Simple modules in the decomposition of modular group algebra KG

Question

Suppose F is a field such that $char(F) = p \ \nmid \ |G|.$ Then we know that in this case (i.e. semisimple), there is a bijection between the irreducible representations of G and the simple components appearing in the wedderbern artin decomposition of FG. Does something similar happen in modular case?
I just know that if $char(K) = p \ | \ |G,|$ then corresponding to each simple module S of $KG,$ there is an indecomposable projective module P such that $\displaystyle \frac{P}{rad(P)} \cong S.$ But how does that S appear in the decomposition of $KG?$

Answer 1

Something a bit similar happens in characteristic $p$, but much more complicated. One may still write $KG$ as a direct sum of indecomposable $2$-sided ideal summands, but they are no longer in bijection with simple modules. These are called the ($p$-)blocks. Each block $B$ is a subalgebra, and when considered as a $KG$-module, $B$ breaks up as a direct sum of projective indecomposable modules $P_1,\dots,P_r$, with simple tops $S_1,\dots,S_r$. If $\dim(S_i)=a$ then $P_i$ is isomorphic to exactly $a$ of the $P_j$, as in the decomposition of $FG$, but each simple module $S$ appears in exactly one block. Occasionally a block $B$ restricts to just $S^{\oplus \dim (S)}$, i.e., it looks exactly as in the characteristic $0$ case. In this case, the block has defect zero, and is isomorphic to a matrix algebra. But in general there are lots of blocks that are not matrix algebras, for example the one with a trivial quotient. This block is called the principal block, and is a matrix algebra if and only if $p\nmid |G|$.