Finding all selforthogonal indecomposable modules

Question

Given a finite dimensional algebra $A$ with finite global dimension such that there are only finitely many basic tilting modules. Then every selforthogonal indecomposable module $M$ (that is a module with $\mathrm{Ext}_A^i(M,M)=0$ for all $i \geq 1$) is a direct summand of a tilting module and thus there are only finitely many such indecomposable selforthogonal modules. See 3.9 in https://arxiv.org/abs/1803.10707.

Question 1: Is there an effective method to obtain every indecomposable selforthogonal module in a nice way? For example using the GAP-package QPA.

(A nice method would for example be an "Auslander-Reiten" theory for indecomposable selforthogonal modules in such algebras, where one can obtain every such module using certain exact sequences starting from just one such module.)

Question 2: Is there an example of an Auslander algebra with infinitely many selforthogonal indecomposable modules?

Answer 1

Concerning your second question, it is claimed in the proof of Corollary 4.8 in a recent preprint by Iyama-Zhang that Kajita showed in his Master's thesis that the Auslander algebra of the linearly oriented path with at least 6 vertices has infinitely many classical tilting modules. So it has in particular infinitely many indecomposable selforthogonal modules.

Let's now turn to the first question. Considering the set $\mathfrak{T}$ of (isomorphism classes of basic) tilting $A$-modules with the partial order induced by the inclusion order of the corresponding perpendicular subcategories, $\mathfrak{T}$ becomes a finite lattice. According to Happel-Unger there is an arrow $T \to T'$ in the Hasse diagram of $\mathfrak{T}$ if and only if $T = X \oplus M$ and $T' = Y \oplus M$ with $X$, $Y$ indecomposable fitting into an exact sequence $$ 0 \to X \xrightarrow{f} M' \xrightarrow{g} Y \to 0 $$ where $f$ is a minimal left and $g$ a minimal right $\mathrm{add}(M)$-approximation.

Given that QPA can compute minimal approximations, we can start with the set $\{P_1,\ldots,P_n\}$ of indecomposable projective $A$-modules and use BFS (or another graph-traversal algorithm) to find all sets $\{T_1,\ldots,T_n\}$ of indecomposable $A$-modules such that $T = \bigoplus_i T_i$ is a tilting module. The union of all these sets $\{T_1,\ldots,T_n\}$ is then the set of indecomposable selforthogonal $A$-modules.