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Problems in the Steenrod algebra
 Bull. London Math. Soc
, 1998
"... This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development ..."
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Cited by 30 (1 self)
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This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below. Contents 1 Historical background 4
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
Differential Operators and the Steenrod Algebra
, 1995
"... This article presents an elementary treatment of the Steenrod algebra from an algebraic point of view in terms of differential operators acting on polynomials. The exposition concentrates on the Steenrod algebra A over the field F 2 of two elements although the approach works for the odd prime field ..."
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Cited by 9 (3 self)
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This article presents an elementary treatment of the Steenrod algebra from an algebraic point of view in terms of differential operators acting on polynomials. The exposition concentrates on the Steenrod algebra A over the field F 2 of two elements although the approach works for the odd prime fields. From a
Invariant Fields of Finite Irreducible Reflection Groups
 Math. Ann
, 1997
"... We prove the following result: If G is a finite irreducible reflection group defined over a base field k, then the invariant field of G is purely transcendental over k, even if jGj is divisible by the characteristic of k. It is well known that in the above situation the invariant ring is in general ..."
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We prove the following result: If G is a finite irreducible reflection group defined over a base field k, then the invariant field of G is purely transcendental over k, even if jGj is divisible by the characteristic of k. It is well known that in the above situation the invariant ring is in general not a polynomial ring. So the question whether at least the invariant field is purely transcendental (Noether's problem) is quite natural. Introduction A linear group G defined over a field k is called a reflection group if it is generated by elements oe with rank(oe \Gamma 1) = 1. If the order of a finite reflection group is not divisible by the characteristic of k, then the invariant ring is isomorphic to a polynomial ring (see, for example, Benson [1, Theorem 7.2.1]). This is no longer true in general if jGj is divisible by char(k). In fact, the authors classified the finite irreducible reflection groups whose invariant rings are polynomial rings in [7]. In view of this, it is natural ...
Sheared algebra maps and operation bialgebras for mod 2 homology and cohomology
 Trans. Amer. Math. Soc
"... Abstract. The mod 2 Steenrod algebra A and DyerLashof algebra R have both striking similarities and dierences, arising from their common origins in \lowerindexed " algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra K, whose module actions are e ..."
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Abstract. The mod 2 Steenrod algebra A and DyerLashof algebra R have both striking similarities and dierences, arising from their common origins in \lowerindexed " algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra K, whose module actions are equivalent to, but quite different from, those of A and R. The exact relationships emerge as \sheared algebra bijections", which also illuminate the role of the cohomology of K. As a bialgebra, K has a particularly attractive and potentially useful structure, providing a bridge between those of A and R, and suggesting possible applications to the Miller spectral sequence and the A structure of Dickson algebras. 1.
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"... This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may nd of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of ..."
Abstract
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This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may nd of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below.
THE ODDPRIMARY KUDOARAKIMAY ALGEBRA OF ALGEBRAIC STEENROD OPERATIONS, AND INVARIANT THEORY
"... Abstract. We describe bialgebras of lowerindexed algebraic Steenrod operations over the …eld with p elements, p an odd prime. These go beyond the operations that can act nontrivially in topology, and their duals are closely related to algebras of polynomial invariants under subgroups of the general ..."
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Abstract. We describe bialgebras of lowerindexed algebraic Steenrod operations over the …eld with p elements, p an odd prime. These go beyond the operations that can act nontrivially in topology, and their duals are closely related to algebras of polynomial invariants under subgroups of the general linear groups that contain the unipotent upper triangular groups. There are signi…cant di¤erences between these algebras and the analogous one for p = 2, in particular in the nature and consequences of the de…ning Adem relations.
A Classification of Polynomial Algebras as Modules over the Steenrod Algebra
"... this paper we give an affirmative answer to the above question and classify the actions defined in this way. More precisely, let S(V ) be the symmetric ..."
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this paper we give an affirmative answer to the above question and classify the actions defined in this way. More precisely, let S(V ) be the symmetric
Extended powers and Steenrod operations in algebraic geometry
, 2007
"... Steenrod operations have been defined by Voedvodsky in motivic cohomology in order to show the Milnor and BlochKato conjectures. These operations have also been constructed by Brosnan for Chow rings. The purpose of this paper is to provide a setting for the construction of the Steenrod operations i ..."
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Steenrod operations have been defined by Voedvodsky in motivic cohomology in order to show the Milnor and BlochKato conjectures. These operations have also been constructed by Brosnan for Chow rings. The purpose of this paper is to provide a setting for the construction of the Steenrod operations in algebraic geometry for generalized cohomology theories whose formal group law has order two. We adapt the methods used by BissonJoyal in studying Steenrod and DyerLashof operations in unoriented cobordism and mod 2 cohomology.