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

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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. ...
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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 ...
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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 ...
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32 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 ...
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39 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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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 ...
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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?
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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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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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29 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, ...
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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 ...
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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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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 ...
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78 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 ...
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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 ...
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1answer
44 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 ...
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63 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 $SO(3)$, ...
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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 ...
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24 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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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 ...
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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 ...
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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 ...
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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 ...
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66 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 ...
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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 := ...
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44 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 ?
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44 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!!
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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 ...
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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
43 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 ...
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1answer
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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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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 ...
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2answers
56 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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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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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 ...
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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 ...
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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 ...
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33 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 : ...
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42 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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15 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 ...
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35 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
43 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 ...
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1answer
45 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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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)$G=\mathbb{Z}/3\mathbb{Z},k=\mathbb{C}$ ...
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79 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 ...
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“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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39 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 $S_n$ ...
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40 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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46 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 ...
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72 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 ...