Shard theory for $g$-fans

Abstract

For a finite dimensional algebra $A$, the notion of $g$-fan $\hat{I}\pounds (A)$ is defined from two-term silting complexes of $A$ in the real Grothendieck group $K_0(\mathsf{proj}A{)}_{\mathbb{R}}$. In this paper, we discuss the theory of shards to $\hat{I}\pounds (A)$, which was originally defined for a hyperplane arrangement. We establish a correspondence between the set of join-irreducible elements of the poset of torsion classes of $\mathrm{mod}A$ and the set of shards of $\hat{I}\pounds (A)$ for $g$-finite algebra $A$. Moreover, we show that the semistable region of a brick of $\mathrm{mod}A$ is exactly given by a shard. We also give a poset isomorphism of shard intersections and wide subcategories of $\mathrm{mod}A$.

We really appreciate the referee for pointing out numerous typos, mistakes in English, and a gap in the proof of the first version, as well as for providing suggestions for improvement. 22 pages