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Results tagged with modular-representation-theory
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user 4231
For questions about modular representation theory, the study of representations over a field of positive characteristic.
5
votes
Module with indecomposable and decomposable reductions mod $p$
[EDIT] Since my various edits got quite long, maybe I can answer the question more directly and refer to previous versions for elaboration.
A key elementary result can be found in Feit's 1982 monogra …
- 52.9k
5
votes
Projectives in the category of modular representations of Lie algebras
It's fairly easy to answer your basic question in the second paragraph: at present there is no guaranteed method for computing these projectives, though quite a bit of work has been done in recent dec …
- 52.9k
3
votes
Accepted
indecomposable modules restricted from $gl_n$ to $sl_n$
This is actually true (in somewhat more generality), as remarked by Jantzen in a recent updating of his unpublished 2011 notes on restrictions of modular representations of $\mathfrak{gl}_n$ to $\math …
- 52.9k
2
votes
Accepted
Well-understood bases for Grothendieck groups of modular representation categories
I'm not sure what sources you are mainly relying on, but there are several points to be made:
1) When you say the Lie algebra is "semisimple", I suspect you mean (as people sometimes do when using sh …
- 52.9k
5
votes
Accepted
Restriction of scalars for the adjoint representation of $SL_2(\mathbb F_q)$
There is some textbook literature which essentially covers the issues raised here, though it often deals with more general situations. (Over finite fields life is simpler, since Schur indices are 1. …
- 52.9k
4
votes
Idempotents and Structure of Simple GL(n,p) modules in the describing characteristic
It's probably safe to say that beyond $n=2$ (and possibly $n=3$), little is known about the representations of these groups in characteristic 2. While there is a lot of general theory aimed at orga …
- 52.9k
1
vote
Composition factors of tensor products of modular representations
Questions along this line have appeared on Math Overflow with some frequency, so it's useful to take a look at some of them and their links, for instance here.
As Geoff points out, it's natural here …
- 52.9k
7
votes
Accepted
What are the irreducible modular representations of $SU(n,p)$?
Yes, it's a theorem of Steinberg from his fundamental 1963 Nagoya Math. J. paper, in the pre-Meataxe era. This is treated in Chapter 2 (especially 2.11) of my LMS Lecture Note Series No. 326
Modular …
- 52.9k
5
votes
Projective modules and tensor products
This is an extended comment, to put Julian's answer and comments in perspective. What Alperin does in his book is a direct but somewhat ad hoc treatment of one suggestive small case, which goes bac …
- 52.9k
5
votes
Innocent question on tensor products of modular representations
The answer to your question is usually no (which is fortunate because the lack of complete reducibility gives modular representation theorists something to do), starting for example with the tensor pr …
- 52.9k
8
votes
two questions in modular representation theory
Mariano has addressed question (1), but let me add that finite representation type is extremely rare especially for interesting classes of groups like the simple nonabelian ones: in characteristic $p$ …
- 52.9k