Tran Nhat Tan (Ruhr-Universität Bochum)

Title: Arrangements arising from digraphs and freeness of arrangements

between Shi and Ish

Abstract: To a given vertex-weighted digraph (directed graph) we associate an arrangement analogous to the notion of Stanley's $\psi$-graphical arrangements and study it from perspectives of combinatorics and freeness. Our arrangement unifies several arrangements in literature including the Catalan arrangement, the Shi arrangement, the Ish arrangement, and especially the arrangements interpolating between Shi and Ish recently introduced by Duarte and Guedes de Oliveira.

It was shown that the arrangements between Shi and Ish all share the same characteristic polynomial with all nonnegative integer roots, thus raising the natural question of their freeness. We introduce two operations on the vertex-weighted digraphs and prove that subject to certain conditions on the weight $\psi$, the operations preserve the characteristic polynomials and freeness of the associated arrangements. In particular, by applying a sequence of these operations to the Shi arrangement, we affirmatively prove that the arrangements between Shi and Ish all are free, and among them only the Ish arrangement has supersolvable cone. Notably, all of the arrangements between Shi and Ish appear as the members in the operation sequence, thus giving a new insight into how they naturally arise and interpolate between Shi and Ish.

This is joint work (arXiv:2108.02518) with T. Abe (Kyushu) and S. Tsujie (Hokkaido).