6 votes
Accepted

Dimension of the space of equivariant homomorphisms.

There are no factorials involved in the dimension of $\mathrm{Hom}_{\Bbb C[G]}(V,V)$. Let's just use Schur's lemma a bunch of times and the fact that $\mathrm{Hom}(V,U\oplus W)=\mathrm{Hom}(V,U)\oplus\...
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  • 14.5k
5 votes
Accepted

Determine the group structure from its character table: a group of order $24$ as an example

As you know that $P$ is normal, all that is left to prove that $Q_1 P = G$ and $Q_1 \cap P =1$, using the characterization of internal semidirect products. Both of these follow from order ...
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  • 14.5k
5 votes

Contradictions about simple modules of matrix rings

In a commutative ring, nilpotent elements annihilate every simple module, because any nilpotent element belongs to the Jacobson radical. This is generally false for noncommutative rings. For instance, ...
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  • 230k
4 votes
Accepted

"Twisted" Invariant Theory

This is exactly the ring of invariants of the commutator subgroup $[G, G]$, with an additional grading given by the character group $\text{Hom}(G, \mathbb{C}^{\times})$ of $G$, so all the standard ...
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3 votes

Contradictions about simple modules of matrix rings

Nilpotent elements of $R$ do not necessarily annihilate $S$. Assume for example that $S=\{ \pmatrix{a & 0 \cr b & 0}, a,b\in k\}$. This is a simple left $R$-module. Then $M=\pmatrix{ 0 & 0 ...
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  • 12.4k
3 votes

Lifts of rotations in $\operatorname{SO}(3)$ to rotations in $\operatorname{SU}(2)$

We have $\mathfrak{so}(3)\cong\mathfrak{su}(2)$, see here or Masahito Hayashi, Group Representation for Quantum Theory, 2017 (See here, equation (3.27) on page 82). The Pauli matrices $\sigma_1$, $\...
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1 vote
Accepted

Is this set of irreducible representations of a compact group finite?

Any nonzero map $\sigma \to \pi \otimes \tau$ dualizes to a nonzero map $\sigma \otimes \tau^{\ast} \to \pi$ and vice versa, so the set in question is exactly the set of irreducibles occuring in $\...
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1 vote

Motivation for studying infinite-dimensional representations of Lie algebras

The following comments are not related to physics, but might give some motivation for studying them. The reason for studying only finite-dimensional representations in the examples you described is ...
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