# understanding the irreducible Brauer characters of a defect-1-block

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)

## 1 Answer

In this book, section 4.12, there is a statement (but no proof) of the structure theorem for blocks of defect one. (The proof can be found in Goldschmidt's book as I mentioned below.) The authors then use this theorem to deduce characters for precisely the group you're interested in. In any case, I think some calculations within the specific group will be needed, specifically involving the normalizer and centralizer of a Sylow $$p$$-group, rather than just using general hypotheses like having a Sylow group of order $$p$$.

Original answer below

I think you've generalized too far, because these statements are not true.

For a counterexample to the first statement, consider the group $$A_5$$ at the prime $$p=5$$. Its principal block has representations of dimensions 1,3,3,4, and the two representation of dimension 3 restrict to the same Brauer character.

For a counterexample to the second statement, consider the group of upper triangular matrices in $$SL(2,p)$$, which has order $$p(p-1)$$. It has four irreducible representations of dimension $$(p-1)/2$$, and each of them decomposes into $$(p-1)/2$$ Brauer characters (over the prime $$p$$) of dimension 1.

Goldschmidt's "Lectures on Character Theory" has a section on blocks of defect one. I don't have the book right now but I recall that there is a fairly concrete statement of the structure of blocks of defect one and you may be able to extract the needed results from there.

• Thank you for pointing out these mistakes! I must have misinterpreted some statements. I will add the correct questions after rethinking. – Dune Feb 26 at 9:18
• The book of Lux and Pahlings is very helpful, but unfortunately, I have no access to Goldschmidt's book (and so to the proof of the structure theorem). Yet, I begin to understand, what is really going on there: I was able to prove the above statement for a very specific group I am currently interested in (other than $J_1$) by using the techniques of section §20B in Curtis and Reiner's book (essentially the Green correspondence). I can very well imagine that these techniques ultimately generalize to blocks with cyclic defect groups. – Dune Mar 11 at 14:36