Jordan--Hölder criterion for bricks to determine maximal green sequences

Let

$$be an Artin algebra, and suppose that every functorially finite torsion class in$$

satisfies the Jordan--Hölder property (JHP). For maximal green sequences

$$and$$

of

$$, write that they have the same set of bricks when their associated brick sets coincide, and write$$

when they are equivalent.

Jordan--Hölder criterion. The maximal green sequences

$$and$$

have the same set of bricks if and only if they are equivalent.

The claim is motivated by Nakayama algebras, whose functorially finite torsion classes satisfy the JHP. The paper gives examples showing that the converse implication in the proposed global characterization cannot hold without the stated direction: preprojective algebras of type $D_4$ have torsion classes failing the JHP even when the brick criterion can hold.