Modulated semi-invariants

TL;DR

This paper establishes fundamental properties of determinantal semi-invariants in the context of hereditary algebras, providing key theorems that connect these invariants to cluster tilting theory.

Contribution

It introduces and proves basic properties of determinantal semi-invariants, including the virtual generic decomposition, stability, and c-vector theorems, linking invariants to cluster tilting objects.

Findings

Proves the virtual generic decomposition theorem.

Establishes the stability theorem for determinantal semi-invariants.

Shows c-vectors correspond to determinantal weights of semi-invariants.

Abstract

We prove the basic properties of determinantal semi-invariants for presentation spaces over any finite dimensional hereditary algebra over any field. These include the virtual generic decomposition theorem, stability theorem and the c-vector theorem which says that the c-vectors of a cluster tilting object are, up to sign, the determinantal weights of the determinantal semi-invariants defined on the cluster tilting objects. Applications of these theorems are given in several concurrently written papers.