Existence of symmetric maximal noncrossing collections of \(k\)-element sets

We investigate the existence of maximal collections of mutually noncrossing \(k\)-element subsets of \(\left\{ 1, \dots, n \right\}\) that are invariant under adding \(k\pmod n\) to all indices. Our main result is that such a collection exists if and only if \(k\) is congruent to \(0, 1\) or \(-1\)...

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Bibliographic Details
Published in:arXiv.org 2019-05
Main Authors: Pasquali, Andrea, Thörnblad, Erik, Zimmermann, Jakob
Format: Article
Language:English
Online Access:Get full text
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Summary:We investigate the existence of maximal collections of mutually noncrossing \(k\)-element subsets of \(\left\{ 1, \dots, n \right\}\) that are invariant under adding \(k\pmod n\) to all indices. Our main result is that such a collection exists if and only if \(k\) is congruent to \(0, 1\) or \(-1\) modulo \(n/\operatorname{GCD}(k,n)\). Moreover, we present some algebraic consequences of our result related to self-injective Jacobian algebras.
ISSN:2331-8422