Poincaré Duality Algebras, Macaulay's Dual Systems, and Steenrod Operations, Volume 13
Poincaré duality algebras originated in the work of topologists on the cohomology of closed manifolds, and Macaulay's dual systems in the study of irreducible ideals in polynomial algebras. Steenrod operations also originated in algebraic topology and they provide a noncommutative tool to study commutative algebras over a Galois field. The authors skilfully bring together these ideas and apply them to problems in invariant theory. A number of remarkable and unexpected interdisciplinary connections are revealed that will interest researchers in the areas of commutative algebra, invariant theory or algebraic topology.
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A-primary irreducible ideal algebra H an-i ancestor ideal basis c/*-invariant catalecticant matrices coinvariants commutative graded connected compute conjugate Wu classes connected sum Corollary corresponding Poincare duality defined degree denoted Dickson polynomials dual principal system duality quotient algebra elementary symmetric polynomials example F. S. Macaulay F[Xi F[zi field F finite extension formal dimension formula Fq[V Fq[Xi Fq[zi Frobenius powers fundamental class Galois field Galois field Fq GL(n ground field Hence Hit Problem homogeneous component Hopf algebra indecomposable integer invariant ideal inverse polynomial inverse system Lemma linear form Macaulay dual matrix module monomials nonzero notation orbits parameter ideal Poincare duality algebra Poincare duality quotient polynomial algebra proof Proposition quotients of F[V regular ideal regular sequence represents a fundamental ring of invariants Section Steenrod algebra Steenrod operations subspace Suppose Theorem II.6.6 Theorem III.3.5 Thom class transition element trivial Wu classes unstable algebra variables Wu(H zero