#### Volume 2, issue 1 (2002)

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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
##### Abstract

This paper provides analogues of the results of [G.Walker and R.M.W. Wood, Linking first occurrence polynomials over ${\mathbb{F}}_{2}$ by Steenrod operations, J. Algebra 246 (2001), 739–760] for odd primes $p$. It is proved that for certain irreducible representations $L\left(\lambda \right)$ of the full matrix semigroup ${M}_{n}\left({\mathbb{F}}_{p}\right)$, the first occurrence of $L\left(\lambda \right)$ as a composition factor in the polynomial algebra $P={\mathbb{F}}_{p}\left[{x}_{1},\dots ,{x}_{n}\right]$ is linked by a Steenrod operation to the first occurrence of $L\left(\lambda \right)$ as a submodule in $P$. This operation is given explicitly as the image of an admissible monomial in the Steenrod algebra ${\mathsc{A}}_{p}$ under the canonical anti-automorphism $\chi$. The first occurrences of both kinds are also linked to higher degree occurrences of $L\left(\lambda \right)$ by elements of the Milnor basis of ${\mathsc{A}}_{p}$.

##### Keywords
Steenrod algebra, anti-automorphism, $p$–truncated polynomial algebra $\mathbf{T}$, $\mathbf{T}$–regular partition/representation
Primary: 55S10
Secondary: 20C20