# Tagged Questions

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### left regular representation of a group thru group action

Let G be a group and let $g\cdot:G\to G$(i.e.$g\cdot g'=gg'$) . This induces a permutation representation of the group. I was trying to walk thru the problems in dummite and foote. One of the problem ...
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### The Heisenberg group over $\mathbb{Z}/2\mathbb{Z}$

This is inspired by a problem from from Dummit and Foote. It asked me to calculate the order of every element in the Heisenberg group over $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$, which is defined as ...
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### Soluble group algebras and centralizers

Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and let $H$ be ...
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### Sum of degrees of irreducible complex characters for certain groups

The sum of the degrees of the irreducible complex characters (not the square sum which is the group order) is relevant do determine the dimension of a maximal torus in the group algebra. I have ...
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### Verify that two linear representations are equivalent

I've a problem in verifying that two linear representations are equivalent. First of all, I have two permutation representations of the group $G=\langle\alpha ,\beta ,\gamma\rangle$ on the set ...
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### Matrix representation associated to a permutation representation

I have just begun studying group representation theory. I don't understand how I can find the matrix representation associated to a permutation representation. Should I identify each permutation ...
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### What are the main differences among representations of $GL(n, \mathbb{R})$, $GL(n, \mathbb{C})$, and $GL(n,k)$?

What are the main differences among representations of $GL(n, \mathbb{R})$, $GL(n, \mathbb{C})$, and $GL(n,k)$? Here $k$ is a non-archimedean local field. Thank you very much.
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### Positive definite functions generated by irreducible representations — what do people call them?

Let $G$ be a group and $\pi:G\to B(H)$ be its irreducible unitary representation (one can endow $G$ with topology and claim that $\pi$ is continuous in some sense, this doesn't matter). For a given ...
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### Irreducible representations of group

I'm basically interested in $C^*$-algebras $A$, where the following conditions for a $^*$-representation $\pi$ on Hilbert space $H$ are all equivalent: 1. $\pi$ is irreducible i.e. there are no ...
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$\begin{matrix} & e& a& b& c& d&f \\ e& e& a & b& c& d&f \\ a& a& b& e& d& f&c \\ b& b& e& ... 0answers 44 views ### Family of equivalent unitary representations is not a set. I have recently come across a statement in the book: Kazhdan's property (T) by B. Bekka, P. de la Harpe, A. Valette at the beginning Appendix F.2. Fell topology on sets of unitary representations. ... 2answers 136 views ### The definition of the right regular representation I'm having difficulties understanding the definition of the right regular representation as it appears in Dummit & Foote's Abstract Algebra text. On page 132 it says Let$\pi:G \to S_G$be the ... 1answer 86 views ### Elements whose orders are multiple of$p$[closed] Let$G$be a non-solvable group,$N$an abelian minimal normal$p$-subgroup of order$p^r$with$p\notin \pi(G/N)$,$N=C_G(N)$and$K=G/N\cong A_5$. By these assumption we can conclude that$G$has ... 1answer 102 views ### Computing values of centralizers in a non-solvable group with a given property A finite group G satisfies property$P_n$if for every prime integer$p$,$G$has at most$(nā1)$non-central conjugacy classes the order of the representative element of which is a multiple of$p$. ... 0answers 70 views ### The order of the representative elements of conjugacy classes Suppose$A$is an arbitrary subset of group$G$and$K_G(A)$be the number of$G$-conjugacy classes contained in$A$. A finite group$G$satisfies property$P_n$if for every prime integer$p$,$G$... 1answer 25 views ### Question in Fulton and Harris regarding induced representation. I'm confused by the following paragraph: I don't see why$g\cdot W$depends only on the left coset$gH$. What does he mean precisely by that? Why is it true that$gh\cdot W = g\cdot(h\cdot W) = ...
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I have a problem deriving the adjoint action $ad_X(Y)=XY-YX$ from the adjoint transformation of the group on the Lie algebra. Background: The adjoint action of the Lie algebra on itself is given by ...
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### Formal proof of Clebsch Gordon sum

physicist here. When looking at the irreducible representations of $so(3)$, i.e. the set of all real valued anti-symmetric matrices, one can parametrize those irreps with an index $j$ which can be ...
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### Why is the Plancherel measure interesting?

One can average a class function $f:G\to\Bbb C$ for a finite group $G$ by interpreting $f$ as a complex-valued function on the space ${\rm cl}(G)$ of conjugacy classes and computing the expectation ...
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