A peak reduction functor for socle projective representations
The authors study representations of \(r\)-peak finite posets, namely of those posets with \(r\) maximal elements. The representations considered mainly by the authors are socle projective modules and prinjective modules over the incidence algebra of the given poset. Recall that a module \(X\) is called prinjective if the kernel of the map from the projective cover of \(X\) to \(X\) is a semisimple projective module. For posets with at most 2 peaks, a characterization of tame prinjective type has been discussed by Nazarova and the authors, respectively, but for posets with arbitrary number of maximal elements, this is an open question. Keeping this in mind, the authors associate with any poset \(J\) and a maximal element \(q\) a socle projective algebra \(A\), and provide a functor from the category of all socle projective representations of \(J\) to the category of all socle projective modules over \(A\). They call this functor a peak reduction functor and show as main result of the paper that a peak reduction functor is always full, dense and preserves and reflects tame representation type. Moreover, some special posets \(J\) are investigated in detail.
scientific article; zbMATH DE number 1014031
scientific article