All Questions
Tagged with rt.representation-theory group-cohomology
72 questions
3
votes
0
answers
130
views
Galois cohomology and Levi subgroups
Let $F$ a field and $G$ a smooth connected reductive group with a Levi subgroup $M$. Under what assumptions is $H^1(F, M) \to H^1(F, G)$ injective? In the case $F$ is nonarchimedean local I believe ...
- 605
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 ...
- 12.9k
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
$$ M_G:=G/\langle g m -m\ |\ g\...
- 5,898
12
votes
2
answers
467
views
Example of a finite group $G$ with low dimensional cohomology not generated by Stiefel-Whitney classes of flat vector bundles over $BG$
In Stiefel-Whitney classes of real representations of finite groups, J. Algebra 126 (1989), no. 2, 327–347, Gunawardena, Kahn and Thomas dealt with the question, whether the cohomology ring $H^...
- 17.4k
5
votes
1
answer
292
views
Extension of base field for modules of groups and cohomology [duplicate]
Let $G$ be a group and let $K/k$ be a field extension. Suppose that $V$ is a $kG$-module, and let $V_K = K \otimes_k V$ be the $KG$-module given by changing the base field.
Is it true that $H^n(G,V_K) ...
- 3,054
4
votes
1
answer
194
views
Uniqueness of the rank of the core of a lattice
In the paper
P.J. Webb: Bounding the ranks of ZG-lattices by their restrictions to elementary abelian groups. J. Pure Appl. Algebra 23 (3) (1982), 311-318.
the author writes in the introduction ...
- 2,821
24
votes
0
answers
813
views
Revising the proof of CFSG
This is an oft-quoted excerpt from John Thompson's article "Finite Non-Solvable Groups":
“... the classification of finite simple groups is an exercise in taxonomy. This is
obvious to the ...
- 1,433
3
votes
1
answer
301
views
The torsion subgroup of the coinvariants for a $G$-module
Let $G$ be a finite group and $M$ be a finitely generated $G$-module,
that is, a finitely generated abelian group on which $G$ acts.
Consider the functor
$$ (G,M)\rightsquigarrow F(G,M):= (M_G)_{\rm ...
- 12.9k
2
votes
1
answer
361
views
Computation of $H^*_c(\mathcal{M}_{1,[2]},\mathbb{V}_1)$
As a term of a Serre spectral sequence, I would like to compute the cohomology group with compact support $H^*_c(\mathcal{M}_{1,[2]},\mathbb{V}_1)$ of the moduli space of genus $1$ curves with $2$ ...
- 40.2k
28
votes
4
answers
3k
views
Yoga of six functors for group representations?
I'm trying to understand how the six functor philosophy applies to representation theory. Consider the category of classifying stacks $BG$ (assume $G$ discrete for simplicity). To every stack we can ...
- 21.3k
3
votes
1
answer
308
views
Projective dimension of group ring
Assume that $G$ is a group and $R$ is a p.i.d. What can we say about the projective dimension of $R[G]$? For example can we say that this dimension is at most $1$ for reductive groups? (I think if $...
CommunityBot
- 1
3
votes
2
answers
207
views
Classification of crossed $G$-algebras
Added later: As Viktor Ostrik points out in a comment, what I'm looking for is a classification of so-called crossed $G$-algebras corresponding to homotopy TQFTs with homotopy target space $K(G, 1)$ ...
- 3,001
4
votes
0
answers
98
views
Is there a cohomological interpretation of the bilinear form arising from Clifford theory?
For this question all groups are finite, and representations are over $\mathbb{C}$. The setup is that we have $N$ a normal subgroup of $G$, with abelian quotient $A$, and an irrep $V$ of $N$ that is ...
- 1,949
3
votes
1
answer
172
views
On the linearizability of the action of a finite group on a formal polydisc
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gl{Gl}$Let $\mathcal{D}=\mathbb{C}[[t_{1},\dotsc , t_{n}]]$ be a formal polydisc over $\mathbb{C}$, and $G$ be a finite group. On Lemma 7.8 of ...
- 148k
18
votes
1
answer
1k
views
Explanation for $\chi(\operatorname{SL}_2(\mathbb{Z})) = -1/12$ with zeta function
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$One can compute the (group cohomological) Euler characteristic of $\SL_2(\mathbb{Z})$ via
$$ \chi(\SL_2(\mathbb{Z})) = \chi(\mathbb{Z}/2) \...
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 ...
- 63.9k
3
votes
1
answer
565
views
Finiteness of cohomology group
Suppose $G$ is a finite Galois group, and $M$ is an infinite $G$-module. When can I say that $H^1(G, M)$ is finite?
I know this not true in general. Is it true under certain assumptions on $M$?
To be ...
- 143
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$-...
- 12.9k
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
9
votes
0
answers
232
views
A conjecture of Adams on the mod p cohomology of classifying spaces of Lie groups
In the paper On the mod p cohomology of BPU(p), the authors say that there is a conjecture of J. F. Adams as follows:
Conjecture (J. F. Adams) Let $G$ be a compact connected Lie group, and let $p$ be ...
- 3,051
20
votes
0
answers
445
views
Which classes in $\mathrm{H}^4(B\mathrm{Exceptional}; \mathbb{Z})$ are classical characteristic classes?
Let $G$ be a compact connected Lie group. Recall that $\mathrm{H}^4(\mathrm{B}G;\mathbb{Z})$ is then a free abelian group of finite rank. Let us say that a class $c \in \mathrm{H}^4(\mathrm{B}G;\...
- 54.6k
4
votes
0
answers
192
views
Extended double 2-cocycle conditions: Mathematical structure behind?
Note: For experts, to save your time, you can just read the highlighted texts and Eqs directly.
The ordinary group 2-cocycle condition:
Let us remind the usual so-called homogeneous group 2-cocycle $...
- 10.5k
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$ ...
- 2,821
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 ...
CommunityBot
- 1
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 ...
- 1,382
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 ...
- 45.7k
5
votes
0
answers
109
views
Restricting projective representations of Lie groups to lattices
Let $G$ be a simple Lie group with trivial center, and let $\Gamma$ be a lattice in $G$. Is it true that an infinite-dimensional projective representation of $G$ restricted to $\Gamma$ can be de-...
- 229
6
votes
1
answer
289
views
Calculation of the Schur multiplier of $\mathbb Z^2$
Consider a projective representation of $\mathbb Z^2$ with $U(1)$ coefficients. I would like to find the covering group corresponding to this representation. For this, one needs to find the ...
- 66.3k
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 ...
- 12.9k
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 \...
- 38.6k
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 ...
- 37.4k
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 ...
- 25.5k
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
13
votes
0
answers
266
views
Group homology $\mathrm{SL}_2$ acting on $\mathrm{Sym}^g$
Let $k$ be a field. We write $\mathrm{Sym}^g(k^2)$ for the $g$-th symmetric power of the (a?) standard representation of $\mathrm{GL}_2(k)$ ($g\geq 0$ an integer). Here I consider $\mathrm{Sym}^g(k^2)$...
- 408
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 ...
- 37.4k
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
0
answers
185
views
Classifications of projective representations of $SO(N)$ and $SO(N)\times Z_2^T$
This question is motivated by some physics questions: what are the classifications of projective representations of groups $SO(N)$ and $SO(N)\times Z_2^T$? This is equivalent to asking what are $H^2(...
- 197
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
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 ...
- 31.5k
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 ...
- 5,039
5
votes
1
answer
419
views
Eisenstein cohomology - explicit computation and relation to Franke's trace formula
Let $G$ be a reductive group over $\mathbb{Q}_p$. Let $X_G$ be a locally symmetric space associated to the group $G$, and let $\partial X_G$ be the Borel-Serre boundary of $X_G$. The space $\partial ...
CommunityBot
- 1
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 ...
- 10.9k
6
votes
1
answer
377
views
(ir)reducibility of projective representation when restricted to abelian subgroup
I am looking for a finite group $G$ and an irreducible projective representation $\rho: G \to PGL(\mathbb C^n)$ such that for any abelian subgroup $A\subset G$, the restricted representation $\rho|_A$ ...
- 44.4k
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)/...
- 8,529
18
votes
0
answers
532
views
A cohomology class associated with a complex representation of a group
$\newcommand\CC{\mathbb C}\newcommand\ZZ{\mathbb Z}\newcommand\ad{\mathsf{ad}}\newcommand\Ext{\operatorname{Ext}}$ Suppose that $G$ is a finite group and that it acts on a finite dimensional complex ...
CommunityBot
- 1
5
votes
0
answers
89
views
Generalization of a 1D unitary representation
Probably a very naive question, but I'd be grateful for any input. Consider 1D representations of finite group $G$: $\chi(g) \chi(h) = \chi(g h)$ with $\chi(g)\in \text{U}(1)$, and $\chi(1)=1$. The ...
- 51
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 ...
- 6,349
10
votes
1
answer
2k
views
The term $H^1(N,A)^{G/N}$ in the inflation-restriction exact sequence
[a repost from SE due to the lack of response]
Given a group $G$, let $A$ be a $G$-module and let $N\trianglelefteq G$.
If I understand it correctly, the superscript "G/N" in the third term of the ...
- 6,792