1. $\tau$-tilting finiteness and $\mathbf{g}$-tameness: Incidence algebras of posets and concealed algebras
- Author
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Børve, Erlend D., Grevstad, Jacob Fjeld, and Rundsveen, Endre S.
- Subjects
Mathematics - Representation Theory - Abstract
We prove that any $\tau$-tilting finite incidence algebra of a finite poset is representation-finite, and that any $\mathbf{g}$-tame incidence algebra of a finite simply connected poset is tame. As the converse of these assertions are known to hold, we obtain characterizations of $\tau$-tilting finite incidence algebras and $\mathbf{g}$-tame simply connected incidence algebras. Both results are proved using the theory of concealed algebras. The former will be deduced from the fact that tame concealed algebras are $\tau$-tilting infinite, and to prove the latter, we show that wild concealed algebras are not $\mathbf{g}$-tame. We conjecture that any incidence algebra of a finite poset is wild if and only if it is not $\mathbf{g}$-tame, and prove a result showing that there are relatively few possible counterexamples. In the appendix, we determine the representation type of a $\tau$-tilting reduction of a concealed algebra of hyperbolic type., Comment: 31 pages. v2: Minor changes to structure and grammatical corrections
- Published
- 2024