Linking first occurrence polynomials over 𝔽p by Steenrod operations

This paper provides analogues of the results of [G.Walker and R.M.W. Wood, Linking first occurrence polynomials over $F_{2}$ 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 $M_{n} (F_{p})$ , the first occurrence of $L (λ)$ as a composition factor in the polynomial algebra $P = F_{p} [x_{1}, \dots, x_{n}]$ 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 $A_{p}$ 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 $A_{p}$ .

Keywords

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

Mathematical Subject Classification 2000

Primary: 55S10

Secondary: 20C20

References

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Publication

Received: 24 January 2002

Accepted: 10 July 2002

Published: 20 July 2002

Authors

Phạm Anh Minh
Department of Mathematics College of Sciences University of Hue Dai hoc Khoa hoc Hue Vietnam

Grant Walker
Department of Mathematic University of Manchester Oxford Road Manchester M13 9PL UK

Volume 2, issue 1 (2002)

Phạm Anh Minh and Grant Walker