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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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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
The cohomology of the Steenrod algebra and representations of the general linear groups
 Trans. Amer. Math. Soc
"... ABSTRACT. Let Trk be the algebraic transfer that maps from the coinvariants of certain GLkrepresentation to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer trk: n~((BVk)+)7 11'~(8°). It has been shown that the ..."
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ABSTRACT. Let Trk be the algebraic transfer that maps from the coinvariants of certain GLkrepresentation to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer trk: n~((BVk)+)7 11'~(8°). It has been shown that the algebraic transfer is highly nontrivial, more precisely, that Trk is an isomorphism for k = 1, 2, 3 and that Tr = ffikTrk is a homomorphism of algebras. In this paper, we first recognize the phenomenon that if we start from any degree d, and apply Sq0 repeatedly at most (k 2) times, then we get into the region, in which all the iterated squaring operations are isomorphisms on the coinvariants of the GLkrepresentation. As a consequence, every finite Sq0family in the coinvariants has at most (k 2) non zero elements. Two applications are exploited. The first main theorem is that Trk is not an isomorphism for k 2: 5. Furthermore, Trk is not an isomorphism in infinitely many degrees for each k> 5. We also show that if Tre detects a nonzero element in certain de
SubHopf algebras of the Steenrod algebra and the Singer transfer
, 2007
"... The Singer transfer provides an interesting connection between modular representation theory and the cohomology of the Steenrod algebra. We discuss a version of “Quillen stratification” theorem for the Singer transfer and its consequences. ..."
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The Singer transfer provides an interesting connection between modular representation theory and the cohomology of the Steenrod algebra. We discuss a version of “Quillen stratification” theorem for the Singer transfer and its consequences.