Representation theory studies (among else) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.

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2
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1answer
47 views

The canonical surjection between the full and the reduced group C^*-algebras

This might be an incredible easy question -- since any reference I've found state it as obvious -- but anyway: Given a group $G$, I can construct the full group-$C^*$-algebra $C^*(G)$ be completing ...
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38 views

How to compute the dimesnion of the image of the tensor product of Young symmetrizers?

The following identity is contained in J. Landsberg, Tensors: Geometry and Applications, Graduate Studies in Mathematics, v.128. (p.152): $$ S^3(A\otimes B\otimes C)= S_3S_{3}S_3\oplus ...
2
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0answers
43 views

*-homomorphism and *-representation

I understand the concept of the unitary representation of G . A unitary representation of G is group homomorphism π:G→U(H) where H is a complex Hilbert space and U(H) is the group of unitary operators ...
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4answers
71 views

Representation of $S_n$ by $V^{\otimes n}$,

Let $V$ be a real and finite dimensional vectorspace. Then $$ \sigma.(v_1 \otimes \cdots \otimes v_n) := (-1)^{\sigma} v_{\sigma(1)} \otimes \cdots \otimes v_{\sigma(n)}. $$ My question: Why is this ...
3
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1answer
60 views

Irreducible representation of $\mathcal{S}_5$ over $\mathbb{C}$ of degree 4

I have come to a point where I need an irreducible representation of $\mathcal{S}_5$ over $\mathbb{C}$ of degree 4. Can somebody help me to find one and explain how to obtain one?
8
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2answers
94 views

the morphism from $SL(2,\mathbb{Z})$ to $SL(2,\mathbb{R})$

For every morphism $\rho: SL(2,\mathbb{Z}) \to GL(2,\mathbb{R})$, then $Im(\rho)\subset SL(2,\mathbb{R})$? Thanks.
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0answers
12 views

affine isometric actions and orthogonal actions

Question on comment made on page 87 in Alain Valettes book Kazhdan property (T). Let H be an affine Hilbert space, that is a set H on which a Hilbert space H^0 acts freely and transitively (as the ...
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0answers
49 views

Adjoint Representation of Lorentz Group

I'm thinking about the image under the adjoint representation $\mathrm{Ad}$ of the proper (identity connected component) Lorentz group $SO^+(1,3)$. Since this group has a trivial centre (it contains, ...
4
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1answer
45 views

restriction of unitary operator is unitary?

Let $\mathcal{U}: \mathcal{H} \rightarrow \mathcal{H}$ be a unitary operator on a Hilbert space $\mathcal{H}$. If $\mathcal{K}\subset \mathcal{H}$ is a closed subspace such that ...
2
votes
1answer
49 views

Decompose the permutation representation into irreducible representations.Construct three non-isomorphic irreducible representations from $S_3$

$S_3$ works on $\mathbb{C^3}$ with the permutation representation. I have to decompose this into irreducible representations and construct three non isomorphic irreducible representations from $S_3$ ...
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0answers
57 views

Understanding the irreducible representations of $D_3$

By the dimensionality theorem, $$\sum_i d_i^2 = |G|,$$ where $d_i$ is the dimension of the $i$th irreducible representation, we can infer that the dihedral group $D_3$ has two one dimensional irreps ...
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1answer
115 views

How can I use Clebsch-Gordan coefficients to decompose this group representation?

Let $G$ be a compact group, $\alpha$ be a unitary irrep of $G$ with carrier space $\mathcal A$, and $\beta$ be a unitary irrep of $G$ with carrier space $\mathcal B$. Then, the action of $G$ on ...
4
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0answers
61 views

Left and Right minimal homomorphisms.

In the literature on representation theory of finite dimensional algebras, a left (and similarly right) minimal homomorphism is defined as the following: For a pair of modules $L $ and $M$ in ...
2
votes
1answer
55 views

The only irrep of a group of order p over a field of characteristic p is trivial

I found an answer to my bigger question here, but I'm curious about my attempted proof in the case where $|G|=p$. I'm nearly certain this does not work, but I can still learn something from it. Do I ...
5
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2answers
113 views

Representations of the Special Orthogonal Group in Three Dimensions.

This will perhaps be an unenlightening question, but here I go. Hopefully someone can varify my thoughts. $\\$ Considering Lie Group Theory and Representation Theory, for the case of the $SO(3)$, ...
3
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0answers
24 views

Character of $N\lhd G$ with $[\chi\downarrow_N,\mathbb{I}]\neq 0$

If we let $G$ be a finite group and take $N\lhd G$. Then take $\chi$ to be an irreducible character on $G$ such that we have $[\chi\downarrow_N,\mathbb{I}]_N\neq 0$. I am trying to show that $N\leq ...
1
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0answers
32 views

What is a Complete Set of Weights of a Representation of a Lie Subalgebra?

In relation to Lie Group and Lie Algebra theory, I am studying about the weights of representations. I have come across the terminology "a complete string of weights" in my lecture course, but it is ...
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0answers
22 views

Example of projective rep being used in Clifford theory

I'm trying to understand the use of projective representations in Clifford theory, and I'd like a small example where projective representations really help, and the ingredients are actually ...
0
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1answer
45 views

Representations of two-dimensional Lie algebra

It is widely known that there is only one $2$-dimensional non-abelian Lie algebra: it can be generated by two vectors $e_1$ and $e_2$ such that $[e_1,e_2]=e_1$. Let us lenote it by $L$. The question ...
5
votes
1answer
44 views

Weight space of a representation of ${\frak sl}(2,\mathbb C)$

Suppose $(\pi,V)$ is a finite representation of $SU(2)$. Then there's an induced representation $(\pi_*,V)$ of the complexified ${\frak su}^\mathbb C(2) = {\frak sl}(2,\mathbb C)$. Show that the ...
2
votes
2answers
86 views

Nilpotent groups are monomial

I'm trying to show that a nilpotent group $G$ is monomial; i.e., that every irreducible representation $\rho$ of $G$ satisfies $\rho = \text{Ind}_H^G(\tau)$ for some $H \leq G$, $\tau$ a one ...
1
vote
1answer
101 views

Representation of $GL(V)$ on exterior algebra

I have a couple ideas for the following problem and would like verification, since I am still shaky with representation theory. Let $V$ be a $n$-dimensional vector space over a field $k$ and let ...
2
votes
0answers
44 views

Divided powers in the context of elements of the Schur algebra

I am currently reading through the paper Presenting Schur algebras as quotients of the universal enveloping algebra of $\mathfrak{gl}_2$. Here it defines the following matrices $e := ...
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1answer
62 views

Every unitary representation is a direct sum of cyclic representations.

Every unitary representation is a direct sum of cyclic representations. it can be proved without the Zorn's Lemma ?
0
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1answer
65 views

Character table from a representation?

Can anybody explain how to contruct a character table. A good explained example will be fantastic to me. For example, the character table of $S_4$. I'm quite desperate about representation theory!!
3
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1answer
31 views

A class function $f$ is a character if and only if $(f,\chi_{q_i})_G $ is a non-negative integer, for all irreducible characters $\chi_{q_i}$

I'm currently revising representation theory and I'm a bit stuck trying to prove the converse of the above statement. $(\Rightarrow)$ is straight forward because if $f$ is the character of a ...
1
vote
1answer
35 views

Showing $V\cong W$ if $dim V^H=dimW^H$

I am trying to show that if $W$ and $V$ are to $\mathbb{Q}[G]$ modules then $V\cong W$ if $dim V^H=dim W^H$ for all cyclic $H\leq G$ ( where $V^H$ denotes the invariant subspace under $H$ So I have ...
1
vote
2answers
50 views

Non-unitary representation

How to prove $\pi :\mathbb R\to \mathbb C^2$, defined by $t\mapsto \begin{pmatrix} 1 & t\\ 0 & 1\end{pmatrix}$ is a non-unitary representation? Is the following correct? $\pi$ is a ...
1
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1answer
24 views

Prove that a representation have a base and it's irreductible

I'm quite new in representations and I'm trying to do next problem: (It's supposed that I don't know anything about characters theory) We want to study $S_3=(\tau=(123),\sigma=(1,2)\,|\, ...
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0answers
32 views

$\mathbb{1}\uparrow_H^{G}$ is the permutation representation on $G/H$

Is the following correct? If we have $G$ is a group with $H\leq G$ and we take $\mathbb{1}$ to be the trivial character on $H$ then I am trying to show that $\mathbb{1}\uparrow_H^{G}$ is then the ...
1
vote
2answers
60 views

the presentation of $SL(2,\mathbb{Z})$

There is a natural presentation $SL(2,\mathbb{Z})\hookrightarrow GL(2,\mathbb{R})$, are there other presentations in real dimension 2? Or there is a classification of all the presentation of ...
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votes
5answers
288 views

$\sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2 $

I came across this result while doing some representation theory of the permutation group $S_n$ $$ \sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2 = 2 n!$$ This can be ...
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0answers
41 views

Showing that $g$ and $g^{-1}$ are conjugate iff $\chi(g)$ is real

I am trying to show that for a finite group $G$ and $g\in G$, $g$ and $g^{-1}$ are conjugate iff $\chi(g)$ is real for all $\chi$ irreducible characters of $G$. I have the following: I first want ...
1
vote
1answer
65 views

Group representations and short exact sequences

Let $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ be a short exact sequence of groups. What can be said about group representations of $B$ if we assume a complete classification of the ...
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2answers
32 views

Showing 1-dim representations factor through $G/G'$

I have a question that is as follows: Show that the 1-dim complex representations of $G$ are those that factor through $G/G'$. Now I am a bit confused by this question, what exactly does it mean ...
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0answers
51 views

Why is character sum of eigenvalues?

Working my way through a first course in Representation theory, I run into some difficulties (due to bad knowledge of linear algebra) with that said I am wondering about the following. Let $\Theta : ...
2
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0answers
46 views

Relationship between O(n)- and SO(n)-representations?

Write $O(n)$ and $SO(n)$ for the orthogonal and special orthogonal group of degree $n$ over the real numbers. Suppose that $V$ and $W$ are real, finite-dimensional and orthogonal ...
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0answers
17 views

restriction of spin representation to block diagonal subgroup

What is the restriction of the (complex) spin representation of $so(n+m)$ to the block diagonal subalgebra $so(n)\times so(m)$? A naive guess is that it is the (complex) tensor product of the two ...
0
votes
1answer
42 views

Question on Frobenious Reciprocity

I have in my notes the statement of frobenoius reciprocity in the following two ways: If $H\leq G$ and suppose that we have $\chi_1$ a character of $G$ and $\chi_2$ a character of $H$. Then: ...
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2answers
44 views

Representation of dense Subset

let $\mathcal B \subset \mathcal A$ a dense subset of a C*-algebra $\mathcal A$. I have a representation for $\mathcal B$. Can I then conclude that this is somehow also a representation for ...
0
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1answer
49 views

Generalized Clifford's Theorem

A typical statement of Clifford's theorem is the following: Let V be a finite dimensional irreducible representation of a group G, and let N be a normal subgroup of finite index in G. Then the ...
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0answers
63 views

decomposition of representation kG of G

Decompose $kG$ in to indecomposable representations and decide which summands are irreducible. (a)$G=S_2,k=\mathbb{C}$ (b)$G=\mathbb{Z}/3\mathbb{Z},k=\mathbb{C}$ ...
0
votes
1answer
91 views

The number of a set of irreducible projective characters vs the number of the ordinary characters of a finite group G.

I need valid references to show that the number of a set of irreducible projective characters with non-trivial factor set is always strictly less than the number of the ordinary characters of a ...
5
votes
1answer
32 views

“twisted” powers in symmetric monoidal categories

Suppose $C$ is a symmetric monoidal category with monoidal product $\wedge$, $X$ is a $G$-object for some finite group $G$ (say), and $T$ is a finite $G$-set of size $n$. The $n$-fold monoidal power ...
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47 views

Representation theory& module

$V$ is a left $R$ module, how do you understand the ring homomorphism $$\rho_{V}:R \to End_Z(V)$$ I know that it is like a group acting on sets, but it is very easy to understand like a group $S_n$ ...
2
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1answer
55 views

An $\mathrm{Ad}$-invariant inner product that agrees with the trace

Let $\mathfrak{g}$ be a real semisimple Lie algebra. Then, we have an obvious $\mathrm{Ad}$-invariant inner product (I don't care about positive definiteness) on $\mathfrak{g}$, namely the Killing ...
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0answers
51 views

Representing natural numbers as matrices by use of $\otimes$

What I am wanting to do is to find a unique matrix representations for Natural numbers. Say I have the number $n$, how can I represent this number as a matrix in which I can do matrix multiplication ...
3
votes
4answers
79 views

nonsemisimple $k$-algebra

Say $k$ is a field and is the $k$-algebra $A:=\prod_{i\in \mathbb N} k$ (multiplication is defined componentwise) semisimple? If not, what would be a submodule of the regular representation , that is ...
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0answers
41 views

Lie group representatiom - quasi-equivalent representations

Let $T$ and $U$ be unitary representations of a conected simply conected nilpotent Lie group, such that all irreducible subrepresentations of $T$ and $U$ are the same. If $T$ and $U$ are finite, then ...
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304 views

Fourier Transform on compact groups

I'm trying to get my head around the concept of Fourier Transform on a compact group. The standard definition is $$\widehat{f}(\pi)=\int_Gdg\,f(g)\pi(g)$$ where $\pi\in\widehat G$, the Pontryagin ...