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.
Highlights
The question of when two skew diagrams yield equal skew Schur functions has been studied in detail
We consider the related question of when two skew diagrams yield skew Schur functions such that when expanded, the same set of Schur functions appear with nonzero coefficient
We consider the question of when two skew diagrams have equal Schur support (Definition 1)
Summary
The question of when two skew diagrams yield equal skew Schur functions has been studied in detail. Being equitable is not a necessary condition for a ribbon to have full equivalence class (see Theorem 25) This definition of full equivalence class is motivated by the result of McNamara that any two skew diagrams with the same Schur support necessarily contain the same number of k × rectangles as subdiagrams, for every k, ⩾ 1 [6, Corollary 3.10]. (A skew shape A with n parts has full Schur support, by definition, when its Schur support is equal to its Schur interval – a subset of the partitions of n defined with respect to the partitions r(A) and c(A) formed by the row lengths and column lengths of A, respectively [1].) We conjecture that our necessary condition for a ribbon to have full equivalence class is sufficient, which we, along with Tran, have shown to be true for ribbons with 3 or 4 rows [3].
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