebra by commutative invariant theory, hence H ( Y G , q ) is a nice rational function by the Hilbert-Serre Theorem.

Example II R := T ( V ) /I where dim( V ) = 2, I := T-id( K 2 × 2 ), G := K × acting on V via z · x 1 = zx 1 , z · x 2 = z − 1 x 2 ( z ∈ K × ). 1 t 1 t 2 H ( R, t 1 , t 2 ) = (1 − t 1 )(1 − t 2 ) + (1 − t 1 ) 2 (1 − t 2 ) 2 (1 − t 1 t 2 ) by [Formanek-Halpin-Li]. Thus H ( R, t 1 q, t 2 q ) is the formal character of the graded polynomial GL ( V )-module 2 2 � � Z = S ( V ) ⊕ ( ( V ) ⊗ S ( V ⊕ V ) ⊗ S ( ( V ))) . 2 2 Z G = S ( V ) G ⊕ ( ( V ) ⊗ S ( V ⊕ V ) G ⊗ S ( � � ( V ))) . We obtain 1 − q 2 + q 2 (1 + q 2 ) 1 H ( R G , q ) = H ( Z G , q ) = (1 − q 2 ) 4 .

Nagata’s example Remark. H ( S ( V ) G , q ) = 1 + 4 q 9 + 7 q 18 + 10 q 27 + 10 q 36 + 4 q 45 (1 − q 18 ) 4 where the representation G → GL ( V ) is Steinberg’s variant of Nagata’s example of a linear group action whose algebra of invariants in not finitely generated. Problem. Does there exist a G -module V such that S ( V ) G = K [ x 1 , . . . , x n ] G has a non-rational Hilbert series?