Questions tagged [irreducible-representation]

An irreducible representation of a group is a group representation that has no nontrivial invariant subspaces.

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A reductive lie algebra $L$ satisfies that $[L,L]$ is semisimple

Let $L$ be a reductive Lie algebra (i.e., $\mathrm{Z}(L) = \operatorname{Rad}(L)$). Then $\mathrm{ad} \colon L \to \mathfrak{gl}(V)$ is completely reducible, $L = [L, L] \oplus \operatorname{Z}(L)$ ...
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-2 votes
0 answers
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A theorem of irreducible character by Borus.

Let $\xi$ be an irreducible character of G and suppose $|G|=mn$ with $(m,n)=1$. Assume that $\xi(x)=0$ for all $1\neq x\in G$ such that $x^n=1$. Suppose $y\in G$ and $ y^m\neq 1$. Show that $\xi(y)=0$....
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1 vote
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Unitary irreps of a semi-direct product

I am wondering how to construct unitary irreps of the group $$(SO(3) \times SO(3)) \rtimes \mathbb{Z}_2$$ where the $\mathbb{Z}_2$ acts by swapping the two copies. This is the isometry group of $\...
3 votes
0 answers
24 views

Example of a primary representation that is not a multiple of an irreducible representation

Let $G$ be a group. A unitary representation of $G$ on a Hilbert space $H$ is called primary if for any invariant subspace $V\subseteq H$ one has that either: $V\in \{0, H\}$. There is a non-zero ...
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1 vote
0 answers
28 views

Ambiguity in unitary and completely reducible representations

I am reading the book "Group Theoretical Methods and Their Applications" by A. Fassler and E. Stiefel. There is a part that I do not understand. Suppose $\mathcal{v}$ is a finite ...
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1 vote
1 answer
31 views

Decomposing $Sym^2\mathbb{C}[X]$ without using characters

There is a nice and not too long way to decompose $Sym^2\mathbb{C}[X]$ (permutation representation of $S_5$ acting on $\{x_1,\cdots,x_5\}$) in irreducible without using characters? I know a way using ...
1 vote
0 answers
43 views

Intuition behing Mackey's irreducibility criterion for irreducibility of the induced representation.

We consider $\mathbb C$-linear representations of finite groups. If $H$ is a subgroup of $G$, and $(W,\theta)$ is an $H$-representation, irreducibility Mackey's criterion states that $Ind_H^G W$ (the $...
3 votes
1 answer
56 views

Irreducible representations of $\operatorname{SO}(3)$ from $\operatorname{SU}(2)$

I‘m having trouble understanding what exactly the irreducible representations of $\operatorname{SO}(3)$ are. I know that the irreducible representations of $\operatorname{SU}(2)$ are given by: $\rho(A)...
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2 votes
1 answer
53 views

How to complete this proof of the Schur-Weyl duality theorem using commutants?

I have a question on a proof from the following book :"Symmetry, representations and invariants by Roe Goodman and Nolan R. Wallach. Specifically on section 4.2.4 on Schur Weyl duality and the ...
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1 vote
0 answers
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A question related to the spectrum of Gelfand-Tsetlin Algebra

Let's take the chain of permutation subgroups $S_1=\{1\}\subseteq S_2\subseteq .....\subseteq S_n$,where $S_{i-1}$ sits inside $S_i$ canonically as the stabilizer of $\{i\}$.Then for this inductive ...
0 votes
0 answers
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Group algebra, cyclic group, ring isomorphism

Consider the cyclic group $G=\langle g\mid g^{3}=1\rangle$,his group algebra $\mathbb{C}G$ and $\mathbb{R}G$. As a complex algebra, how to construct an isomorphism between $\mathbb{C}G$ and $\mathbb{C}...
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1 vote
1 answer
42 views

Non atomic ring example

Can someone provide me with an example of a ring that is not atomic, that is, there is an element that has no finite factorization in irreducibles?
5 votes
0 answers
55 views

Decomposition of the representation generated by words with m copies of n letters

Let $V$ be the (complex) vector space generated by words of length $nm$ where each letter from $1$ to $m$ appears exactly $n$ times. For example, if $m=2$ and $n=3$, then $$ V = \mathbb{C}\{111222, ...
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0 votes
0 answers
49 views

Unique irreducible complex representation of Clifford algebra implies isomorphism with matrix algebra

Consider the Clifford algebra $\mathrm{Cl}(n)$ over Euclidean space $\mathbb{R}^n$ (with the standard inner product). Now, in the case that $n$ is even, it is known (cf. [1]), then $\mathrm{Cl}(n)$ ...
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1 vote
0 answers
25 views

Tensor product of two representations of $D_4$

Let $(\tau,\mathbb C^2)$ be the irreducible representation of $D_4$ by matrix multiplication, namely for every $v\in\mathbb{C}^2$: $$\begin{bmatrix}\tau(s)\end{bmatrix}v=\begin{bmatrix}-1 & 0\\ 0 &...
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