I have recently been studying with colleagues the representation theory of certain finite monoids that come up in probability theory and combinatorics, see Ken Brown's beautiful survey here.

These monoid have a split basic algebra over any field which is directed quasi-hereditary (standard modules are projective/costandards are simple). Thus they have acyclic quivers and can be written as the quotient of the path algebra of their quiver by an admissible ideal.

We can compute $\mathrm{Ext}^n(S,S')$ between simple modules $S,S'$ and so can get the quiver by taking dimensions when $n=1$. We can also get the number of relations from $n=2$.

Question.Is there some homological way to get at a quiver presentation of a split basic algebra with an acyclic quiver? You can assume the field is algebraically closed if it helps.

We do have explicit complete sets of primitive idempotents for these algebras, but they are sufficiently complicated that we don't even know how to recover the description of the quiver using the primitive idempotents.

The only cases where quiver presentations are known are for oriented matroids (in particular hyperplane arrangements) and of course the hereditary case.

The situation is so embarrassing that we have global dimension 2 examples with a single relation and we can't find the relation.