All Questions
11 questions
16
votes
3
answers
1k
views
Conjectures in the representation theory of the symmetric group
Question: What are current open conjectures about the representation theory of the symmetric group?
I am interested mostly in the characteristic 0 case, but conjectures for the modular case can also ...
- 26.5k
13
votes
4
answers
3k
views
What is a "block" in an abelian category?
In the literature and in some posts here, there has been variation in the undefined use of the term "block" for a category of modules over a ring, or more abstractly an abelian category (all of which ...
- 52.9k
8
votes
1
answer
534
views
Representation theory of $\mathrm{GL}_n(\mathbb{Z})$
I want to understand the (complex) representation theory of $\mathrm{GL}_n(\mathbb{Z})$, the general linear group of the integers. I have gone through several representation theory texts but all of ...
- 81
6
votes
1
answer
186
views
Endomorphism ring of trivial source modules for abelian p-groups
Bernhard Böhmler (who is also on MO) and myself had the following idea:
Let $G$ be a finite group and $k$ a field of characteristic $p$ (algebraically closed when it is needed) such that $p$ divides ...
- 26.5k
6
votes
0
answers
328
views
When does a group and its pro-algebraic completion have equivalent categories of arbitrary representations?
In the following everything is over some field $k$.
Let $G$ be a discrete group. We write $G^{\text{alg}}$ for its pro-algebraic completion. The latter is a pro-affine pro-algebraic group which arises ...
- 1,635
4
votes
0
answers
227
views
Orbits of group representation over $\mathbb{F}_2$
Recently I have been considering a problem that has had me venture into some mathematical territory that is new to me, specifically group representations, words, and character theory. I would ...
- 161
3
votes
1
answer
70
views
2-periodic modules over p-group algebras
Given the group algebra of a p-group over a field of characteristic p. Can the 2-periodic indecomposable modules $M$ ($M$ with $\Omega^{2}(M)=M$) be classified? I am not experienced much with modular ...
- 26.5k
3
votes
0
answers
83
views
Connection between certain finite groups and Frobenius algebras
This question is maybe not so precise yet, but contains some ideas that maybe just search for the right definition.
Let $G=F/U$ be a finite group with presentation $F/U$ (here the presentation really ...
- 26.5k
2
votes
1
answer
395
views
Projectivity of torsion-free modules over integral group rings
Let $G$ be a torsion-free group and assume that the integral group ring $\mathbb{Z}G$ is torsion-free as well. Let $M$ be a torsion-free, finitely generated module over $\mathbb{Z}G$.
If we assume ...
- 2,998
2
votes
1
answer
206
views
Extensions of lattices
Let $G$ be a finite group and $\phi:M\to N$ be a surjective homomorphism of $G$-lattices (i.e. finitely generated $\mathbb{Z}[G]$-modules, free as $\mathbb Z$-modules), with kernel $K$. For every $n\...
- 23
2
votes
0
answers
70
views
Short exact sequences in p-group algebras
Given a group algebra of a finite p-group over a field of characteristic p.
Tachikawa proved that in this case $Ext^{1}(M,M) \neq 0$ for any finite dimensional non-projective module $M$.
Can one give ...
- 26.5k