All Questions
Tagged with finite-groups modular-representation-theory
38 questions
24
votes
0
answers
813
views
Revising the proof of CFSG
This is an oft-quoted excerpt from John Thompson's article "Finite Non-Solvable Groups":
“... the classification of finite simple groups is an exercise in taxonomy. This is
obvious to the ...
- 1,433
18
votes
2
answers
1k
views
The mysterious significance of local subgroups in finite group theory
EDIT 21/12: Even if there are no conclusive answers to these questions, I would very much like to know if anyone has noted and attempted to explain the mysterious significance of local subgroups: are ...
- 1,433
15
votes
1
answer
2k
views
Which finite groups have no irreducible representations other than characters?
A classical result states that all the irreducible representations of a finite group over $\mathbb{C}$ are characters if and only if $G$ is abelian. I would like to know what happens if we consider a ...
- 11.3k
12
votes
0
answers
340
views
Does every finite group have a small projective representation (over some ring)?
Question. Let $G$ be a finite group. Can we find some (commutative) ring $R$ and some positive integer $d=O(\log\lvert G\rvert)$ such that $G$ can be found as a subgroup of $\operatorname{PGL}_d(R)$?
...
- 602
12
votes
0
answers
420
views
Non-isomorphic groups with same character tables and different Brauer character tables
Let $A$ be a discrete valuation ring with perfect residue field $k$ of characteristic $p$ and field of fractions $K$ of characteristic $0$. Let $G$ and $H$ be two finite groups and assume that $K$ is ...
11
votes
1
answer
364
views
Can we glue characteristic 0 and characteristic p representations of a finite group given equality of (Brauer) characters?
Suppose I have a prime $p$ and a finite group $G$ together with representations $\sigma: G \to GL_n(\mathbb{Q}_p)$ and $\pi: G \to GL_n(\mathbb{F}_p)$. My question is:
When does there exist a ...
- 45.7k
10
votes
3
answers
734
views
Low-dimensional irreducible 2-modular representations of the symmetric group
I apologize if this question is a little too basic for MathOverflow, but it's somewhat outside of my background and I'm frustrated that the answer doesn't seem to be explicit in the literature even ...
- 1,298
8
votes
1
answer
703
views
When does a finite group have finitely many indecomposable representations?
Let $G$ be a finite group and let $k=\mathbb F_p$. Then it is well-known that $G$ has finitely many irreducible modules.
However, in general $G$ does not have finitely many indecomposable ...
- 3,054
8
votes
2
answers
448
views
The radical of $kG$-modules
$\DeclareMathOperator\Rad{Rad}$Let $k$ be a finite field of $p$ elements. Let $G$ be an elementary abelian p-group and $V$ a $kG$-module corresponding to the representation $\alpha:G\rightarrow \...
- 393
7
votes
3
answers
783
views
Exact sequences of permutational representations?
Let $R$ be a commutative ring, like the ring of integers $\mathbb Z$ or the ring of $p$-adic integers $\mathbb Z_p$. Let $G$ be a finite group; let us consider permutational representations of $G$ ...
- 15.6k
7
votes
3
answers
524
views
Restriction of scalars for the adjoint representation of $SL_2(\mathbb F_q)$
Let $p$ be a prime number, $q=p^e$ a power of $p$, and $G=SL_2(\mathbb F_q)$. Let $V$ be the adjoint representation of $G$, i.e. $V$ is the 3-dimensional $\mathbb F_q$-space of of (2,2)-matrices of ...
- 26k
7
votes
1
answer
322
views
Why is Nagao's theorem the "Module theoretic version of Brauer's second main theorem"?
Let $G$ be a finite group, $p\in\mathbb{P}$ a prime, $\mathbb{F}$ an algebraically closed field of characteristic $p$, and $D\leq G$ a $p$-subgroup.
Brauer second main theorem states
If $\chi\in ...
- 9,650
7
votes
2
answers
330
views
Representations of $\operatorname{Sp}(2g,\mathbb{Z}_3)$
Let $V$ be a $2g$-dimensional vector space over $\mathbb{Z}_3 := \mathbb{Z}/3\mathbb{Z}$. First, $\operatorname{Sp}(2g,\mathbb{Z}_3)$ acts on $\Lambda^2(V)$, and this decomposition is reducible, as ...
- 75
6
votes
1
answer
366
views
Group of order $5p^aq^b$
In Lectures by Dan Bump on Modular representation theory,
Theorem 13.14 states that whenever $G$ is a non-abelian simple group of order $|G|=p^aq^br$ for distinct primes $p$,$q$, and $r$, every $r$-...
- 367
6
votes
1
answer
579
views
first group cohomology for the standard representation of $S_n$ over $\mathbb{F}_2$
Let $g \geq 2$ be an integer and consider the symmetric group $S_n$ where $n = 2g+1$ or $n = 2g+2$ as a subgroup of the symplectic group $\mathrm{Sp}_{2g}(\mathbb{F}_2)$ via the standard ...
- 1,298
6
votes
1
answer
790
views
What is the Grothendieck group of the category of $\mathbf{Z}_p[G]$-modules?
Let $G$ be a finite group. Let $\mathcal{O}$ be a suitably large finite extension of the $p$-adic integers, with residue field $\mathbf{F}_q$.
The Grothendieck group of the category of finitely-...
- 63
6
votes
1
answer
934
views
Finite groups with all irreducible representations one dimensional
Let $k$ be a field of characteristic $p$ and $G$ a finite group. This question might be a dulicate of this question:
Which finite groups have no irreducible representations other than characters?
...
- 137
6
votes
0
answers
266
views
What is the Fourier transform in modular representation theory?
For a finite group $G$ there is the Fourier transform $\displaystyle \hat{f}(\rho)=\sum_{g \in G} f(g)\rho(g)$ with inverse $$\displaystyle f(g)=\frac{1}{|G|}\sum_{\rho}d_{\rho}\operatorname{Tr}\left(\...
- 571
5
votes
2
answers
1k
views
Decomposing representations of finite groups
Let $G$ be a finite group, $p$ a prime number. We denote by $\mathbb{F}_p$ the field of cardinality $p$. Let $V$ be an infinite dimensional representation of $G$ over $\mathbb{F}_p$.
Must there be $G$...
- 11.3k
5
votes
2
answers
1k
views
two questions in modular representation theory
I have two questions:
Let $G$ be a finite group. Because complex representations of $G$ are completely reducible to know all representations is same as knowing irreducible ones. In case of modules ...
- 81
5
votes
2
answers
450
views
Module with indecomposable and decomposable reductions mod $p$
Let $G$ be a finite group and $p$ a prime dividing the order of $G$. Let $V$ be an irreducible $\mathbb{C}[G]$-module.
Let $F$ be the finite field $\mathbb{Z} / p \mathbb{Z}$. Suppose that there ...
- 2,821
5
votes
0
answers
205
views
Can modular representation theory be used to prove Sylow's existence theorem?
Edit 20/12: I added a more precise question at the bottom of the post.
Given a finite group $G$ and a prime $p$, we want to prove that $G$ has a $p$-subgroup $P$ such that $|G:P|$ is not divisible by $...
- 1,433
5
votes
0
answers
130
views
subgroups of $\mathrm{Sp}_{2g}(\mathbb{Z}_2)$ whose mod-2 image is the symmetric group
Let $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ be a closed subgroup of the symplectic group over the $2$-adic integers whose image under the mod-$2$ homomorphism $\pi : \mathrm{Sp}_{2g}(\mathbb{Z}_2)...
- 1,298
4
votes
1
answer
364
views
the number of indecomposable modules of finite groups over finite fields of a fixed dimension
I am interested in determining the the number of indecomposable modules of finite groups over finite fields of a fixed dimension. Specifically, I have the following conjecture:
Conjecture. Suppose we ...
- 565
4
votes
1
answer
267
views
Fourier transform on finite groups in characteristic $p>0$
Is there a Fourier theory for finite groups in characteristic $p>0$? Assume that $p$ divides the order $|G|$ of finite groups (or just work with $p$-groups), i.e., in a modular representation-...
- 41
3
votes
1
answer
444
views
Defect groups and subgroups
I have asked this question on Stack Exchange but had no response; it's been bugging me for a few days. I am struggling to see how to apply Mackey's theorem to prove a certain Lemma in Local ...
- 159
3
votes
1
answer
122
views
Idempotents and Structure of Simple GL(n,p) modules in the describing characteristic
Let $F_2$ denote the finite field of two elements, and $GL(n,2)$ the general linear group of degree $n$ over $F_2$. I have a promising line of inquiry into identifying the structure of simple $F_2\ GL(...
- 31
3
votes
1
answer
288
views
Representation of a finite group over a finite field from rational representations
Suppose $G$ is a the cyclic group of $n$ elements and $p$ is a prime not in $n$. $G$ has an action on the cyclotomic field as a $\mathbb{Q}$-vector space $\mathbb{Q}[X] / \langle \Phi_n(X) \rangle$ by ...
- 885
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
3
votes
0
answers
109
views
Exact structures on representations of a finite group
For simplicity assume $G$ is a (finite) $p$-group, and $k$ is field of characteristic $p$, so that there exists a unique simple $kG$-module the trivial module $k$. I am looking for a class of short ...
- 542
3
votes
0
answers
133
views
Isomorphism of certain irreducible representations over finite fields
We are given a faithful representation of a cyclic group of order 5 $\rho: C_5=G \rightarrow End_{\mathbb{F}_3}(V) $ with $dim_{\mathbb{F}_3}V=8$ as vector space. It is also known that $V=U\oplus W$ ...
- 269
2
votes
0
answers
57
views
Brauer pairs associated to a normalizer subsystem in the fusion system of a block of a finite group
Let $G$ be a finite group and let $k$ be an algebraically closed field of positive characteristic $p$. Let $b$ be a block of $kG$ and let $(P,e)$ be a maximal $(G,b)$-Brauer pair. For every subgroup $...
- 81
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
529
views
Representations of semidirect product over $C_p$
Hi,
I am wondering if anything is known about irreducible representations of a semidirect product over $C_p:=\mathbb{Z} / p \mathbb{Z}$ in general or at least in special cases. For example of $C_q \...
1
vote
0
answers
129
views
$G$-invariant bilinear maps
Let $G$ be a finite group and $M$ a finitely generated $\mathbb{Z}G$-module. Then $M \otimes k$ is a representation of $G$ for any field $k$. I am interested in the number of ways we can turn $M \...
1
vote
0
answers
54
views
Largest almost quasisimple group that acts on a spin module
I'd like to know the largest almost quasisimple group $G$ which acts faithfully and irreducibly on the spin module for an odd dimensional orthogonal group $\Omega(2k+1,q)$, or on each of the two half-...
- 121
0
votes
1
answer
143
views
Faithful representation of group of order $p^4$
In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book, "Theory of groups of finite order". The group ($\mathbb{Z}_{p^{2}}\rtimes \mathbb{Z}_{p^{}}) ...
- 381
0
votes
1
answer
197
views
Analogon to Brauer characters, if K not algebraically closed
Hello,
is there a theory for characters of a finite group over a field $K$ with prime characteristic, if $K$ is not algebraically closed? For algebraically closed fields $K$ for example Brauer found ...
- 1