Finiteness and growth of brick chain filtrations

Comment

We study Ringel’s brick chain filtrations over finite-dimensional algebras: filtrations whose factors are filtered by copies of individual bricks, ordered so that morphisms from earlier bricks to later ones vanish.

• Using submodule varieties, we prove that the largest submodule of a fixed module belonging to a torsion class takes only finitely many values as the class varies, answering Pavón’s question. We deduce finiteness of brick chain filtrations and the bound $2^{d^2}$ for modules of dimension $d$.
• For $\tau$-tilting finite algebras, we bound their number by the multinomial coefficient determined by simple composition multiplicities.
• We construct families of bricks over the three-arrow Kronecker algebra whose filtration counts grow exponentially in the square of composition length, using a Littlewood–Richardson formula for submodule counts and the hook-length formula. In particular, the counts eventually exceed the factorial of composition length.

These results answer two questions of Ringel.