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

This detailed two-volume reference on the Steenrod algebra and its various applications presents more than thirty years of research. Developing the structure of the Steenrod algebra from an algebraic viewpoint, this first volume is recommended for researchers or postgraduates in pure mathematics and can be used as a graduate textbook.

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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.