# Tag Info

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### Are linear representations of finite groups in one-to-one correspondence with modules over the group algebra?

Converting my comment into an answer, for a group $G$ (not necessarily finite) and a field $K$, the following categories are equivalent (actually I think even isomorphic but this doesn't really matter)...
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### Let $a$ be the reflection of the plane $\mathbb{R}^2$ over the bisector of the odd quadrants

For part (a), note that $G = \langle a, b \rangle$ indicates the subgroup generated by $a$ and $b$ (in which larger group?). So, in general, this can be rather large, since any word we write down in ...
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### The action of $SL(n,\Bbb{R})$ on the tangent space of $SL(n,\Bbb{R})/SO(n)$.

You are making certain (very interesting !) mistakes. This is not quite my field, so I apologize if I say something imprecise. The overall issue is the question of "the tangent space of $P$ at $p$...
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### Are linear representations of finite groups in one-to-one correspondence with modules over the group algebra?

There is a bijection between representations of $G$ and a $\mathbb C$-vector space $V$ with a homomorphism $\xi\colon\mathbb C[G]\to \mathrm{End}_{\mathbb C}(V)$ of $\mathbb C$-algebras. However, as ...
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### Grothendieck ring of $Rep(\mathfrak{sl}_2)$

This is a version of the formal character. It will be easier to first think about a less formal version of the character. Let's work over $\mathbb{C}$. If $V$ is a finite-dimensional representation of ...
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### Irreps of $SU(3)/\mathbb{Z}_3$ from irreps of $SU(3)$

(All representations are complex and finite-dimensional throughout except for at a handful of points where I talk about real representations.) In general, if $V$ is an irreducible representation of a ...
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### On Frobeniusâ€“Schur indicator of real/complex representations

It means both in different contexts, as far as I know. It's unfortunate but they can be disambiguated based on whether the author is discussing real or complex representations. The Frobenius-Schur ...
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### Locally compact group whose unitary irreducible reps are one dimensional

This is always true, no assumptions concerning direct integral decomposition are needed. More precisely: if $G$ is a locally compact, Hausdorff topological group and all irreducible representations of ...
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### For two irreducible modules $V$ and $W$, $f : V\to W$ is a $G$-module isomorphism $\iff \text{span}(f)\subset V^*\otimes W$ is trivial

Perhaps it is better to identify $V^* \otimes W$ with $Hom(V,W)$. The action of $G$ on $Hom(V,W)$ is then seen to be $$g \cdot f := (g \cdot f)(v) = g \cdot f(g^{-1} \cdot v)$$ Recall now the ...
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### What is the connection between double commutant theorem and group representation theory?

They are not the same result. The von Neumann double commutant theorem applies to infinite-dimensional Hilbert spaces and infinite-dimensional operator algebras acting on them, which are not ...
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