Reverse mathematics, Young diagrams, and the ascending chain condition

TL;DR

This paper explores the reverse mathematical strength of a theorem about the ascending chain condition in group algebras, linking it to the well ordering of ordinal $\, ext{ extomega}^ ext{ extomega}$ and properties of Young diagrams.

Contribution

It establishes the equivalence of the ascending chain condition theorem with the well ordering of $\, extomega^ extomega$ in reverse mathematics, connecting algebraic and combinatorial principles.

Findings

The ascending chain condition for $K[S]$ is equivalent to $\, extomega^ extomega$ being well ordered.

Young diagrams form a well partial ordering, crucial for the equivalence proof.

The result links algebraic properties to foundational combinatorial and ordinal principles.

Abstract

Let $S$ be the group of finitely supported permutations of a countably infinite set. Let $K[S]$ be the group algebra of $S$ over a field $K$ of characteristic $0$. According to a theorem of Formanek and Lawrence, $K[S]$ satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over RCA$_0$ (or even over RCA$_0^*$) to the statement that $\omega^\omega$ is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.