Symmetric powers, Steenrod operations and representation stability (Q2001437)

Symmetric powers \(S^n(V)\) of a finite-dimensional \(k\)-vector space \(V\) give a rich and far from understood source of representations working over a finite field \(k\). The power \(S^n(V)\) is a representation of the general linear group \(\mathrm{GL}(V)\) and the multiplicities of its composition factors are not known in general. The mod \(p\) algebra \(\mathcal{A}(p)\) of Steenrod reduced powers, acts upon the symmetric algebra \(S^\ast(V)\) naturally with respect to \(V\) and one can consider the representations given by the indecomposables \(Q^\ast(V)= \mathbb{F}_p\otimes_{\mathcal{A}(p)} S^\ast(V)\). The author, working over the prime field \(\mathbb{F}_p\), exploits the theory of strict polynomial functors to study the structure of the indecomposables \(Q^\ast(V)\) for the action of \(\mathcal{A}(p)\) on the symmetric power functors \(S^\ast(V)\). Then, the last section puts everything together at the prime \(2\), stating the main results and two conjectures which provide a new approach, using methods developed by \textit{G. Walker} and \textit{R. M. W. Wood} [Polynomials and the mod 2 Steenrod algebra. Volume 1: The Peterson hit problem. Cambridge: Cambridge University Press (2018; Zbl 1387.55001)], to understanding the structure of the indecomposables \(Q^\ast(V)\) and hence of the symmetric power functors \(S^\ast(V)\) as well.