Simple-mindedness: negativity and positivity

Given an acyclic quiver Q and an integer \(w\), the orbit category \[ C_w(Q) = \mathcal{D}^b(\mathbf{k} Q)/\Sigma^{-w} \mathbb{S} \] is a \(w\)-Calabi-Yau triangulated category, where \(\Sigma\) is the shift functor on \(\mathcal{D}^b(\mathbf{k} Q)\) and \(\mathbb{S}\) is the Serre functor. When \(w \geq 2\), these orbit categories are called (higher) cluster categories and the key generators are (higher) cluster-tilting objects; they are in bijection with silting objects in the fundamental domain. Amongst other connections, the combinatorics of (higher) cluster-tilting objects serve as a categorical model of those of (higher) noncrossing partitions.

Simple-minded systems were introduced by Koenig-Liu as an abstraction of nonprojective simple modules in stable module categories. In this talk, we will see that simple-minded systems in \(C_w(Q)\) for \(w \leq -1\) are in bijection with simple-minded collections in the fundamental domain. We will show further that they are in bijection with positive (higher) noncrossing partitions in the Weyl group of Q. This provides further evidence supporting the view that \(C_w(Q)\) for \(w \leq -1\) are negative cluster categories in which simple-minded systems play the role of cluster-tilting objects, providing categorical models of combinatorial objects, such as positive (higher) noncrossing partitions.

This talk is based on joint work with Raquel Coelho Simoes and David Ploog.