Polynomials and the mod 2 Steenrod Algebra: Volume 1, The Peterson Hit Problem

The first of two volumes covering the Steenrod algebra and its various applications. Suitable as a graduate text.

Grant Walker (Author), Reginald M. W. Wood (Author)

9781108414487, Cambridge University Press

Paperback / softback, published 9 November 2017

370 pages
22.7 x 15.1 x 2.2 cm, 0.56 kg

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.

Preface
1. Steenrod squares and the hit problem
2. Conjugate Steenrod squares
3. The Steenrod algebra A2
4. Products and conjugation in A2
5. Combinatorial structures
6. The cohit module Q(n)
7. Bounds for dim Qd(n)
8. Special blocks and a basis for Q(3)
9. The dual of the hit problem
10. K(3) and Q(3) as F2GL(3)-modules
11. The dual of the Steenrod algebra
12. Further structure of A2
13. Stripping and nilpotence in A2
14. The 2-dominance theorem
15. Invariants and the hit problem
Bibliography
Index of Notation for Volume 1
Index for Volume 1
Index of Notation for Volume 2
Index for Volume 2.

Subject Areas: Combinatorics & graph theory [PBV], Algebraic topology [PBPD], Algebra [PBF], Discrete mathematics [PBD]