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3 votes
0 answers
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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 ...
Mare's user avatar
  • 26.5k
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 ...
Mare's user avatar
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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 ...
Kenji's user avatar
  • 81
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 ...
healynr's user avatar
  • 161
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 ...
Mare's user avatar
  • 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 ...
Patrick Elliott's user avatar
2 votes
1 answer
207 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\...
Tetawo's user avatar
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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 ...
Mare's user avatar
  • 26.5k
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 ...
Mare's user avatar
  • 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 ...
AlexE's user avatar
  • 2,998
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 ...
Jim Humphreys's user avatar