Representation Type and Schurian Finiteness of Tits Algebras
Abstract
We determine the representation type and Schurian finiteness of the Tits algebra of a finite central real hyperplane arrangement over an algebraically closed field. The algebra is representation-finite when the arrangement has at most two hyperplanes, tame when it has three, and wild when it has at least four. Moreover, it is Schurian-finite if and only if the arrangement has at most two hyperplanes. Both classifications hold over algebraically closed fields of arbitrary characteristic. The proof combines Saliola's bound-quiver presentation with explicit module constructions, convex subquivers, essentialization, and deletion. In the ℝ2 case, we describe all eleven indecomposable modules for two lines and construct a full exact embedding of modules over a free algebra for at least four lines.
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BibTeX
@misc{chen2026representation,
title = {{Representation Type and Schurian Finiteness of Tits Algebras}},
author = {Siyuan Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16214},
eprint = {2610.16214},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16214},
}
RIS
TY - UNPB
TI - Representation Type and Schurian Finiteness of Tits Algebras
AU - Siyuan Chen
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16214
UR - https://arxiv.org/abs/2610.16214
AB - We determine the representation type and Schurian finiteness of the Tits algebra of a finite central real hyperplane arrangement over an algebraically closed field. The algebra is representation-finite when the arrangement has at most two hyperplanes, tame when it has three, and wild when it has at least four. Moreover, it is Schurian-finite if and only if the arrangement has at most two hyperplanes. Both classifications hold over algebraically closed fields of arbitrary characteristic. The proof combines Saliola's bound-quiver presentation with explicit module constructions, convex subquivers, essentialization, and deletion. In the ℝ^2 case, we describe all eleven indecomposable modules for two lines and construct a full exact embedding of modules over a free algebra for at least four lines.
ER -