Let $A,B$ be a pair of quasi-hereditary algebras and assume that $A$ and $B$ are both standard Koszul. Further assume that the graded decomposition matrices of $A$ and $B$ coincide (that is, the multiplicities of standard modules in projective modules coincide, as do their grading shifts). Under what circumstances could I conclude that these algebras $A$ and $B$ are graded Morita equivalent?


It feels to me that one should be able to make a general statement. If not, are there any counterexamples to the following claim? If $A$ and $B$ are standard Koszul and have the same decomposition matrices, then they are Morita equivalent.



Are standard Koszul algebras with the same Kazhdan-Lusztig polynomials Morita equivalent?
Let $k$ be an infinite field. I will descibe an infinite family of standard Koszul
algebras with the same graded decomposition matrix.


They will be algebras given by a quiver with relations. The quiver
will be the same in all cases: nine vertices $a\_{1}, a\_{2}, a\_{3},
b\_{1}, b\_{2}, b\_{3}, c\_{1}, c\_{2}, c\_{3}$, with an arrow from $a\_{i}$
to $b\_{j}$ and an arrow from $b\_{i}$ to $c\_{j}$ for all choices of
$(i,j)$, so $18$ arrows in total.


For the relations, for each $(i,j)$ pick one of the three paths from
$a\_{i}$ to $c\_{j}$ and for each of the other two paths, make it a
scalar multiple of the first path. All $18$ scalars may be chosen
independently. So this gives an $18$ parameter family of sets of
relations.


Different choices may give isomorphic algebras. But since there is at
most one arrow between each pair of vertices, the only ambiguity is
that we can permute vertices (but only in finitely many ways) and
multiply arrows by nonzero scalars. However, some choices of these
scalars (for example, choosing them all to be equal) do not affect the
relations, so we have at most $17$ parameter families of choices of
relations giving the same algebra (in fact, less than $17$
parameters).


So we must have infinitely many nonisomorphic algebras.


Since the quiver is acyclic, we can order the vertices so that the
standard modules are just the simple modules.


If we do this, it is straightforward to check that the algebras are
all standard Koszul and have the same graded decomposition matrices.


