Fans and polytopes in tilting theory II: $g$-fans of rank 2

TL;DR

This paper classifies complete $g$-fans of rank 2 in tilting theory, showing they correspond to fans of finite dimensional algebras, and explores their structural properties and algebraic realizations.

Contribution

It provides a complete classification of rank 2 $g$-fans and characterizes the associated algebras, introducing fundamental theorems that connect fan operations with algebraic structures.

Findings

Every complete sign-coherent fan of rank 2 is a $g$-fan of some finite dimensional algebra.

Theorems (Gluing, Rotation, Subdivision) connect fan operations with algebraic constructions.

Existence of algebras with a specified number of connected components in silting complexes.

Abstract

The $g$-fan of a finite dimensional algebra is a fan in its real Grothendieck group defined by tilting theory. We give a classification of complete $g$-fans of rank 2. More explicitly, our first main result asserts that every complete sign-coherent fan of rank 2 is a $g$-fan of some finite dimensional algebra. Our proof is based on three fundamental results, Gluing Theorem, Rotation Theorem and Subdivision Theorem, which realize basic operations on fans in the level of finite dimensional algebras. For each of 16 convex sign-coherent fans $\Sigma$ of rank 2, our second main result gives a characterization of algebras $A$ of rank 2 satisfying $\Sigma(A)=\Sigma$. As a by-product of our method, we prove that for each positive integer $N$, there exists a finite dimensional algebra $A$ of rank 2 such that the Hasse quiver of the poset of 2-term silting complexes of $A$ has precisely $N$ connected components.