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

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 ${\mathcal{A}}_p$ 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 ${\mathcal{A}}_p$.