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

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$G$-invariant symmetric, nondegenerate form is unique up to scalar

Let $V$ be a f.d. representation of a finite group $G$ over a field $F$. A standard argument shows there is a $G$-invariant, symmetric, nondegenerate bilinear form on $V$. If $(-,-)$ is any such ...
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What are the good references on tame hereditary algebras?

I have Thomas Brustle's Typical Examples of Tame Algebras, but I still do not have a systemic understanding of what tubes are and what regular roots of a tame hereditary algebra are. I'm also looking ...
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65 views

Finite-dimensional unitary representations of $SL_n(\mathbb{R})$

In Proposition 2.6.4 of his book Automorphic Forms and Representations, Bump is trying to prove that $SL_n(\mathbb{R})$ has no non-trivial finite-dimensional unitary representations. His argument is ...
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47 views

Characters of permutation representations for $S_4$

I am going through the lecture note How to get character tables of symmetric groups. On page 2, it computes the character table of $S_4$. The procedure starts with building the table of the ...
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67 views

Character table of $S_3 \times C_2$

I need get of character table of $S_3 \times C_2$. How to make this character table? The representation is a $\psi (g,h) = \rho_1 (g) \rho_2 (h)$ with $\deg (\rho _2) = 1$ and $\rho _1 $ ...
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1answer
31 views

Galois of character

I am readying a paper and can not understand a concept. What is $\text{Gal}(\mathbb Q(\chi)/\mathbb Q)$ where $\chi$ is a character? I know what the Galois group is like $\text{Gal}(\mathbb ...
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98 views

Reference for Harmonic Analysis?

I'm looking primarily for references for Harmonic Analysis. I'm mostly considering Doran&Fell or Deitmar, but I have access to lectures using Stein as well. The important thing is covering ...
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41 views

Faithful irreducible character and Sylow subgroup

I am trying to solve the (very nice) exercise 5.25 from Isaacs, character theory. Assume that every Sylow subgroup of $G$ has a faithful irreducible character. Show that $G$ has one also. The ...
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20 views

Equivalence between Representations

Asseume that $k$ is an algebraically closed field of a strictly positive characteristic $p$, G is a finite group of order $p$ and that $p:G \rightarrow GL(V)$ is a representation of $G$. Then $p(g)$ ...
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1answer
60 views

Representation of $Q_8$ over $\mathbb{R}$

I'm trying to solve the following problem, Give an example of a finite group $G$ and its irreducible representation $L$ over $\mathbb{R}$ such that the division algebra $Hom_G(L, L)$ is isomorphic ...
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73 views

On the converse of Schur's Lemma

Let $G$ be a finite group and $F$ a field with $\mathrm{char}(F)=0$ or coprime to $|G|$. Let $V$ be a $FG$-module in a way that every $ FG$-homomorphism $ f : V \to V $ is given by $f(x)= \lambda x ...
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73 views

some confusions about the concepts of algebra

Recently I tried to learn Algebra(Revised third edition) with the book written by Serge Lang. Since I have not covered all topics in the elegant book but now just view it as a reference for some ...
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130 views

Representation theory of locally compact groups

My knowledge about representation theory of locally compact groups is rather scattered. As I got more interested with this subject, I would like to know some good references, where I could learn the ...
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1answer
56 views

Decompose induced representation of $S_2$ and $S_3$

Let $ H=S_2 \subset G=S_3 $. Then use Frobenius reciprocity to decompose $ \operatorname{Ind}_H^G(\operatorname{sgn}_H) $ into irreducibles. $ G=S_3 $ has $ 3 $ irreps $ 1_G, ...
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47 views

Ordering on the weight lattice

When given a finite dimensional complex Lie algebra $\mathfrak{g}$ that is also semisimple and a choice of Cartan subalgebra $\mathfrak{h}$ we may talk about its weight lattice $\Lambda_{W} $ in ...
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89 views

Faithful representation of a $p$-group

Suppose $G$ is a nontrivial $p-group$. Let $H$ be the intersection of the center of $G$ and the set of elements in $G$ of exponent $p$. Let $\rho: G\rightarrow GL(V)$ be a representation. Show that if ...
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35 views

How to write $R_{ij}$ as a matrix?

Suppose that $V$ is a vector space of dimension $n$ and $R: V \otimes V \to V \otimes V$ a linear map. Then we can write $R$ as a $n^2 \times n^2$ matrix. Let $R_{ij}: V^{\otimes m} \to V^{\otimes m}$ ...
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Considering $Res^G_{H_\rho}$ instead of $G$ in quantum Fourier sampling

I am going through the proof of theorem 4 in Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. Here, they are trying to calculate the probability of measuring the ...
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1answer
15 views

Classify subrepresentations in finite dimensional semisimple representations

Quoted from "forgetfulfunctor": I'm following the notes by Prof. Etingof, linked here, and am stuck on a detail from Prop. 2.2, on page 23. To briefly recap what is in the notes, we have a ...
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1answer
22 views

How to find normal subgroups from a character table?

I know that normal subgroups are the union of some conjugacy classes Conjugacy classes are represented by the the columns in a matrix How could we use character values in the table to determine ...
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19 views

The socle of cokernel of irreducible monomorphisms in the AR quiver of type An is simple

The socle of cokernel of irreducible monomorphisms in the AR quiver of type An is simple. I believe that this result is hidden in a more general result in some articles, I tried to find a lot but ...
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12 views

Represent of multilinear function (map)

$$ f:R^{k_1}\times ...\times R^{k_n} \rightarrow R $$ is a $n$ multilinear function , $k_i$ is positive integer.Then $f$ must can be represented as $$ f(x_1...x_n)=C\prod\limits_{i=1}^n<x_i,u_i> ...
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1answer
26 views

Non-zero fixed point of some linear action on any finite group

Let $G$ be a group , $F$ be a field , $n$ be a positive integer , a map $h:G \times F^n \to F^n$ is called a linear action if there is a group homomorphism $f:G \to GL(n,F)$ such that ...
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100 views

Polynomials with $S_n \times \mathbb{Z}_2$ symmetry

Suppose that a polynomial $p(x_1\ldots x_n, y_1\ldots y_n)$ in $2n$ variables is invariant under the following operations: 1) $p(x_1\ldots x_n, y_1\ldots y_n)=p(y_1\ldots y_n, x_1\ldots x_n)$ 2) ...
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Decomposing representations

The problem I am trying to do is the following: Show that vector representation 5 and adjoint representation 10 in SO(5) decompose respectively into representations of SO(4) as: 5 →4⊕1 10→6⊕4 I ...
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25 views

Dimension of a direct sum of characters (example with $S_3$)

Here is the character table of $S_3$: I was wondering how one can determine the dimension of for example the sign character $sgn$. Could we get it from the character table? Also, if we define $A$ ...
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47 views

How does the sum of the absolute values of the diagonal entries of a matrix change when the matrix is written in a random basis?

The set-up is as follows: I have a complex, Hermitian matrix $H$ with $\mbox{Tr }H=0$, and such that the trace norm $\|H\|_1=1$ (i.e. the sum of the singular values $=1$). Let me define the function ...
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24 views

Left exact functors and long exact sequences

I wonder whether in any Abelian category $\mathcal{C}$ when we have a long exact sequence $0\to M_1\to M_2\cdots\to M_n\to 0$ and a (covariant) left exact functor $F$ we have $0\to FM_1\to FM_2\to ...
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Subrepresentation of invariants in hom space between irreducible representations

Let $\mathfrak{g}_1, \mathfrak{g}_2$ be semisimple lie algebras with irreducible representations $U$ and $W$. Write $\mathfrak{g} = \mathfrak{g}_1 \oplus \mathfrak{g}_2$ and consider both of the ...
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19 views

Lie algebra homomorphism and representation

I am solving a multiple part problem on Lie algebra representations. I have done the first three parts, but am stuck on part (iv) as follows: Define a linear map $\phi : \mathbb{g} \rightarrow ...
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What does it mean for a representation to be one-dimensional?

For example, take the Heisenberg Lie Algebra with the following basis: $X=\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$ $Y=\begin{bmatrix} 0 ...
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Degenerations of affine Hecke algebras

Consider an affine Hecke algebra $H$ corresponding to some semisimple algebraic group $G$. Let $H_{deg}$ denote the corresponding degenerate affine Hecke algebra. The algebra $H_{deg}$ can be obtained ...
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Dimension of induced representation in $S_3$

Let $G=S_3$. It has 3 irreducible representations: $1, sgn$ and $V$; the trivial rep, sign rep and rep $V$ where $dimV=2$ Consider the subgroup $H=S_2$ with irreps $1_H$ and $sgn_H$ What is the ...
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Character of a representation on $S_3$ and irreducible representations

Here is the character table of S3: Consider $V=\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ with basis $e_{ijk} := e_i \otimes e_j \otimes e_k $ Let $\pi$ be the representation of ...
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What does the set of dominant integral elements in a Cartan sub algebra look like?

I'm reading about the theorem of the highest weight: Irreducible finite dimensional representations of a complex semisimple Lie algebra (with a fixed Cartan sub algebra, $\frak{h}$ and choice of ...
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Row in the character table of $D_{10}$

Give the values of one row of the character table of $D_{10}$ corresponding to a character of degree $2$ I know the conjugacy classes of $D_{10}$, the dimensions of the irreducible ...
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45 views

Representation of Sylow which does not extend

Let $H$ be a subgroup of a finite group $G$ and $\rho$ a representation of $G$ such that the restriction of $\rho$ to $H$ is invariant under conjugation in $G$, in the sense that its character is ...
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80 views

Character of an $\mathbb{R}G$-module constructed from a $\mathbb{CG}$-module

I have been reading Representations and Characters of Groups by Gordon James and Martin Liebeck. I encountered the following construction of an $\mathbb{R}G$-module from a $\mathbb{C}G$-module. ...
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1answer
19 views

Irreducible representation of $S_3$

How can I show that this representation of $S_3$ is irreducible? $$\rho\left(e\right)=\left(\begin{array}{cc} 1 & 0\\ 0 & 1 ...
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Decompose $V \otimes V \otimes V$ into irreducible representations of $SL_2(\mathbb{R})$

Let $V=\mathbb{C^2}$ be the standard representation of $SL_2(\mathbb{R})$ Decompose $V \otimes V \otimes V$ into irreducible representations of $SL_2(\mathbb{R})$ I will just consider ...
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A question of the paper“Crystallizing the q -Analogue of Universal Enveloping Algebras”?

I'm reading the paper "Crystallizing the q -Analogue of Universal Enveloping Algebras" written by Masaki Kashiwara. But there is something I don't know. Can anyone tell me how to use the ...
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2answers
64 views

Prove $\exp(\mathrm{Tr}(X))=\det(\exp(X))$

Show that $\exp(\mathrm{Tr}(X))=\det(\exp(X))$ where $X$ is a matrix using the concept of the Jordan normal form I realised this formula by considering that: $\det(\exp(X))=\exp(\lambda_1) ...
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elments of a linear algebraic group agreeing on a vector

Let $G \subset \mathrm{GL}_n(k)$ be a connected affine algebraic group over a field $k$ with the following property: for any two distinct elements $g,h \in G$ there exists a vector $x \in k^n, x\neq ...
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Bi-character for finite, commutative monoids?

If I have a finite commutative monoid $M$ (which is not a group), is it possible to get a bi-character on this? By bi-character, I mean a map $\beta:M\times M\rightarrow \mathbb{C}^*$ such that, ...
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40 views

What is the space $\operatorname{Sym}^2(V)$ and how does it act on the vector space $V$?

If $V$ is a vector space over $\mathbb{C}$ with basis vectors $e_i$, what is the space $\operatorname{Sym}^2(V)$? I am hoping someone can give me some insight into this space; perhaps by ...
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Representation decomposition over $GL_2(\mathbb{C})$

I have found that $Sym^2(V) \otimes Sym^2(V)$ can be decomposed over the special linear group as follows: $Sym^2(V) \otimes Sym^2(V) \simeq Sym^4(V) \oplus Sym^2(V) \oplus 1$ This is done using the ...
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1answer
24 views

Permutation and Linear Representation of Finite Group

By a permutation representation of a finite group $G$, we mean a homomorphism from $G$ to $S_n$, the (full) permutation group on $n$ letters. By a linear representation of a finite group $G$, we mean ...
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Find an irreducible extension of an irreducible representation

Let $G$ be a finite group and $C$ the center of $G$. Let$μ:C→F^×$ be a character of $C$. Prove that there is an irreducible representation $ρ : G → GL(V )$ such that $ρ(c)(v) = μ(c)v$ for all $c ∈ C$ ...
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Semidirect Products and Representations

Assume that $G$ and $H$ are two $p$-groups and $k$ a field of $char(k)=p > 0$. Also, assume that i denote by $G \rtimes H$ the semidirect product of the above groups. Do you know if there is any ...
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1answer
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Witness for $2-$dimensional irreducible representation of $Q_8$ over an algebraically closed field

Show that the $2-$dimensional irreducible representation of $Q_8$ can be realized in the space $V$ of functions $f : Q_8 → F$ such that $f(gi)= \sqrt{−1}f(g)$ (the action of $G$ is by right ...