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

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3
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45 views

Continuous subgroup of SO(3)?

I read from a paperarXiv: cond-mat/0602109 by a theoretical physicist, Prof. Frank Bais, close subgroups of $SO(3)$ is given by ${C_n,D_n,T,O,I,SO(2)\rtimes Z_2}$, where $C_n$ is the cyclic group of ...
0
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0answers
13 views

Assign a root to a irreducible representation

Given a root, e.g. $(-1 0 1 00)$ of $\text{SO}(10)$, how can I see/find to which representation of the Lie group it belongs?
1
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0answers
38 views

References to Lie algebras representaions

Could you give me a reference to a brief introduction to representations of Lie algebras, especially $\mathrm{sl}_2(\mathbb{C})$. I mean some basic Verma modules, Weyl groups etc.
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0answers
21 views

What are the relations among Whittaker functions, Whittaker functionals, Whittaker models, and Whittaker vectors?

What are the relations among Whittaker functions, Whittaker functionals, Whittaker models, and Whittaker vectors? Are there some examples which describe the relations among Whittaker functions, ...
0
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0answers
26 views

Suppose that $n\geq1$, find a representation of $G$ over $F$ with degree $n$.

Suppose that $n\geq1$, find a representation of $G$ over $F$ with degree $n$. For which groups $G$ and which values of $n$ is this representation faithful? So the representation is going to be an ...
0
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0answers
31 views

functions of positive type

We say that $∅∈\mathbb L^∞(G)$ is of positive type, iff $$ \iint f(x)\overline{f(y)} \phi(y^{-1} x)\;dy\;dx≥0 $$ for all $f∈ \mathbb L^1(G)$ The set of all continuous functions of positive type ...
1
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1answer
80 views

What are the Pontryagin duals of additive and multiplicative group of complex number?

What are the Pontryagin duals of additive and multiplicative group of complex number? So basically what are all characters of $(\mathbb{C},+$) and $(\mathbb{C^*},.)$?
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2answers
45 views

Is showing a matrix A is injective the same as showing that $A^3$=e?

Is showing a matrix A is injective the same as showing that $A^3$=e? Let G=$C_6$={1,$g$,$g^2$,$g^3$,$g^4$,$g^5$}. Is the representation determined by $$p(g)=\begin{matrix}0 & 0 & 1\\ 1 ...
1
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0answers
72 views

maximal irreducble subgroups of $SL(2,q)$

If $H$ is a maximal solvable irreducible subgroup of $GL(2,q)$ then intersection $H \cap SL(2,q)$ is maximal solvable irreducible subgroup of $SL(2,q)$. Why is it true? Maybe this is not true, but ...
0
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0answers
15 views

What is an intertwining matrix?

Sorry if this has been asked already but I am really struggling to understand what an intertwining matrix is. My current thinking is that if we have two equivalent representations $\pi$ and $\sigma$ ...
0
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1answer
15 views

Write down the irreducible representations of $G = C_2 \times C_3 = \langle g_1, g_2 \, : \, g_1^2 = g_2^3 = 1, g_1g_2 = g_2g_1 \rangle$

Write down the irreducible representations of $$ G = C_2 \times C_3 = \langle g_1, g_2 \, : \, g_1^2 = g_2^3 = 1, g_1g_2 = g_2g_1 \rangle $$ The answer I am given is: There are six $1$ dimensional ...
0
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1answer
24 views

Let G = S3 and let V be the regular F G-module. Find irreducible submodules U1, U2, U3, U4 of V such that V =U1 ⊕U2 ⊕U3 ⊕U4

Let $G = S_3$ and let $V$ be the regular $F$ $G$-module. i) Find irreducible submodules $U_1, U_2, U_3, U_4$ of $V$ such that $V =U_1$ ⊕$U_2$ ⊕ $U_3$ ⊕ $U_4$ where dim($U_1$) = dim($U_2$) = 1 and ...
0
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0answers
24 views

Give an example of a faithful representation of $D_8$ of degree 3.

Give an example of a faithful representation of $D_8$ of degree 3. So $D_8$=<$a,b : a^4=b^2=1, ab=ba^{-1}$>. A representation is faithful if ker(p)=e. The solution to this question i am given is ...
1
vote
1answer
23 views

How to show that a representation is supercuspidal?

Let $G$ be a reductive group. If we know a representation $\pi$ of $G$ explicitly, how could we determine that $\pi$ is supercuspidal or not? Are there some references about this? Thank you very much. ...
0
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1answer
43 views

Distributivity of tensor products over direct sums for group representations

I'm sure that tensor products for group representations are defined such that the typical properties are satisfied, but it would be nice to have an explicit proof, for group representations that: $$A ...
2
votes
1answer
32 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 ...
0
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0answers
33 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
votes
0answers
40 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 ...
1
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4answers
69 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
votes
2answers
89 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.
0
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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
30 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
votes
1answer
29 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
24 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
42 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 ...
0
votes
1answer
79 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
47 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
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1answer
45 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
64 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
votes
0answers
23 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
25 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 ...
4
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0answers
20 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
33 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
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1answer
42 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
65 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
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1answer
67 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
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0answers
42 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 := ...
0
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1answer
45 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
45 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
votes
1answer
23 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
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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 ...
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2answers
44 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
22 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
31 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
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2answers
57 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 ...
18
votes
5answers
261 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
37 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
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1answer
57 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 ...
0
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2answers
27 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 ...