Symmetric powers, Steenrod operations and representation stability

TL;DR

This paper investigates the structure of indecomposables under Steenrod algebra actions on symmetric powers over F_p, revealing representation stability phenomena and proposing conjectural descriptions for related quotients.

Contribution

It introduces a new approach using strict polynomial functors to analyze indecomposables and demonstrates representation stability at prime 2, with conjectures for further structures.

Findings

Representation stability observed for certain functors at prime 2

Conjectural description of quotients of Q^* from polynomial filtrations

Application of strict polynomial functors to Steenrod algebra actions

Abstract

Working over the prime field F_p, the structure of the indecomposables Q^* for the action of the algebra of Steenrod reduced powers A(p) on the symmetric power functors S^* is studied by exploiting the theory of strict polynomial functors. In particular, working at the prime 2, representation stability is exhibited for certain related functors, leading to a conjectural representation stability description of quotients of Q^* arising from the polynomial filtration of symmetric powers.