Polynomials and the mod 2 Steenrod Algebra

Grant Walker; Reginald M. W. Wood

doi:10.1017/9781108333368

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Volume 1: The Peterson Hit Problem
Grant Walker, University of Manchester, Reginald M. W. Wood, University of Manchester

Publisher:: Cambridge University Press
Online publication date:: November 2017
Print publication year:: 2017
Online ISBN:: 9781108333368
DOI:: https://doi.org/10.1017/9781108333368

Subjects:: Mathematics (general), Mathematics, Discrete Mathematics Information Theory and Coding, Geometry and Topology
Series:: London Mathematical Society Lecture Note Series (441)

80.99 (GBP)

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This is the first book to link the mod 2 Steenrod algebra, a classical object of study in algebraic topology, with modular representations of matrix groups over the field F of two elements. The link is provided through a detailed study of Peterson's 'hit problem' concerning the action of the Steenrod algebra on polynomials, which remains unsolved except in special cases. The topics range from decompositions of integers as sums of 'powers of 2 minus 1', to Hopf algebras and the Steinberg representation of GL(n,F). Volume 1 develops the structure of the Steenrod algebra from an algebraic viewpoint and can be used as a graduate-level textbook. Volume 2 broadens the discussion to include modular representations of matrix groups.

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Polynomials and the mod 2 Steenrod Algebra

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Frontmatter
pp i-iv

Contents of Volume 1
pp v-x

Contents of Volume 2
pp xi-xiv

Preface
pp xv-xxiv

1 - Steenrod squares and the hit problem
pp 1-18

2 - Conjugate Steenrod squares
pp 19-32

3 - The Steenrod algebra A2
pp 33-53

4 - Products and conjugation in A2
pp 54-66

5 - Combinatorial structures
pp 67-95

6 - The cohit module Q(n)
pp 96-122

7 - Bounds for dim Qd(n)
pp 123-138

8 - Special blocks and a basis for Q(3)
pp 139-163

9 - The dual of the hit problem
pp 164-188

10 - K(3) and Q(3) as F2GL(3)-modules
pp 189-210

11 - The dual of the Steenrod algebra
pp 211-233

12 - Further structure of A2
pp 234-253

13 - Stripping and nilpotence in A2
pp 254-278

14 - The 2-dominance theorem
pp 279-296

15 - Invariants and the hit problem
pp 297-324

Bibliography
pp 325-336

Index of Notation for Volume 1
pp 337-338

Index for Volume 1
pp 339-342

Index of Notation for Volume 2
pp 343-344

Index for Volume 2
pp 345-346

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Polynomials and the mod 2 Steenrod Algebra

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