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

Andrea Pasquali; Erik Thörnblad; Jakob Zimmermann

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

Combinatorics 2019-05-28 v2 Representation Theory

Authors: Andrea Pasquali , Erik Thörnblad , Jakob Zimmermann

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.

Keywords

noncrossing sets symmetric collections existence theorem jacobian algebras modular arithmetic

Cite

@article{arxiv.1808.03556,
  title  = {Existence of symmetric maximal noncrossing collections of $k$-element sets},
  author = {Andrea Pasquali and Erik Thörnblad and Jakob Zimmermann},
  journal= {arXiv preprint arXiv:1808.03556},
  year   = {2019}
}

Comments

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

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

Abstract

Keywords

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