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Computational Invariant Theory
 Encyclopaedia of Mathematical Sciences, SpringerVerlag
, 1998
"... This article is an expanded version of the material presented there. The main topic is the calculation of the invariant ring of a finite group acting on a polynomial ring by linear transformations of the indeterminates. By "calculation" I mean finding a finite system of generators for the ..."
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Cited by 36 (2 self)
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This article is an expanded version of the material presented there. The main topic is the calculation of the invariant ring of a finite group acting on a polynomial ring by linear transformations of the indeterminates. By "calculation" I mean finding a finite system of generators for the invariant ring, and (optionally) determining structural properties of it. In this exposition particular emphasis is placed on the case that the ground field has positive characteristic dividing the group order. We call this the modular case, and it is important for several reasons. First, many theoretical questions about the structure of modular invariant rings are still open. I will address the problems which I consider the most important or fascinating in the course of the paper. Thus it is very helpful to be able to compute modular invariant rings in order to gain experience, formulate or check conjectures, and gather some insight which in fortunate cases leads to proofs. Furthermore, the computation of modular invariant ring can be very useful for the study of cohomology of finite groups (see Adem and Milgram [1]). This exposition also treats the nonmodular case (characteristic zero or coprime to the group order), where computations are much easier and the theory is for the most part settled. There are also various applications in this case, such as the solution of algebraic equations or the study of dynamical systems with symmetries (see, for example, Gatermann [11], Worfolk [26]).
Some Algorithms in Invariant Theory of Finite Groups
 Computational Methods for Representations of Groups and Algebras, Euroconference in Essen, April 15 1997, Progress in Mathematics 173
, 1998
"... We present algorithms which calculate the invariant ring K[V ] G of a finite group G. Our focus of interest lies on the modular case, i.e., the case where jGj is divided by the characteristic of K. We give easy algorithms to compute several interesting properties of the invariant ring, such as ..."
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Cited by 18 (7 self)
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We present algorithms which calculate the invariant ring K[V ] G of a finite group G. Our focus of interest lies on the modular case, i.e., the case where jGj is divided by the characteristic of K. We give easy algorithms to compute several interesting properties of the invariant ring, such as the CohenMacaulay property, depth, the finumber and syzygies. Introduction This paper presents various algorithms for invariant theory of finite groups, which were implemented in the computer algebra system Magma [4 or 6] during a visit of the first author to Sydney. We focus on those algorithms which are new or which have never been written up before, and only sketch those that can already be found in the literature. Due to improvements of existing algorithms and a better usage of computational resources by the Magma system, this recent implementation generally produces much better timings than the Invar package which was implemented by the first author in Maple (see Kemper [8]). For ge...
Symmetric Powers of Modular Representations, Hilbert Series and Degree Bounds
, 1999
"... Let G = Zp be a cyclic group of prime order p with a representation G ! GL(V ) over a field K of characteristic p. In 1976, Almkvist and Fossum gave formulas for the decomposition of the symmetric powers of V in the case that V is indecomposable. From these they derived formulas for the Hilbert seri ..."
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Cited by 12 (1 self)
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Let G = Zp be a cyclic group of prime order p with a representation G ! GL(V ) over a field K of characteristic p. In 1976, Almkvist and Fossum gave formulas for the decomposition of the symmetric powers of V in the case that V is indecomposable. From these they derived formulas for the Hilbert series of the invariant ring K[V ] G . Following Almkvist and Fossum in broad outline, we start by giving a shorter, selfcontained proof of their results. We extend their work to modules which are not necessarily indecomposable. We also obtain formulas which give generating functions encoding the decompositions of all symmetric powers of V into indecomposables. Our results generalize to groups of the type Zp \Theta H with jHj coprime to p. Moreover, we prove that for any finite group G whose order is divisible by p but not by p 2 , the invariant ring K[V] G is generated by homogeneous invariants of degrees at most dim(V ) \Delta (jGj  1).
Loci in Quotients by Finite Groups, Pointwise Stabilizers and the Buchsbaum Property
, 2000
"... Let K[V ] G be the invariant ring of a finite linear group G GL(V ), and let GU be the pointwise stabilizer of a subspace U V . We prove that the following numbers associated to the invariant ring decrease if one passes from K[V ] G to K[V ] GU : the minimal number of homogeneous generators, ..."
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Let K[V ] G be the invariant ring of a finite linear group G GL(V ), and let GU be the pointwise stabilizer of a subspace U V . We prove that the following numbers associated to the invariant ring decrease if one passes from K[V ] G to K[V ] GU : the minimal number of homogeneous generators, the maximal degree of the generators, the number of syzygies and other Betti numbers, the complete intersection defect, the difference between depth and dimension, and the type. From this, theorems of Steinberg, Serre, Nakajima, Kac and Watanabe, Smith, and the author follow, which say that if K[V ] G is a polynomial ring, a hypersurface, a complete intersection, or CohenMacaulay, then the same is true for K[V ] GU . Furthermore, K[V ] GU inherits the Gorenstein property from K[V ] G . We give an algorithm which transforms generators of K[V ] G into generators of K[V ] GU . Let P be one of the properties mentioned above. We consider the locus of P in V==G := Spec \Gamma K[V ...
Symmetric Powers of Modular Representations for Groups with a Sylow Subgroup of Prime Order
 J. of Algebra
, 2000
"... Let V be a representation of a finite group G over a field of characteristic p. If p does not divide the group order, then Molien's formula gives the Hilbert series of the invariant ring. In this paper we find a replacement of Molien's formula which works in the case that jGj is divisible ..."
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Let V be a representation of a finite group G over a field of characteristic p. If p does not divide the group order, then Molien's formula gives the Hilbert series of the invariant ring. In this paper we find a replacement of Molien's formula which works in the case that jGj is divisible by p but not by p 2 . We also obtain formulas which give generating functions encoding the decompositions of all symmetric powers of V into indecomposables. Our methods can be applied to determine the depth of the invariant ring without computing any invariants. This leads to a proof of a conjecture of the second author on certain invariants of GL2 (p). Contents Introduction 1 1 Groups with a normal Sylow psubgroup of order p 3 1.1 The representation ring and species . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Exterior and symmetric powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Averaging operators and Hilbert series . . . . . . . . . . . . . . . . . . ...
Non CohenMacaulay invariant rings of infinite groups
 J. Algebra
, 2006
"... We give explicit examples of invariant rings that are not CohenMacaulay for all classical groups SLn(K), GLn(K), Sp2n(K), SOn(K) and On(K), where K is an algebraically closed field of positive characteristic. We prove that every nontrivial unipotent group over K has representations such that the i ..."
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We give explicit examples of invariant rings that are not CohenMacaulay for all classical groups SLn(K), GLn(K), Sp2n(K), SOn(K) and On(K), where K is an algebraically closed field of positive characteristic. We prove that every nontrivial unipotent group over K has representations such that the invariant ring is not CohenMacaulay.
The Depth of Invariant Rings and Cohomology
, 1999
"... Let G be a finite group acting linearly on a vector space V over a field K of positive characteristic p, and let P G be a Sylow psubgroup. Ellingsrud and Skjelbred [20] proved the lower bound depth(K[V ] G ) minfdim(V P ) + 2; dim(V )g for the depth of the invariant ring, with equality if ..."
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Let G be a finite group acting linearly on a vector space V over a field K of positive characteristic p, and let P G be a Sylow psubgroup. Ellingsrud and Skjelbred [20] proved the lower bound depth(K[V ] G ) minfdim(V P ) + 2; dim(V )g for the depth of the invariant ring, with equality if G is a cyclic pgroup. Let us call the pair (G; V ) flat if equality holds in the above. In this paper we use cohomological methods to obtain information about the depth of invariant rings and in particular to study the question of flatness. For G of order not divisible by p 2 it ensues that (G; V ) is flat if and only if dim(V P ) dim(V )\Gamma2 or H 1 (G; K[V ]) 6= 0. We obtain a formula for the depth of the invariant ring in the case that G permutes a basis of V and has order not divisible by p 2 . In this situation (G; V ) is usually not flat. Moreover, we introduce the notion of visible flatness of pairs (G; V ) and prove that this implies flatness. For example, the gro...
On the Depth of Cohomology Modules
, 2003
"... We study the cohomology modules H (G; R) of a pgroup G acting on a ring R of characteristic p, for i > 0. In particular, we are interested in the CohenMacaulay property and the depth of H (G; R) regarded as an R module. We rst determine the support of H (G; R), which turns out ..."
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Cited by 2 (0 self)
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We study the cohomology modules H (G; R) of a pgroup G acting on a ring R of characteristic p, for i > 0. In particular, we are interested in the CohenMacaulay property and the depth of H (G; R) regarded as an R module. We rst determine the support of H (G; R), which turns out to be independent of i. Then we study the CohenMacaulay property for H (G; R). Further results are restricted to the special case that G is cyclic and R is the symmetric algebra of a vector space on which G acts. We determine
Depth and Cohomological Connectivity in Modular Invariant Theory
 Trans. Amer. Math. Soc
, 2003
"... Let G be a nite group acting linearly on a nite dimensional vector space V over a eld K of characteristic p. Assume that p divides the order of G so that V is a modular representation and let P be a Sylow psubgroup for G. De ne the cohomological connectivity of the symmetric algebra S(V ) ..."
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Let G be a nite group acting linearly on a nite dimensional vector space V over a eld K of characteristic p. Assume that p divides the order of G so that V is a modular representation and let P be a Sylow psubgroup for G. De ne the cohomological connectivity of the symmetric algebra S(V ) to be the smallest positive integer m such that H (G; S(V )) 6= 0. We show that min ) +m+ 1; dimK (V ) is a lower bound for the depth of S(V . We characterize those representations for which the lower bound is sharp and give several examples of representations satisfying the criterion. In particular, we show that if G is pnilpotent and P is cyclic then, for any modular representation, the depth of S(V is min .
Symmetric Powers of Modular Representations for Groups with a Sylow Subgroup of Prime Order
, 2000
"... Abstract Let V be a representation of a finite group G over a field of characteristic p. If p does not divide the group order, then Molien's formula gives the Hilbert series of the invariant ring. In this paper we find a replacement of Molien's formula which works in the case that jGj is d ..."
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Abstract Let V be a representation of a finite group G over a field of characteristic p. If p does not divide the group order, then Molien's formula gives the Hilbert series of the invariant ring. In this paper we find a replacement of Molien's formula which works in the case that jGj is divisible by p but not by p2. We also obtain formulas which give generating functions encoding the decompositions of all symmetric powers of V into indecomposables. Our methods can be applied to determine the depth of the invariant ring without computing any invariants. This leads to a proof of a conjecture of the second author on certain invariants of GL2(p). Contents Introduction 1 1 Groups with a normal Sylow psubgroup of order p 3