Linking first occurrence polynomials over 𝔽p by Steenrod operations

Minh, Phạm Anh¹ ; Walker, Grant²

¹Department of Mathematics, College of Sciences, University of Hue, Dai hoc Khoa hoc, Hue, Vietnam
²Department of Mathematic, University of Manchester, Oxford Road, Manchester M13 9PL, UK

Algebraic and Geometric Topology, Tome 2 (2002) no. 1, pp. 563-590

Résumé

This paper provides analogues of the results of [G.Walker and R.M.W. Wood, Linking first occurrence polynomials over F2 by Steenrod operations, J. Algebra 246 (2001), 739–760] for odd primes p. It is proved that for certain irreducible representations L(λ) of the full matrix semigroup Mn(Fp), the first occurrence of L(λ) as a composition factor in the polynomial algebra P = Fp[x1,…,xn] is linked by a Steenrod operation to the first occurrence of L(λ) as a submodule in P. This operation is given explicitly as the image of an admissible monomial in the Steenrod algebra Ap under the canonical anti-automorphism χ. The first occurrences of both kinds are also linked to higher degree occurrences of L(λ) by elements of the Milnor basis of Ap.

Détail

DOI : 10.2140/agt.2002.2.563

Keywords: Steenrod algebra, anti-automorphism, $p$–truncated polynomial algebra $\mathbf{T}$, $\mathbf{T}$–regular partition/representation

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