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

TL;DR

This paper characterizes the conditions under which symmetric maximal noncrossing collections of k-element sets exist, linking combinatorial structures with algebraic consequences, specifically in the context of self-injective Jacobian algebras.

Contribution

It provides a complete characterization of the existence of symmetric maximal noncrossing collections based on modular arithmetic conditions.

Findings

Existence of collections is characterized by k ≡ 0, 1, or -1 mod n/GCD(k,n).

The results have algebraic implications for self-injective Jacobian algebras.

The paper connects combinatorial and algebraic structures through these conditions.

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 $k\pmod n$ 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.