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Assume that the Schur algebra $S(n,r)$ with $n \geq r$ is not representation-finite.

Question: For which $n$, $r$ is the quiver and relations of the blocks of $S(n, r)$ explicitly known?

I just found the cases $n=r=4$ and $n=r=5$ in Xi - On representation types of $q$-Schur algebras.

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Erdmann's article Schur algebras of finite type shows that $S(n,r)$ has finite representation type in prime characteristic $p$ if and only if $n=2$ and $r < p^2$ or $n \ge 3$ and $r \le 2p$ or $p=2$, $n=2$ and $r=5$ or $7$. In these cases the quiver and relations for (the basic algebra Morita equivalent to) each block are found explicitly. Quivers and relations are also found for some blocks of Schur algebras of infinite representation type. For example, Proposition 5.2 gives a basic algebra Morita equivalent to the principal block of $S(2,p^2)$ or $S(2,p^2+1)$ when $p > 2$. There are further results in Section 5 on blocks of $S(3,r)$ of infinite type.

More recently Doty, Erdmann, Martin and Nakano have classified all the tame Schur algebras. As one would expect, their paper gives some information about the Ext quivers. For example, see page 153 for the quiver for the basic algebra of $S(2,6)$ in characteristic $2$: it has wild representation type.

I can't find any explicit results in these papers for $n \ge r$. Since the authors' main interest is in the finite/tame/wild distinction, they concentrate on the blocks at the `threshold', where typically $n < r$. The corollary on page 143 of the second paper outlines a method for embedding the module category of $S(n,d)$ in the module category of $S(n',d)$, whenever $n' \ge d$.

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  • $\begingroup$ Thank you, that is an interesting survey. But note that my assumption were that the algebras are not representation-finite and $n \geq r$. $\endgroup$ – Mare Aug 2 at 20:18

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