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16 votes
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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 ...
Ehud Meir's user avatar
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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, ...
Piotr Pstrągowski's user avatar
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 ...
Qiaochu Yuan's user avatar
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 ...
user125639's user avatar
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 $G$...
S. Carnahan's user avatar
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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 $N$-...
Patrick Elliott's user avatar
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 ...
Clay Cordova's user avatar
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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 ...
Yassine Guerboussa's user avatar
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 $\alpha(s,t)\alpha(st,r)=\alpha(s,tr)\alpha(t,r)$ ...
geometricK's user avatar
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