Shard theory for $g$-fans

TL;DR

This paper develops a shard theory for $g$-fans of finite-dimensional algebras, linking torsion classes, semistable regions, and wide subcategories through combinatorial and geometric structures.

Contribution

It introduces a shard framework for $g$-fans, establishing correspondences with torsion classes, semistable regions, and wide subcategories in the context of finite-dimensional algebras.

Findings

Established a correspondence between join-irreducible torsion elements and shards.

Showed that semistable regions of bricks are exactly shards.

Provided a poset isomorphism between shard intersections and wide subcategories.

Abstract

For a finite dimensional algebra $A$, the notion of $g$-fan $\Sigma(A)$ 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 $\Sigma(A)$, 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 $\Sigma(A)$ 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 $\mathrm{mod} A$.