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.