All Questions
Tagged with group-cohomology rt.representation-theory
72 questions
5
votes
3
answers
1k
views
classifying space and cohomology of integer general linear group
I have obtained that the classifying space
$$
BGL(\mathbb{R}^n)=BO(\mathbb{R}^n)=G_n(\mathbb{R}^\infty)
$$
is the Grassmannian.
I have also obtained that the mod 2 cohomology is the polynomial ...
- 1,990
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
0
answers
130
views
Are Zariski-dense representations of a cocompact complex hyperbolic lattice non-obstructed?
Question
Suppose that $\Gamma < \text{SU}(n,1)$ is a cocompact lattice, and let $\rho \colon \Gamma \to G$ be a representation to a non-compact simple Lie group (most interesting case for me: $G = \...
- 301
2
votes
1
answer
318
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
4
votes
0
answers
177
views
Rational cohomology of $S$-arithmetic groups over function fields and Gauss-Bonnet
I have a question on the ranks of rational cohomology groups of
$S$-arithmetic groups over function fields. To fix the situation, $G$
is a simple Chevalley group of rank $r$, $k=\mathbb{F}_q$ a finite
...
- 17.4k
10
votes
2
answers
405
views
H*(braid group, irrep of symmetric group) = ?
As in the title, say $\lambda$ is some irrep of the symmetric group $S_n$, and $Br_n$ the braid group on $n$ strands,
What is $H^*(Br_n, \lambda)$?
- 8,723
5
votes
0
answers
264
views
Group cohomology in dimension $-1$
This may seem like a pie-in-the-sky speculation question, but I have good reasons for asking this.
Is there any sense in which $H^{-1}(G;M)$ is defined for a group $G$ and a $G$-module $M$?
The ...
- 4,898
4
votes
1
answer
658
views
p-adic Lie group vs Lie algebra cohomology with mod p coefficients
My question concerns the cohomology of a compact $p$-adic Lie group $G$ (wich is pro-$p$).
Let $M$ be a finite dimensional $\mathbb{Q}_p$-vector space with continuous linear $G$-action.
Lazard ...
- 991
4
votes
1
answer
194
views
Uniqueness of the rank of the core of a lattice
In the paper
P.J. Webb: Bounding the ranks of ZG-lattices by their restrictions to elementary abelian groups. J. Pure Appl. Algebra 23 (3) (1982), 311-318.
the author writes in the introduction ...
- 115
5
votes
1
answer
417
views
Identifying projective representations using "gauge-invariant" traces tr[V_g V_h V_k ... ]
Background
A projective representation $V_g\in \mathrm{GL}_n(\mathbb{C})$ of a group $G$ is characterized by $V_gV_h=\omega(g,h)V_h$, where $\omega(g,h)\in\mathrm{U}(1)$ is a 2-cocycle. Changing the ...
- 244
1
vote
1
answer
274
views
Dimension of fixed points of Galois group actions
I have a question about fixed points of Galois group actions.
I am hoping that this is easy for the experts.
Let $k$ be a field of characteristic $0$. Let $K$ be a finite
Galois extension of $k$ ...
3
votes
1
answer
252
views
Large modules with non-trivial cohomology
Let $p$ be a prime and $F$ algebraic closer of $F_p$.
I want to know if it is possible to construct family of groups $\{G_i\}_{i=1}^{\infty}$ and a family of simple modules $V_i$ over $F[G_i]$ of ...
- 81
2
votes
0
answers
186
views
A local-global question on group representations
Let $G$ be a group, and let $V$ be a finite dimensional $\mathbb Q$-linear representation of $G.$ By extension of scalars we obtain the $\mathbb Q_l$-linear representation $V_l=V\otimes\mathbb Q_l.$
...
- 4,265
13
votes
4
answers
2k
views
metaplectic group does not split
I'm trying to understand the Weil representation and hope there are some experts around who can set me straight. Let $F$ be a non-Archimedean local field (I don't mind assuming that the characteristic ...
- 3,605
10
votes
2
answers
902
views
An explicit description of $\operatorname{gr}(k \cdot G)$ for the filtration induced by the augmentation ideal?
Let $A$ be any bialgebra (associative, unital, etc.) over a ring $k$. Then among other things it has a counit $\epsilon : A \to k$, and hence an augmentation ideal $I = \ker \epsilon$, which is a ...
- 54.6k
14
votes
1
answer
790
views
Noether-Deuring for injections and surjections?
Noether-Deuring theorem (not in the strongest form, but in the one I usually need):
Let $L\diagup K$ be a field extension. Let $A$ be a $K$-algebra which is finite-dimensional as a vector space over $...
- 33.8k
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
8
votes
0
answers
449
views
Relation between group representations and elements of group cohomology groups
Having already seen group cohomology, I was just introduced to the formula $U \otimes Ind W = Ind(Res(U) \otimes W)$ from representation theory. This seems oddly like the formula $\mathrm{Cor}(u) \cup ...
- 15.4k
2
votes
2
answers
529
views
Interesting representations/cohomology of surface groups?
For purposes of my own, I'm interested in constructing connected spaces, without recourse to geometric realisation or the like, that have non-trivial homotopy groups in dimension 1 and 2 and are not ...
- 35.5k
14
votes
2
answers
2k
views
Hilbert 90 for algebras
Let $L\diagup K$ be a Galois extension of fields satisfying $\left[L:K\right] < \infty$. Let $B$ be a finite-dimensional (as a $K$-vector space) $K$-algebra. Then, the Galois group $G$ of $L\diagup ...
- 33.8k
12
votes
3
answers
3k
views
Zariski tangent spaces to representation varieties
In Bill Goldman's paper "The Symplectic Nature of the Fundamental Groups of Surfaces" (Advances, 54, 200-225, '84) it is stated that the "Zariski tangent space" to a representation space Hom$(\pi, G)/...
- 8,803