$\tau$-tilting finiteness and $\mathbf{g}$-tameness: Incidence algebras of posets and concealed algebras

TL;DR

This paper characterizes when incidence algebras of finite posets are representation-finite or tame based on $ au$-tilting finiteness and $ extbf{g}$-tameness, using concealed algebra theory.

Contribution

It establishes new criteria linking $ au$-tilting finiteness and $ extbf{g}$-tameness to representation type for incidence algebras, with proofs involving concealed algebras.

Findings

$ au$-tilting finite incidence algebras are representation-finite

$ extbf{g}$-tame incidence algebras are tame

Wild concealed algebras are not $ extbf{g}$-tame

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.