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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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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]).
MacMahon's Partition Analysis V: Bijections, Recursions, and Magic Squares
"... . 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, MacMah ..."
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. 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 partitiontheoretic 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...
Lectures on Deformation Theory
, 2004
"... My goal in these notes is to give an introduction to deformation theory by doing some basic constructions in careful detail in their simplest cases, by explaining why people do things the way they do, with examples, and then giving some typical interesting applications. The early sections of these n ..."
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My goal in these notes is to give an introduction to deformation theory by doing some basic constructions in careful detail in their simplest cases, by explaining why people do things the way they do, with examples, and then giving some typical interesting applications. The early sections of these notes are based on a course I gave in the Fall of 1979. Warning: The present state of these notes is rough. The notation and numbering systems are not consistent (though I hope they are consistent within each separate section). The crossreferences and references to the literature are largely missing. Assumptions may vary from one section to another. The safest way to read these notes would be as a loosely connected series of short essays on deformation theory. The order of the sections is somewhat arbitrary, because the material does not naturally fall into any linear order. I will appreciate comments, suggestions, with particular reference to where I may have fallen into error, or where the text is confusing or misleading.
COMPUTING WITH MATRIX INVARIANTS
"... 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 re ..."
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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.
Origins and breadth of the theory of higher homotopies
, 2007
"... Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. The purpose of the talk is to recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least ..."
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Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. The purpose of the talk is to recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least as far back as the 1940’s. Prompted by the failure of the AlexanderWhitney multiplication of cocycles to be commutative, Steenrod developed certain operations which measure this failure in a coherent manner. Dold and Lashof extended Milnor’s classifying space construction to associative Hspaces, and a careful examination of this extension led Stasheff to the discovery of Anspaces and A∞spaces as notions which control the failure of associativity in a coherent way so that the classifying space construction can still be pushed through. Algebraic versions of higher homotopies have, as we all know, led Kontsevich eventually to the proof of the formality conjecture. Homological perturbation theory (HPT), in a simple form first isolated by Eilenberg and Mac Lane in the early 1950’s, has nowadays become a standard tool to handle algebraic incarnations of higher homotopies. A basic
POLYNOMIAL BOUNDS FOR RINGS OF INVARIANTS
, 2000
"... Hilbert proved that invariant rings are finitely generated for linearly reductive groups acting rationally on a finite dimensional vector space. Popov gave an explicit upper bound for the smallest integer d such that the invariants of degree ≤ d generate the invariant ring. This bound has factorial ..."
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Hilbert proved that invariant rings are finitely generated for linearly reductive groups acting rationally on a finite dimensional vector space. Popov gave an explicit upper bound for the smallest integer d such that the invariants of degree ≤ d generate the invariant ring. This bound has factorial growth. In this paper we will give a bound which depends only polynomially on the input data.
Polynomial Invariants of Finite Groups: A Survey of Recent Developments
 Bull. Amer. Math. Soc
, 1997
"... 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 o ..."
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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