SLIDE 1
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
SLIDE 2 Basic setup
◮ G a group, V a K-vector space (char(K) = 0)
with basis x1, . . . , xn ρ : G → GL(V ) a representation
◮ Induced action of G on
–the tensor algebra T(V ) = Kx1, . . . , xn (free associative K-algebra) –symmetric tensor algebra S(V ) = K[x1, . . . , xn] (commutative polynomial algebra)
◮ The algebra of G-invariants: for R = T(V ) or S(V )
RG := {f ∈ R | g · f = f ∀g ∈ G}
SLIDE 3 Relatively free algebras
◮ Let A be a K-algebra, f ∈ Kx1, . . . , xn.
f = 0 is a polynomial identity (PI) on A if f(a1, . . . , an) = 0 ∈ A ∀ a1, . . . , an ∈ 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)
SLIDE 4
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|.
SLIDE 5
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 AG 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) = supV {β(G, R, V ) | V is a G-module}.
SLIDE 6 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.
SLIDE 7 Answer1
Let R be a weakly noetherian variety properly containing the variety
- f commutative algebras and G a finite group. Then
β(G, R) ≤ (ν(n(R)) − 1) · ν (2β(G)ℓ(R, |G|)) − 1 where
◮ R satisfies a multihomogeneous identity
x2xn(R)+1
1
x3 + x1h1(x1, x2, x3) + h2(x1, x2, x3)x1 = 0 (see [L’vov])
◮ ν(n) = min{d ∈ N | x1 · · · xd ∈ (xn)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.]
SLIDE 8 Hilbert series
F(R, V ) = ∞
d=0 F(R, V )d is graded
H(F(R, V )G, q) =
∞
dim(F(R, V )G
d )qd ∈ Z[[q]].
———————————————————————————— 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
- f GL(V ). Assume I(R) = 0. Then
H(F(R, V )G, q) = P(q)
m
(1 − qdj) for some m, d1, . . . , dm ∈ N and P ∈ Z[q].
- 2M. Domokos and V. Drensky, arXiv:1512.06411v2
SLIDE 9 Example 1
Let dim(V ) = 2, so H(T(V ); t1, t2) =
1 1−t1−t2 .
Then H(T(V )SL(V ), q) =
∞
1 n + 1 2n n
1 2q2 (1 −
by [Almkvist-Dicks-Formanek].
SLIDE 10 Theorem of Belov and Berele
F(R, V ) is Zn-graded (n = dim(V )), H(F(R, V ), t1, . . . , tn) =
dim(F(R, V )γ)tγ1
1 · · · tγn n .
————————————————————————— Supose I(R) = 0. Then H(F(R, V ); t1, . . . , tn) = P(t1, . . . , tn)
(1 − tα1
1 · · · tαn n )
where α ranges over a finite subset of Nn
0 and P ∈ Z[t1, . . . , tn].
SLIDE 11 Corollary. H(F(R, V ); qt1, . . . , qtn) = P(t1, . . . , tn, q)
(1 − tα1
1 · · · tαn n qk)
where P ∈ Z[t1, . . . , tn][q] is a polynomial in q and the product (α, k) range over a finite multiset of pairs with α ∈ Zn, k ∈ N0.
- Def. An element of Z[t1, . . . , tn]Sn[[q]] of the above form is called a
nice rational function.
SLIDE 12 Characterization of nice rational functions
The formal character of a polynomial GLn-representation on Y is chY (t1, . . . , tn) = Tr(diag(t1, . . . , tn)|Y ) When Y = ∞
d=0 Yd is graded:
chY (t1, . . . , tn, q) =
chYd ∈ Z[t1, . . . , tn]Sn[[q]].
SLIDE 13 C := graded polynomial GLn-representations Y = ∞
d=0 Yd, such that
– Y is a finite module over S(W) for some finite dimensional graded polynomial GLn-representation W; – for g ∈ GLn, f ∈ S(W), and m ∈ Y , we have g · (fm) = (g · f)(g · m).
- Proposition. f ∈ Z[t1, . . . , tn][[q]] is a nice rational function if and
- nly if it belongs to the Z-submodule generated by
{chZ | Z ∈ C}.
SLIDE 14
- Corollary. H(F(R, V ); qt1, . . . , qtn) can be expressed as an
alternating sum of formal characters chY with Y ∈ C. For any polynomial GLn-module Z and subgroup G ≤ GLn, dim(ZG) depends only on chZ. 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.
SLIDE 15 Example II
R := T(V )/I where dim(V ) = 2, I := T-id(K2×2), G := K× acting on V via z · x1 = zx1, z · x2 = z−1x2 (z ∈ K×). H(R, t1, t2) = 1 (1 − t1)(1 − t2) + t1t2 (1 − t1)2(1 − t2)2(1 − t1t2) by [Formanek-Halpin-Li]. Thus H(R, t1q, t2q) is the formal character
- f the graded polynomial GL(V )-module
Z = S(V ) ⊕ (
2
2
ZG = S(V )G ⊕ (
2
2
We obtain H(RG, q) = H(ZG, q) = 1 1 − q2 + q2(1 + q2) (1 − q2)4 .
SLIDE 16 Nagata’s example
Remark. H(S(V )G, q) = 1 + 4q9 + 7q18 + 10q27 + 10q36 + 4q45 (1 − q18)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[x1, . . . , xn]G has a non-rational Hilbert series?
