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]).

. A signicant portion of MacMahon's famous book \Combinatory Analysis " is devoted to the development of \Partition Analysis" as a computational method for solving problems in connection with linear homogeneous diophantine inequalities and equations, respectively. Nevertheless, MacMahon's ideas have not received due attention with the exception of work by Richard Stanley. A long range object of a series of articles is to change this situation by demonstrating the power of MacMahon's method in current combinatorial and partition-theoretic research. The renaissance of MacMahon's technique partly is due to the fact that it is ideally suited for being supplemented by modern computer algebra methods. In this paper we illustrate the use of Partition Analysis and of the corresponding package Omega by focusing on three dierent aspects of combinatorial work: the construction of bijections (for the Rened Lecture Hall Partition Theorem), exploitation of recursive patterns (for Cayley composit...

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This article deals with a topic on the borderline of commutative ring theory, computer algebra and combinatorics. We study canonical decompositions of commutative Noetherian rings. These techniques are based on earlier results of Rees [Ree], Stanley [St2], and Baclawski & Garsia [BGa], and they generalize the well-known Hironaka decomposition of Cohen-Macaulay rings. Here it is our main objective to give explicit algorithms for computing these decompositions.

We use the theory of resolutions for a given Hilbert function to investigate the multiplicity conjectures of Huneke and Srinivasan and Herzog and Srinivasan. To prove the conjectures for all modules with a particular Hilbert function, we show that it is enough to prove the statements only for elements at the bottom of the partially ordered set of resolutions with that Hilbert function. This enables us to test the conjectured upper bound for the multiplicity efficiently with the computer algebra system Macaulay 2 [9], and we verify the upper bound for many Artinian modules in three variables with small socle degree. Moreover, with this approach, we show that though numerical techniques have been sufficient in several of the known special cases, they are insufficient to prove the conjectures in general. Finally, we apply a result of Herzog and Srinivasan on ideals with a quasipure resolution to prove the upper bound for Cohen-Macaulay quotients by ideals with generators in high degrees relative to the regularity. 1

For a polyhedral cone C = pos{

An immediate generalization of the classical McKay correspondence for Gorenstein quotient spaces C r /G in dimensions r ≥ 4 would primarily demand the existence of projective, crepant, full desingularizations. Since this is not always possible, it is natural to ask about special classes of such quotient spaces which would satisfy the above property. In this paper we give explicit necessary and sufficient conditions under which 2-parameter series of Gorenstein cyclic quotient singularities have torus-equivariant resolutions of this specific sort in all dimensions. 1.

. The theorem of Hochster and Roberts says that, for every module V of a linearly reductive group G over a field K, the invariant ring K[V ] G is Cohen--Macaulay. We prove the following converse: if G is a reductive group and K[V ] G is Cohen--Macaulay for every module V , then G is linearly reductive. Introduction Linearly reductive groups play a prominent role in invariant theory. Hilbert's original proof of the finiteness theorem [3] works for linearly reductive groups. Much later, Hochster and Roberts [5] proved that if G is a linearly reductive group and V a G- module, then the invariant ring K[V ] G is Cohen--Macaulay, i.e., a free module over the subalgebra generated by a homogeneous system of parameters. The goal of this note is to prove the converse. Namely, we will show that if G is a reductive group and K[V ] G is Cohen--Macaulay for every G-module V , then G must be linearly reductive. This becomes false if the hypothesis that G be reductive is dropped (see Remar...

Abstract. The polynomial invariants of finite groups have been studied for more than a century now and continue to find new applications and generate interesting problems. In this article we will survey some of the recent developments coming primarily from algebraic topology and the rediscovery of old open problems. It has been almost two decades since the Bulletin of the AMS published the marvelous survey article [111] of R. P. Stanley. Since then the invariant theory of finite groups has taken on a central role in many problems of algebraic topology, such as e.g. [22], [2], [101], [65], [105], [84], [106] chapter 11, and the references there. It has received new impetus as a subject of study in its own right, [72]–[81], [3], [43], and several textbooks with varying viewpoints [9], [114], and [106], as well as a reprint of venerable old lecture notes [48], have recently appeared. In this survey article I will try to discuss some of these developments as seen through the eyes of one who came to the subject from algebraic topology. That means that finite groups and finite fields will play a central role, and the modular case, i.e. where the

Abstract. This is an improved version of the talk of the author given at the Antalya Algebra Days VII on May 21, 2005. We present an introduction to the theory of the invariants under the action of GLn(C) by simultaneous conjugation of d matrices of size n × n. Then we survey some results, old or recent, obtained by a dozen of mathematicians, on minimal sets of generators, the defining relations of the algebras of invariants and on the multiplicities of the Hilbert series of these algebras. The picture is completely understood only in the case n = 2. Besides, explicit minimal sets of generators are known for n = 3 and any d and for n = 4, d = 2. The multiplicities of the Hilbert series are obtained only for n = 3, 4 and d = 2. For n> 2 most of the concrete results are obtained with essential use of computers.