Finiteness and growth of brick chain filtrations
Abstract
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. First, 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 2d2 for modules of dimension d. For τ-tilting finite algebras, we bound their number by the multinomial coefficient determined by simple composition multiplicities. Finally, we construct families of bricks over the three-arrow Kronecker algebra whose filtration counts grow exponentially in the square of composition length. We prove this growth 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 Ringel's two questions.
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BibTeX
@misc{enomoto2026finiteness,
title = {{Finiteness and growth of brick chain filtrations}},
author = {Haruhisa Enomoto},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10217},
eprint = {2609.10217},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10217},
}
RIS
TY - UNPB
TI - Finiteness and growth of brick chain filtrations
AU - Haruhisa Enomoto
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.10217
UR - https://arxiv.org/abs/2609.10217
AB - 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. First, 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 τ-tilting finite algebras, we bound their number by the multinomial coefficient determined by simple composition multiplicities. Finally, we construct families of bricks over the three-arrow Kronecker algebra whose filtration counts grow exponentially in the square of composition length. We prove this growth 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 Ringel's two questions.
ER -