All Questions
9 questions
11
votes
1
answer
159
views
Are all indecomposable $\mathbb{Z}_+$-modules over the character ring of a group, character rings of a subgroup?
A $\mathbb Z_+$-algebra is an algebra $A$ over $\mathbb C$ with given basis $\{v_i\}$ such that
$$v_iv_j=\sum_k n_{ijk}v_k,\hspace{10mm}n_{ijk}\in\mathbb Z_{\geq0}.$$
An example of such an object is ...
- 301
5
votes
1
answer
244
views
What are the indecomposable modules over $\mathbb{F}_2(C_2\times C_2)$?
Let $C_2$ be the cyclic group of order $2$ and $\mathbb{F}_2$ the field with $2$ elements. Consider the group algebra $A:= \mathbb{F}_2 (C_2\times C_2)$. It is well-known that $A$ has infinite ...
- 696
5
votes
0
answers
221
views
Group cohomology of $\mathbb{Z}$ vs $\mathbb{Z}_p$
Let $M$ be a continuous representation of $\mathbb{Z}_p$ over $\mathbb{F}_p$, likely infinite-dimensional.
There is the inflation map of group cohomology $H^*_{\text{cts}}(\mathbb{Z}_p, M) \rightarrow ...
- 689
2
votes
1
answer
160
views
MAGMA-question concerning the transformation of a $kG$ -module $M$ into a right ideal of the group algebra
Let $G$ be a finite group and $k$ be a finite field of characteristic $p>0$ such that $p\mid |G|$.
Let $M$ be a $kG$-module which has an embedding $M\hookrightarrow kG^{reg}$ into the regular $kG$-...
- 1,755
1
vote
0
answers
77
views
Existence of a certain direct summand
Let $P$ be a cyclic group of order $p^n$. Let $M$ be a $\mathbb{Z}_p[P]$-module and $M_p :=\mathbb{Q}_p\otimes M$ be the associated $\mathbb{Q}_p[P]$-module. When will $M_p$ have a direct summand ...
- 1,283
1
vote
1
answer
271
views
Rank of a locally free $\mathbb Z[G]$-module
This is a basic question. Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$, i.e. is locally free.
...
- 255
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
3
answers
2k
views
A semisimple group ring
Let $n \in \mathbb{N}$, $p$ a prime number, and $G$ a finite group of order coprime to $p$. Let $R = \mathbb{Z} /p^n \mathbb{Z}$ be the ring of integers mod $p^n$. Must $R[G]$ be semisimple?
As noted ...
- 11.3k
7
votes
2
answers
848
views
Does every irreducible representation of a finite group G embed into its group algebra?
Let $G$ be a finite group, and let $F$ be a field. Is there a simple proof that every irreducible representation of
$G$ embeds into the group algebra $F[G]$? I am specially interested in the case when ...
- 71