In section XII of his famous paper from 1966, Janko investigated the principal 11-block of his group $J_1$ (and thereby finally proved existence and uniqueness of this group). I would like to learn the right amount of modular representation theory to understand Janko's reasoning from a modern perspective (currently, I am using the books of Curtis and Reiner). So far, I have a basic understanding of the Brauer-Cartan-triangle and of block theory. But there are still some details in Janko's reasoning which are unclear to me, and it seems very hard to extract these details from the textbooks.
It would be very helpful if someone could explain (only approximately of course), which theories / statements I need to know in order to understand Janko's reasoning which I am reciting in the following:
Janko observed by counting arguments that there are 11 irreducible (ordinary) characters and 10 irreducible Brauer characters in the principal $11$-block $B$ (this part is clear to me). Then he partitioned the irreducible characters of $B$ into two sets $L_1, L_2$ such that $\chi(1) \equiv 1 \mod 11$ for all $\chi \in L_1$ and $\chi(1) \equiv -1 \mod 11$ for all $\chi \in L_2$. Now he claims the following statements which are unclear to me:
- No two characters of a fixed set $L_i$ have a Brauer constituent in common.
- Every irreducible Brauer character of $B$ appears as a constituent of precisely two irreducible characters of $B$.
- Every irreducible character of $B$ has at most two irreducible Brauer constituents.
(In summary, one may visualize the irreducible (ordinary/Brauer) characters of $B$ as a tree, which in Janko's case is actually a path.)
It seems to me that nowadays the situation of $p$-blocks with cyclic defect groups is very well understood. On the other hand, the literature treating this generality seems utterly complicated to me. For now, I would be completely satisfied to understand the $p$-blocks of a finite group with cyclic $p$-Sylow groups of order $p$, as in Janko's context.
(As Ted pointed out, Janko's assertions are not true in general for finite groups with cyclic $p$-Sylow groups of order $p$. Unfortunately, I don't recognize which other hypotheses Janko was using in the above reasoning)