Support Equalities Among Ribbon Schur Functions

TL;DR

This paper investigates conditions under which ribbons have identical Schur support after permuting row lengths, advancing the classification of support equalities among ribbon Schur functions.

Contribution

It provides a sufficient condition for ribbons to have full equivalence class and proposes a necessary condition, advancing understanding of support equalities.

Findings

A sufficient condition for a ribbon to have full equivalence class.

A necessary condition for support equality, conjectured to be sufficient.

Progress towards classifying when permuted ribbons share the same Schur support.

Abstract

In 2007, McNamara proved that two skew shapes can have the same Schur support only if they have the same number of $k\times \ell$ rectangles as subdiagrams. This implies that two ribbons can have the same Schur support only if one is obtained by permuting row lengths of the other. We present substantial progress towards classifying when a permutation $\pi \in S_m$ of row lengths of a ribbon $\alpha$ produces a ribbon $\alpha_{\pi}$ with the same Schur support as $\alpha$; when this occurs for all $\pi \in S_m$, we say that $\alpha$ has "full equivalence class." Our main results include a sufficient condition for a ribbon $\alpha$ to have full equivalence class. Additionally, we prove a separate necessary condition, which we conjecture to be sufficient.