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Linking first occurrence polynomials over $\mathbb{F}_p$ by Steenrod operations

Phạm Anh Minh and Grant Walker

Algebraic & Geometric Topology 2 (2002) 563–590

arXiv: math.AT/0207213


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.

Steenrod algebra, anti-automorphism, $p$–truncated polynomial algebra $\mathbf{T}$, $\mathbf{T}$–regular partition/representation
Mathematical Subject Classification 2000
Primary: 55S10
Secondary: 20C20
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Received: 24 January 2002
Accepted: 10 July 2002
Published: 20 July 2002
Phạm Anh Minh
Department of Mathematics
College of Sciences
University of Hue
Dai hoc Khoa hoc
Grant Walker
Department of Mathematic
University of Manchester
Oxford Road
Manchester M13 9PL