The odd-primary Kudo-Araki-May algebra of algebraic Steenrod operations and invariant theory

TL;DR

This paper explores algebraic Steenrod operations over finite fields for odd primes, revealing new algebraic structures and their connections to invariant theory, with notable differences from the p=2 case.

Contribution

It introduces bialgebras of algebraic Steenrod operations for odd primes and links their duals to polynomial invariants under certain subgroups of general linear groups.

Findings

Bialgebras of algebraic Steenrod operations are described for odd primes.

Duals of these bialgebras relate to polynomial invariants under subgroup actions.

Distinct differences from the p=2 case are analyzed.

Abstract

We describe bialgebras of lower-indexed algebraic Steenrod operations over the field with p elements, p an odd prime. These go beyond the operations that can act nontrivially in topology, and their duals are closely related to algebras of polynomial invariants under subgroups of the general linear groups that contain the unipotent upper triangular groups. There are significant differences between these algebras and the analogous one for p=2, in particular in the nature and consequences of the defining Adem relations.