Maximal finite semibricks consist only of open bricks

TL;DR

This paper proves that all modules in a maximal finite semibrick over a finite dimensional algebra are open bricks, meaning their orbit closures form irreducible components in the representation schemes.

Contribution

It establishes that maximal finite semibricks are composed solely of open bricks, providing a structural characterization in representation theory.

Findings

Maximal finite semibricks consist only of open bricks.

Open bricks have irreducible orbit closures in representation schemes.

The result applies to any finite dimensional algebra over an algebraically closed field.

Abstract

A semibrick is a set of modules satisfying Schur's Lemma, and it is said to be maximal if it is not properly contained in another semibrick. For any finite dimensional algebra $\varLambda$ over an algebracally closed field $K$, we prove that any maximal finite semibrick $\mathcal{S}$ consists only of open bricks $B$, that is, bricks whose orbit closures $\overline{\mathcal{O}_B}$ are irreducible components in the representation schemes.