All Questions
10 questions
3
votes
1
answer
355
views
A generalisation of induced representations
Let $G$ be a finite group, and $H\subseteq G$ a subgroup. Let $F$ be a field. Let $W$ be a finite-dimensional $F[H]$-module. Let $T$ be a left transversal of $H$ in $G$. Then we can define:
$W^G=\sum_{...
- 1,433
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
1
vote
0
answers
65
views
Is it possible to describe a $k$-basis for $M\otimes_{kH}N$ when $M$ is a $k[G\times H]$-module and $N$ is a $k[H\times K]$-module?
Suppose $k$ is a field for Let $M$ be a finitely-generated a $k[G\times H]$-module and let $N$ be a finitely-generated $k[H\times K]$-module. Then in particular, $M$ and $N$ are finite-dimensional $k$-...
- 41
1
vote
1
answer
395
views
Making use of extra symmetries; more examples?
TL; DR.
In representation theory, it's nice to decompose a given representation into smaller ones. One technique is by utilizing extra symmetries. Explicit examples come from compact groups, and I ...
- 5,230
2
votes
0
answers
116
views
Loewy structure of $S_4$
How to deduce the Loewy Structure of $kS_4$ where $k$ has characteristic 2. I can compute the Cartan matrix and Decomposition matrix with Brauer Characters without difficulties. But when it comes to ...
- 51
2
votes
1
answer
197
views
Top and bottom composition factors of $M$ are isomorphic
Let $k$ be a field and $N$ a finite group. Let $M$ be a projective indecomposable $kN$-module. Since the algebra $kN$ is symmetric, it follows that the top and bottom composition factors of $M$ are ...
- 51
3
votes
1
answer
286
views
Why we study Endo-Trivial Modules?
I made the exact same question on MSE some days ago and there wasn't any response whatsoever so far, so thought probably I have to ask here to get an answer. So the question goes as follows:
Recently ...
- 441
4
votes
1
answer
382
views
Non-existence of projective covers
I didn't get the argument of Example 7.3.11, page 123, from the representation theory book of Peter Webb, available also online at:
http://www-users.math.umn.edu/~webb/RepBook/RepBookLatex.pdf
In ...
- 137
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
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