On Schur algebras, Doty coalgebras and quasi-hereditary algebras

Motivated by Doty's Conjecture we study the coalgebras formed from the coefficient spaces of the truncated modules. We call these the Doty Coalgebras D_(n,p)(r). We prove that D_(n,p)(r) = A(n,r) for n = 2, and also that D_(n,p)(r) = A(\pi,r) with \pi a suitable saturated set, for the cases; i) n = 3, 0 \leq r \leq 3p-1, 6p-8\leq r \leq n^2(p-1) for all p; ii) p = 2 for all n and all r; iii) 0\leq r \leq p-1 and nt-(p-1)\leq r\leq nt for all n and all p; iv) n = 4 and p = 3 for all r. The Schur Algebra S(n,r) is the dual of the coalgebra A(n,r), and S(n,r) we know to be quasi-hereditary. Moreover, we call a finite dimensional coalgebra quasi-hereditary if its dual algebra is quasi-hereditary and hence, in the above cases, the Doty Coalgebras D_(n,p)(r) are also quasi-hereditary and thus have finite global dimension. We conjecture that there is no saturated set \pi such that D_(3,p)(r) = A(\pi,r) for the cases not covered above, giving our reasons for this conjecture. Stepping away from our main focus on Doty Coalgebras, we also describe an infinite family of quiver algebras which have finite global dimension but are not quasi-hereditary.