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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.

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I'm afraid I can't help with the question, but out of curiosity, are these monoids certain types of band? – Yemon Choi Mar 8 '12 at 5:10
They are bands. – Benjamin Steinberg Mar 8 '12 at 12:39
Can you do the Yoneda product of these Ext groups? The quiver presentation of the algebra in particular contains the information about products of $Ext^1$-groups. – Dag Oskar Madsen Mar 16 '12 at 13:20
I don't know how to compute the Yoneda product. How does it relate to the quiver presentation? – Benjamin Steinberg Apr 6 '12 at 0:27
I think Dag Madsen is suggesting the homological approach of computing the $A_\infty$-structure of the $\operatorname{Ext}$-algebra of the simples. This gives the quiver and relations. The relations are given by the image of a map $Dm:D\operatorname{Ext}^2(S,S)\to \bigoplus (D\operatorname{Ext}^1(S,S))^m$, where $S$ is the direct sum of all the simples. – Julian Kuelshammer Jul 2 '14 at 15:58

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