Existence of symmetric maximal noncrossing collections of $k$-element sets

Abstract

We investigate the existence of maximal collections of mutually noncrossing $k$-element subsets of $\left\{1,\dots ,n\right\}$ that are invariant under adding $k(\mathrm{mod}n)$ 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 $n/\mathrm{GCD}(k,n)$. Moreover, we present some algebraic consequences of our result related to self-injective Jacobian algebras.

12 pages, 1 figure. Final version, to appear in Journal of Algebraic Combinatorics