Reference request: basic algebras of a group algebra I need a reference to the topic of the Basic Algebra of a group algebra, in the context of modular representation theory... Is there some book? 
Sadly, googling it only gives a lot of "basic algebra" stuff. Bad name choice I guess!
 A: A good reference on the subject seems to be Karin Erdmann's book Blocks of tame representation type and related algebras, Lecture Notes in Mathematics 1428.  In particular, the tables at the end of the book list quivers with relations for (Morita-equivalence classes of) blocks of certain group algebras, including dihedral and semi-dihedral groups.
Other references concerning symmetric groups would be


*

*S.Martin, On the ordinary quiver of the principal block of certain symmetric groups, Quart. J. Math. (2) 40 (1989), p.209-223.  (The case of symmetric groups over $2p$ elements is treated here).

*S.Martin, Ordinary quivers for symmetric groups II, Quart. J. Math. (2) 41 (1990), p.79-92. (Some more symmetric groups and some alternating groups are treated here).

A: In general the basic algebra of a finite-dimensional algebra $A$ over a field $F$ may be obtained (non-canonically!) as follows:
Start (!) with a complete set of primitive orthogonal idempotents $e_1,\dots,e_n$ for $A$. Choose a subset $S \subseteq \{1,2,\dots,n\}$ in such a way that the modules $e_i A$ for $i \in S$ are non-isomorphic and each $e_i A$, for $1 \leq i \leq n$, is isomorphic to some $e_i A$ for $i \in S$. 
Define $$e=\sum_{i \in S} e_i.$$ The basic algebra $B$ corresponding to our choice of $S$ is then
$$B=eAe.$$ The functor $M \mapsto Me$ gives a Morita equivalence from mod-$A$ to $B$-mod.
A reference for this is section 6 of chapter 1 of Elements of the representation theory of associate algebras, by Assem-Simson-Skowronski.
For the specific case of finite groups in characteristic $p$, the difficult step here is calculating a complete set of primitive orthogonal idempotents. Knowing these explicitly one knows, for example, the dimensions of all the irreducible modules. For example, for the case of the group algebra of the symmetric group over a field of characteristic $p$, we know an indexing set for the irreducible modules ("p-restricted partitions"), and which ones belong to the same blocks (via "$p$-cores"), but we do not have a good algorithm for computing dimensions. 
For instance, I would be somewhat surprised (and very impressed!) if a computer algebra package could correctly compute the dimensions of the irreducible modules for the symmetric group $S_{35}$ over the field $F_3$ with three elements.
A: There is considerable work by Jon Carlson, Klaus Lux, and collaborators on constructing and working computationally with basic algebras. See for example
http://ci.coastal.edu/~thoffman/basic/basic.html and 
http://magma.maths.usyd.edu.au/magma/handbook/text/967
Best wishes.
Eamonn 
