Pavón's finiteness question and Ringel's brick-chain questions

Let $A$ be a finite-dimensional algebra and let $M$ be a finite-dimensional $A$ -module. A brick-chain filtration of $M$ is a filtration whose factors are filtered by copies of bricks $B_1,\ldots,B_r$ , ordered so that $\operatorname{Hom}_A(B_i,B_j)=0$ whenever $i<j$ . The questions are: (1) whether $M$ has only finitely many brick-chain filtrations; (2) whether the number of such filtrations admits a universal bound in terms of the composition length $d=\ell_A(M)$ , in particular whether it is at most $d!$ ; and (3) for a torsion class $\mathcal{T}$ , if $t_{\mathcal{T}}(M)$ denotes the largest submodule of $M$ belonging to $\mathcal{T}$ , whether the set $\{t_{\mathcal{T}}(M):\mathcal{T}\text{ is a torsion class of }A\}$ is finite.

References

Primary source

arXiv

Additional references

Finiteness and growth of brick chain filtrations — arXiv

algebraic-combinatorics representation-theory

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A September 2026 preprint claims to settle both questions, but its conclusions have not yet been independently verified.

The problem asks whether brick-chain filtrations satisfy the finiteness properties posed by Pavón and Ringel, and how rapidly their possible lengths or types can grow.

Known results

Earlier work established that every module has a brick-chain filtration, while a related finiteness question was explicitly reported as open; special classes were known to have bounded complexity.

September 9, 2026 preprint

The preprint Finiteness and growth of brick chain filtrations claims finiteness, a $2^{d^2}$ bound, sharper bounds for $\tau$ -tilting-finite algebras, and examples with superfactorial growth, thereby claiming answers to Pavón’s and Ringel’s questions. It is unrefereed and remains unverified.

Current status (as of September 2026): An unrefereed preprint claims a complete resolution with quantitative bounds and growth examples, but independent verification is not recorded.

Sources

No solutions have been posted yet.