τ-tilting finiteness and g-tameness: Incidence algebras of posets and concealed algebras

We prove that any τ-tilting finite incidence algebra of a finite poset is representation-finite, and that any 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 τ-tilting finite incidence algebras and 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 τ-tilting infinite, and to prove the latter, we show that wild concealed algebras are not g-tame. We conjecture that any incidence algebra of a finite poset is wild if and only if it is not g-tame, and prove a result showing that there are relatively few possible counterexamples. In the appendix, we determine the representation type of a τ-tilting reduction of a concealed algebra of hyperbolic type.