All Questions
26 questions
2
votes
0
answers
157
views
Cohomologically trivial modules over finite $p$-groups
Let $A$ be a finitely generated $\mathbb{Z}_pG$-module, where $G$ is a finite $p$-group and $\mathbb{Z}_p$ is the ring of $p$-adic integers; assume moreover that $A$ is cohomologically trivial, that ...
8
votes
1
answer
312
views
Decomposing the homology of a finite-index subgroup into isotypic components
$\newcommand\C{\mathbb{C}}$Let $\Gamma$ be a discrete group and let $M$ be a $\C[\Gamma]$-module. Let $G \lhd \Gamma$ be a finite-index normal subgroup with quotient $Q = \Gamma/G$. The conjugation ...
- 83
1
vote
1
answer
232
views
Transfer for the group of coinvariants: a reference request
Let $G$ be a group and $M$ be a $G$-module,
that is, an abelian group written additively on which $G$ acts:
$$ (g,m)\mapsto g m.$$
We consider the group of coinvariants
$$ M_G:=G/\langle g m -m\ |\ g\...
- 14.2k
5
votes
1
answer
292
views
Extension of base field for modules of groups and cohomology [duplicate]
Let $G$ be a group and let $K/k$ be a field extension. Suppose that $V$ is a $kG$-module, and let $V_K = K \otimes_k V$ be the $KG$-module given by changing the base field.
Is it true that $H^n(G,V_K) ...
- 884
3
votes
1
answer
301
views
The torsion subgroup of the coinvariants for a $G$-module
Let $G$ be a finite group and $M$ be a finitely generated $G$-module,
that is, a finitely generated abelian group on which $G$ acts.
Consider the functor
$$ (G,M)\rightsquigarrow F(G,M):= (M_G)_{\rm ...
- 14.2k
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
3
votes
0
answers
67
views
Standard terminology for "observable" subgroups of discrete groups
I've encountered in Bass, Lubotzky, Magid, and Mozes - The proalgebraic completion of rigid groups (Remark 1. p. 7) the following terminology:
A normal subgroup $N$ of $G$ is observable if every $N$-...
- 1,635
2
votes
0
answers
90
views
Regular conjugacy classes and irreducible representations in the infinite, projective case
Let $k$ be an algebraically closed field and $G$ a (not necessarily finite) group. Let $\alpha\colon G\times G\to k^*$ be a multiplier, meaning that
$\alpha(s,t)\alpha(st,r)=\alpha(s,tr)\alpha(t,r)$ ...
- 1,903
3
votes
1
answer
148
views
Vanishing of first co-homology with coefficients modular representations of small dimension
Is the following true:
For any $n$ there exists $p_0$ s.t. for any finite group $G$ of Lie type of rank $\leq n$ and characteristic $p\geq p_0$ and any (irreducible) $\mathbb F_p$ representation $V$ ...
- 2,649
15
votes
1
answer
629
views
Characteristic classes of symmetric group $S_4$
For the symmetric group $S_3$, it is classically known that \begin{equation} H^*(S_3;\mathbb{Z})\cong \mathbb{Z}[x,y]/(2x,6y,x^2-3y), \end{equation} where $|x|=2$ and $|y|=4$. Moreover, $x$ can be ...
- 439
9
votes
0
answers
315
views
Colimit of continuous cohomology over subgroups
Suppose $G$ is a profinite group, in fact in the applications I'm interested in it would be a $p$-adic analytic group similar to $GL_{n}(\mathbb{Z}_{p})$. Say $M$ is a profinite $G$-representation, ...
- 2,197
6
votes
1
answer
220
views
Relationship between irreducible representations of the Schur covering group and elements of $H^2(G,U(1))$
Let $G$ be a finite group and let $D(g)$ be a projective representation of $G$ i.e.
\begin{equation}
D(g) D(h) = e^{i \omega(g,h)} D(gh)
\end{equation}
These can be classified by the equivalence ...
- 113
9
votes
1
answer
308
views
Projective resolutions of finite-dimensional representations of infinite groups
Let $G$ be a group and let $V$ be a finite-dimensional complex representation of $G$. Question: Under what circumstances can I find a projective resolution
$$ \cdots \longrightarrow P_3 \...
- 91
6
votes
2
answers
399
views
A finite group that splits and does not split
Is there an example of a finite group $A$ that acts on a finite group $C$ irreducibly (that is, $C$ has no proper nontrivial $A$-invariant subgroup)
such that there exists an epimorphism $$\tau \colon ...
- 11.3k
7
votes
2
answers
521
views
Kazhdan constant and finite index subgroups
I am wondering if there is some general relation between Kazhdan constants of a group and it finite index subgroups?
Let $G$ be a finitely generated group with a generating set $\Sigma$ that ...
- 165
5
votes
0
answers
97
views
Is there a composite-order generalization of the homomorphism on Rep(Z/p) giving total dimension of Tate cohomology?
Let $p$ be a prime, let $\mathbb{Z}_p$ be the ring of $p$-adic integers, and let $G$ be a cyclic group of order $p$. It is rather well-known that finite rank $\mathbb{Z}_p$-free representations of $G$...
- 45.7k
4
votes
1
answer
243
views
Second cohomology of the adjoint representation
Let $p$ be a prime and let $M_p$ be the $\mathrm{GL}_2(\mathbb{F}_p)$-module of $2 \times 2$ matrices over $\mathbb{F}_p$ with trace $0$ (the action is by conjugation).
Is it true that for $p$ large ...
- 11.3k
3
votes
0
answers
184
views
Mackey Obstruction Class with Integral Coefficients
Consider an exact sequence of groups
\begin{equation}
1\rightarrow H\rightarrow K\rightarrow G \rightarrow1~.
\end{equation}
Mackey theory enables us to understand representations of $K$ in terms of ...
- 2,087
3
votes
1
answer
372
views
Reference for real and complex projective representation of finite group
I'm not a mathematician. I've only learnt about irreducible representation of finite group, symmetric group and simple Lie group. In fact, I don't know projective representation belong to which part ...
- 977
16
votes
0
answers
382
views
Representation categories and homology
Let $G$ be a finite group.
Let $\mathcal{C}=Rep-G$ be the rigid $\mathbb{C}$-linear symmetric monoidal category of finite dimensional complex representations of $G$.
Can we recover some homological ...
- 5,039
14
votes
1
answer
704
views
What is the first Pontryagin class of the $n$-dimensional representation of $S_n$?
The symmetric group $S_n$ has an $n$-dimensional defining representation, which splits as $n = (n-1) + 1$. Although this representation exists integrally, I would like to think of this as a real ...
- 54.6k
7
votes
0
answers
330
views
What's the name of the cohomology class associated to a projective representation?
Suppose $\rho : G \to PGL_n(k)$ is a projective representation of a group $G$ over a field $k$. It's classical that the obstruction to lifting this to a linear representation $G \to GL_n(k)$ is a ...
- 118k
5
votes
2
answers
1k
views
symmetric 2-cocycle / many projective representations
Let $G$ be a finite group, $k$ the field of complex numbers.
Are there (cohomologically nontrivial) group 2-cocycles $\sigma\in Z^2(G,k^\times)$ such that for all $g,h\in G$:
$$\sigma(g,h)=\...
- 1,846
5
votes
2
answers
332
views
Embedding $G$ in a $Z(G)$ extension of $\operatorname{Aut}G$
This question follows up a question I asked on math.SE. This is a refinement and a reference request.
For what groups $G$ does there exist a $Z(G)$-extension of $\operatorname{Aut}G$ (call it $\tilde ...
- 2,851
2
votes
1
answer
319
views
Successive Schur covers
Let $G_0$ be a finite group and $G_j$ a Schur cover of $G_{j-1}$ for $j=1,2,3\ldots$. Is $G_2$ equal to $G_1$? If not, will the sequence stop after finite steps in general?
- 533
5
votes
1
answer
2k
views
Is it useful to consider cohomology of group representations?
In group representation theory, one attempts to explain and classify (some of) the modules over the group ring $k[G]$, for some field $k$. In group cohomology, one develops the machinery of the ...
- 15.4k