Duality, rigidity, and peeling for multisegments: on hypotheses of Mitra, Offen, and Sayag

Source abstract

Mitra, Offen, and Sayag introduced distinguished multisegments and multisegments of Speh type, and proposed that every distinguished multisegment is of Speh type, together with a related duality hypothesis (see A. Mitra, O. Offen, and E. Sayag, Klyachko Models for Ladder Representations, Documenta Math. 22 (2017), 611-657). They proved these statements for sets of segments and when at most two segments share an endpoint. We develop a combinatorial theory of relevant decompositions and prove that relevance is preserved under the involution $Δ\mapstoΔ^\vee$ together with reversal of standard order. Hence, $\mathfrak m$ is distinguished if and only if $\mathfrak m^\vee$ is distinguished, so the two hypotheses are equivalent. We define $S_{\mathfrak m}(Δ)=\sum_{t\ge0}(-1)^t\mathfrak m(ν^tΔ)$ and show that $\mathfrak m$ is of Speh type if and only if $S_{\mathfrak m}(Δ)\ge0$ for every segment $Δ$. Using rigidity properties, we obtain a peeling theorem and a numerical tameness condition, and prove both hypotheses for every tame multisegment, extending the previously known classes. Finally, we construct a five-segment multisegment that is not distinguished, although every standard order with non-increasing endpoints admits a non-trivial relevant decomposition. Thus, endpoint-ordered witnesses alone cannot prove the hypothesis in general. The remaining case reduces to multisegments for which both $\mathfrak m$ and $\mathfrak m^\vee$ are non-tame.