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

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41 views

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

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
55 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
26 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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34 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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67 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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307 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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48 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
103 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
37 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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70 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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0answers
52 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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23 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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1answer
60 views

Question on Frobenius Reciprocity

I have in my notes the statement of Frobenius 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
48 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
58 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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1answer
73 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)$G=\mathbb{Z}/3\mathbb{Z},k=\mathbb{C}$ ...
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98 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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38 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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55 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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1answer
84 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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64 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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91 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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45 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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324 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 ...
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29 views

Representing numbers in unique p-adic to matrix representation, is there a way?

What I am wanting to do, if it is possible is find unique matrix representations for the p-adic representation of numbers. So for example, say I have the number 1365 = 3*5*7*13. Now I could take ...
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1answer
53 views

The sign representation of the Symmetric Group

I am currently trying to learn some of the basics of Representation Theory through Sagan's The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. On page 11, after ...
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0answers
42 views

What is a weight vector in layman's terms?

I am trying to understand what a weight vector is but the wikipedia is in greek. What is the jist?
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22 views

Representation vector expansion in subgroup representation vectors

Imagine I have a Group $G$ and a representation of it acting on a vector space $V_G$. Imagine that I have a subgroup of $G$ named $H$ and a representation of $H$ acting on $V_H$. Consider a vector ...
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1answer
52 views

isomorphism $\mathrm{Hom}_G(k,V)\to V^G$

Let $V$ be a finite dimensional representation of $G$ over a field $k$ and let $G$ act on $k$ trivially. The evaluation at $1\in k$ gives an isomorphism $\mathrm{Hom}_G(k,V)\to V^G$, where ...
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1answer
15 views

Consider the action of $S_3$ on $C^3 = \{ (x,y,z) | x + y + z = 0\}$. Show that $\rho$ is irreducible.

The action is defined as $\rho_g (x_1, x_2, x_3) = (x_{g(1)}, x_{g(2)}, x_{g(3)})$. For example: if $g=(12)$, then $g(2,3,-5) = (3,2,-5)$. I understand that the action just permutes the elements, ...
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66 views

Representations of Nilpotent Lie Algebras

Let $\mathfrak{g}$ be a rational, nilpotent Lie algebra. Then its adjoint representation will consist of elements which are nilpotent matrices over rationals. But this representation generally is not ...
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1answer
32 views

Dimension of intertwining space of unitary representation

I'm currently trying to read through an article by Poguntke, to be found here. The main theorem of the article is the following: Theorem. Let $\pi$ and $\pi'$ be irreducible unitary ...
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0answers
35 views

The embedding $L^2(\Gamma(N)\backslash\text{SL}_2(\mathbb{R})) \hookrightarrow L^2(\text{SL}_2(\mathbb{Q})\backslash \text{SL}_2(\mathbb{A})))$?

Let $\mathbb{A}$ be the ring of adeles. Let $N$ be a natural number. Let $\Gamma(N)=\ker(\text{SL}_2(\mathbb{Z})\rightarrow\text{SL}_2(\mathbb{Z}/N\mathbb{Z})))$. I saw the following embedding of ...
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1answer
111 views

Sum of irreducible character values in a row of the character table

If $\chi$ is a nontrivial irreducible character of $G$ (a finite group), define $S_{\chi}:= \sum_{x \in G} \chi(x)$. In terms of conjugacy classes $\mathcal{C}$, this is $\sum_{\mathcal{C}} ...
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28 views

Universality of restricted representations

Suppose that $H$ is a subgroup of a finite group $G$. Given an irreducible representation $\rho$ of $G$, this creates a (possibly reducible) representation $\rho'$ of $H$ obtained by restricting ...
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1answer
85 views

Intertwining map in Schur's Lemma

I am learning Schur's Lemma from page 4 here. It says Schur's Lemma 1. If $(\rho_1, V_1)$ and $(\rho_2, V_2)$ are irreducible representations of a group $G$, then any nonzero homomorphism $\phi : ...
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1answer
36 views

When does a sequence of finitely generated $k[G]$ modules split?

I am self studying some non-commutative algebra, and I want to make sure I don't confuse myself. Here is what I am thinking: Let A and B be finitely generated $k[G]$-algebras, for $G$ a finite group ...
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0answers
44 views

Quotient braid group as a representation of SU(n)

I am working with the quotient braid group $B_3 (3) = B_3 / \langle\sigma_1 ^3\rangle$, where I construct a vector space $V$ so that every element $a \in B_3 (3)$ has a corresponding basis vector ...
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1answer
59 views

Group algebra is a tensor product?

Am I correct in describing the group algebra $R[G]$ as $R \otimes_{Z} G$? (As a tensor product of $Z$-algebras.) There is clearly a map $R \times G$ to $R[G]$, just by sending $(r,g)$ to $rg$, and ...
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1answer
44 views

Can the equality $e^{-tY}Me^{tY} = e^{tX}M $ be shown by showing it only to 1st order? (Lie representations)

We have that A and B belong to different representations of the same Lie group. The representations have the same dimension. X and Y are elements of the respective Lie algebra representations. $$A = ...
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1answer
62 views

Representing Groups as matrices

How to represent any group as group of matrices ? Like how to represent dihedral (4) group (order $8$) as group of $2$ by $2$ matrices ? How to represent direct product of $Z_2$ and $Z_2$ as a group ...
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1answer
46 views

Injection from the permutation representation of $S_4$ to $\uparrow^{S_4}_{S_2 \times S_2}$?

Let $V$ denote the permutation representation of $S_4$. I want to know if there is an injection $\alpha: V \rightarrow \space \uparrow^{S_4}_{S_2 \times S_2} 1$. My Answer: I don't think we can find ...
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1answer
61 views

$\mathbb{C} [G] \longrightarrow \prod_{\rho} \text{End}(V_{\rho})$ an intertwining isomorphism

Consider the vector space $\mathbb C[G]$ of functions $f: G \longrightarrow \mathbb{C}$ where $G$ is a finite group, or equivalently a vector space of all formal linear combinations of elements of $G$ ...
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1answer
102 views

Conjugacy classes and centralizers of a SmallGroup

What is the complete lists of conjugacy classes and centralizers of SmallGroup(64,138)? Would someone be willing to provide the complete lists of conjugacy classes and centralizers of ...
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1answer
81 views

Which non-Abelian finite groups contain the two specific centralizers? - part II

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers isomorphic to both of these two groups (but may contain other ...
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89 views

Which finite groups contain the two specific centralizers?

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers both of these two groups: i. the elementary group $Z_2^4$, ...
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17 views

About representations and transformations under an $SU(n)$ Lie Group

I think my problem is that I misunderstand what "transforms under" really means. Let's take $SU(3)$, for the $\mathbf{3}$ with Dynkin indices $(1,0)$, a state transforms like : $ψ→gψ$. For the ...
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1answer
58 views

Obtaining representations of $G$ from $\mathrm{Lie}(G)$.

Suppose $\mathfrak{g}$ is a semisimple Lie algebra over $\mathbb{C}$, and $\tilde{G}$ is the unique connected, simply connected Lie group whose Lie algebra is $\mathfrak{g}$. Let $C$ be any discrete ...
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48 views

A construction of $\mathfrak{e}_8$ in Fulton and Harris

In section $22.4$ of "Representation Theory: A First Course" by Fulton and Harris, the exceptional Lie algebra $\mathfrak{e}_8$ is constructed using a method of Freudenthal. For background, I will ...
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2answers
170 views

What are the finite groups with 8 or 16 conjugacy classes?

What are the list of finite groups with 8 or 16 conjugacy classes? I learned that dihedral groups $D_{10}$ and $D_{13}$ have 8 conjugacy classes. (Here the order of these groups are $|D_{10}|=20$, ...