It is a classical problem to compute a minimal set of invariant polynomial generating the invariant ring of a finite group as an algebra. We present here an algorithm for the computation of minimal generating sets in the non-modular case. Apart from very few explicit computations of Groebner bases, the algorithm only involves very basic operations, and is thus rather fast.
As a test bed for comparative benchmarks, we use transitive permutation groups on 7 and 8 variables. In most examples, our algorithm implemented in Singular works much faster than the one used in Magma, namely by factors between 50 and 1000. We also compute some further examples on more than 8 variables, including a minimal generating set for the natural action of the cyclic group of order 11 in characteristic 0 and of order 15 in characteristic 2.
We also apply our algorithm to the computation of irreducible secondary invariants.
We show that the rings of invariants for the three dimensional modular representations of an elementary abelian $p$-group of rank four are complete intersections with embedding dimension at most five. Our results confirm the conjectures of Campbell, Shank and Wehlau.
Using the description of the Frobenius limit of modules over the ring of invariants under an action of a finite group on a polynomial ring over a field of characteristic $p>0$ developed by Symonds and the author, we give a characterization of the ring of invariants with a positive dual $F$-signature. Combining this result and Kemper's result on depths of the ring of invariants under an action of a permutation group, we give an example of an $F$-rational, but non-$F$-regular ring of invariants under the action of a finite group.
We study the rings of invariants for the indecomposable modular representations of the Klein four group. For each such representation we compute the Noether number and give minimal generating sets for the Hilbert ideal and the field of fractions. We observe that, with the exception of the regular representation, the Hilbert ideal for each of these representations is a complete intersection.
Consider a finite dimensional vector space $V$ over a finite field $\mathbb{F}_q$. We give a minimal generating set for the ring of invariants $\mathbb{F}_q[V \oplus V^*]^{\text{GL}(V)}$, and show that this ring is a Gorenstein ring but is not a complete intersection. These results confirm a conjecture of Bonnafé and Kemper.
We determine necessary and sufficient conditions on the ring of differential operators of a finite purely inseparable field extension of positive characteristic for determining whether the extension is modular.
We investigate the presence of Cohen-Macaulay ideals in invariant rings and show that an ideal of an invariant ring corresponding to a modular representation of a $p$-group is not Cohen-Macaulay unless the invariant ring itself is. As an intermediate result, we obtain that non-Cohen-Macaulay factorial rings cannot contain Cohen-Macaulay ideals. For modular cyclic groups of prime order, we show that the quotient of the invariant ring modulo the transfer ideal is always Cohen-Macaulay, extending a result of Fleischmann.
We initiate a study of the rings of invariants of modular representations of elementary abelian p-groups. With a few notable exceptions, the modular representation theory of an elementary abelian p-group is wild. However, for a given dimension, it is possible to parameterise the representations. We describe parameterisations for modular representations of dimension two and of dimension three. We compute the ring of invariants for all two dimensional representations; these rings are generated by two algebraically independent elements. We compute the ring of invariants of the symmetric square of…
We describe "quasi canonical modules" for modular invariant rings $R$ of finite group actions on factorial Gorenstein domains. From this we derive a general "quasi Gorenstein criterion" in terms of certain 1-cocycles. This generalizes a recent result of A. Braun for linear group actions on polynomial rings, which itself generalizes a classical result of Watanabe for non-modular invariant rings. We use an explicit classification of all reflexive rank one $R$-modules, which is given in terms of the class group of $R$, or in terms of $R$-semi-invariants. This result is implicitly contained in a p…
Let $G$ be a finite group, and $V$ a finite dimensional vector space over a field $k$ of characteristic dividing the order of $G$. Let $H \leq G$. The transfer map $k[V]^H \rightarrow k[V]^G$ is an important feature of modular invariant theory. Its image is called a transfer ideal $I^G_H$ of $k[V]^G$, and this ideal, along with the quotients $k[V]^G/I^G_H$ are widely studied. In this article we study $k[V]^G/I$, where $I$ is any sum of transfer ideals. Our main result gives an explicit regular sequence of length $\dim(V^G)$ in $k[V]^G/I$ when $G$ is a $p$-group. We identify situations where th…
We exhibit a set of generating relations for the modular invariant ring of a vector and a covector for the two-dimensional general linear group over a finite field.
Let $G$ be a finite $p$-group and $k$ a field of characteristic $p>0$. We show that $G$ has a \emph{non-linear} faithful action on a polynomial ring $U$ of dimension $n=\mathrm{log}_p(|G|)$ such that the invariant ring $U^G$ is also polynomial. This contrasts with the case of \emph{linear and graded} group actions with polynomial rings of invariants, where the classical theorem of Chevalley-Shephard-Todd and Serre requires $G$ to be generated by pseudo-reflections. Our result is part of a general theory of "trace surjective $G$-algebras", which, in the case of $p$-groups, coincide with the Gal…
It is known that a finite group G can only act freely on affine n-space if K has positive characteristic p and G is a p-group. In that case the group action is "non-linear" and the ring of regular functions must be a trace-surjective G-algebra. Now let K be an arbitrary field of characteristic p>0 and let G be a finite p-group. In this paper we study the category Ts of all finitely generated trace-surjective K-G algebras. In a previous paper we have shown that the objects in Ts are precisely those finitely generated K-G algebras A such that the extension of A over the invariant ring A^G is a G…
We compute some numerical invariants of local cohomology of the ring of invariants by a finite group, mainly in the modular case. Also, we present some applications. In particular, we study Cohen-Macaulay property of modular invariants from the viewpoints of depth, Serre's condition and the relevant generalizations (e.g., the Buchsbaum property, etc). The situation in the local case is different from the global case.
In this article we present a parallel modular algorithm to compute all solutions with multiplicities of a given zero-dimensional polynomial system of equations over the rationals. In fact, we compute a triangular decomposition using Möller's algorithm (cf. [Mö93]) of the corresponding ideal in the polynomial ring over the rationals using modular methods, and then apply a solver for univariate polynomials.
We construct a number of different examples of non-Noetherian graded rings that are injective as modules over themselves (or have some related but weaker properties). We discuss how these are related to the theory of triangulated categories, and to Freyd's Generating Hypothesis in stable homotopy theory.
We consider the ring of coinvariants for modular representations of cyclic groups of prime order. For all cases for which explicit generators for the ring of invariants are known, we give a reduced Gröbner basis for the Hilbert ideal and the corresponding monomial basis for the coinvariants. We also describe the decomposition of the coinvariants as a module over the group ring. For one family of representations, we are able to describe the coinvariants despite the fact that an explicit generating set for the invariants is not known. In all cases our results confirm the conjecture of Harm Derks…
For a prime number p, we construct a generating set for the ring of invariants for the p+1 dimensional indecomposable modular representation of a cyclic group of order p^2. We then use the constructed invariants to describe the decomposition of the symmetric algebra as a module over the group ring, confirming the Periodicity Conjecture of Ian Hughes and Gregor Kemper for this case.
We use the gluing construction introduced by Jia Huang to explore the rings of invariants for a range of modular representations. We construct generating sets for the rings of invariants of the maximal parabolic subgroups of a finite symplectic group and their common Sylow $p$-subgroup. We also investigate the invariants of singular finite classical groups. We introduce parabolic gluing and use this construction to compute the invariant field of fractions for a range of representations. We use thin gluing to construct faithful representations of semidirect products and to determine the minimum…
For a field $\mathbb{F}$, let $R(n, m)$ be the ring of invariant polynomials for the action of $\mathrm{SL}(n, \mathbb{F}) \times \mathrm{SL}(n, \mathbb{F})$ on tuples of matrices -- $(A, C)\in\mathrm{SL}(n, \mathbb{F}) \times \mathrm{SL}(n, \mathbb{F})$ sends $(B_1, \dots, B_m)\in M(n, \mathbb{F})^{\oplus m}$ to $(AB_1C^{-1}, \dots, AB_mC^{-1})$. In this paper we call $R(n, m)$ the \emph{ring of matrix semi-invariants}. Let $β(R(n, m))$ be the smallest $D$ s.t. matrix semi-invariants of degree $\leq D$ generate $R(n, m)$. Guided by the Procesi-Razmyslov-Formanek approach of proving a strong d…
