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

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Solvability and representation of finite groups

Let $G$ be a finite solvable group, and let $G=G^{(0)}\unrhd G^{(1)}\unrhd...G^{(n)}=1$ be its derived series. Is it true that any irreducible representation of $G$ has dimension at most $n$?
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Let $\Delta$ be a group representation. Then there exists a $\Delta$-invariant $F$-subspace $V \subseteq F^n$ such that $F^n= U \oplus V$.

Assume $\left|G \right|= \left| G \right| \cdot 1_F$ is invertible in $F$. Let $\Delta:F \to GL_n(F)$ be a representation and $U \subseteq F^n$ be an $F$-subspace that is $\Delta$-invariant. Then ...
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45 views

Why all irreducible representations appeear in the regular representation?

Let $G$ be a finite group and $R$ the regular representation. That is, as a vector space $R = F(G)$ is the free vector space with basis $G$. If the basis is $\{e_g : g \in G\}$ the action is defined ...
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102 views

A Question on integration formula on $KAK$ decomposition

The following proposition appears in page 141 in Knapp's book, representation theory of semisimple groups. Let $G$ be linear connected reductive, and fix a positive system $\Sigma^+$ of restricted ...
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31 views

Weyl's theorem confusion

Weyl's theorem states that given a semisimple Lie algebra $\mathfrak{g}$, any $\mathfrak{g}$-module $V$ is completely reducible. If we consider the case of $\mathfrak{g}= \mathfrak{gl}(1)$, then ...
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18 views

Proof of unicity of decomposition of a representation

I'm studying representation theory and in the book the author makes the following proposition with the following proof: Proposition: For any representation $V$ of a finite group $G$, there is a ...
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2answers
16 views

Equivalent group representations commutative diagram

I am confused by the commutative diagram displayed above. Why is $\varphi:V \to V$ and not $G \to GL(V)$? Analogous question for the mapping $\psi: W \to W$.
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1answer
39 views

Representations of an abelian group

Let $V$ be an $F$-vector space, and let $f:G\to GL(V)$, where $G$ is a group. For $g\in G$, how can we show that if $G$ is abelian then the eigenspace of $f(g)$ is a $G$-invariant space? Moreover, ...
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30 views

Constraint for matrix representation for general irreducible permutation group.

Say I have a matrix $\bf P$ for which is ensured that $P_{ij} \in \{0,1\}$. Then consider this requirement: $$\sum_{k=0}^{n-1}{\bf P}^k[1,0,\dots,0]^T = [1,1,\dots,1]$$ Should this be enough to make ...
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1answer
103 views

Completely Reducible, Irreducible, Decomposable, Indecomposable Representation

Let $V$ be a vector space over $F$. Let $\varphi:G\to GL(V)$ be a representation. If $G$ is infinite or $\text{char }F$ divide $|G|$ or $\dim{V}=\infty$, then an irreducible representation of $G$ is ...
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91 views

Anti-involution on universal enveloping algebra of a Lie algebra.

Let $\mathfrak{g}$ finite dimentional semisimple Lie algebra and $\sigma$ the usual chevalley anti-involution that fixes the Cartan subalgebra $\mathfrak{h}$ sends the weight space ...
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Constructing an explicit isomorphism between automorphism group of bijective $F$-linear mappings and group of intertible $n \times n$ matrices

I'm going over some class notes: In the literature, sometimes a representation of $G$ over $F$ is defined as a pair $(V, \rho)$ where $V$ is a finite-dimensional $F$-vector space and $\rho: G \to ...
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1answer
27 views

Non-rational G-modules

Let me recall the definition of a rational $G$-module from M. Brions notes Introduction to actions of algebraic groups (Def. 1.6) Let $G$ be an affine group scheme over $\mathbb{C}$. A rational ...
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23 views

How to find a basis of weight vectors

I have to following Lie Algebra $L=\{x\in End(\mathbb{C}^6)\colon x^tS+Sx=0\}$, where $S=[\begin{smallmatrix} 0&I_3 \\ I_3&0 \end{smallmatrix}]$, and the subalgebra $H$ given by the diagonal ...
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75 views

algebras without identity

This problem is an exercise from Drozd-Kirichenko's book Finite Dimensional Algebras, page 29. Let $k$ be a field. Let $A$ be a $k$-algebra not necessarily with identity. Let $\overline A$ be the ...
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30 views

Problem with Molien's formula for covariants

If $E$ and $H$ are finite-dimensional faithful representations (over $\mathbb{C}$) of a finite group $G$, with $H$ irreducible. The Molien formula describer the Poincaré series of the covariants as $$ ...
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34 views

Classification of the irreducible group representations of the dihedral groups

Let $D_n$ be the dihedral group of order $2n$. Show that all irreducible representations have vector space dimension $1$ or $2$, and describe them up to isomorphism. Any hints how to even start? ...
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41 views

Computing the characters of $\mathbf{1}_{D_n} \uparrow^{S_n}_{D_n}$

How can I compute the characters of the induced representation $\mathbf{1}_{D_n} \uparrow^{S_n}_{D_n}$? Here, $S_n$ is the symmetric group over $n$ symbols and $D_n$ is the dihedral group of order $2 ...
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54 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 functiona ...
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21 views

Reduction of a representation of the Symmetric Group $S_3$

I have this representation of $S_3$ obtained in the usual way $$\varrho\left(\sigma\right)e_i=e_{\sigma_i}$$. Being more explicit the representation is this one: ...
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Differences between realizations and representation of a group

I am studying an introduction to group representation theory on my relativity class' lecture notes. I've previously learned in other classes and also on the Wikipedia article that a representation $T$ ...
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26 views

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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$G$-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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1answer
49 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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72 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
34 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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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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71 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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76 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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74 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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131 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
57 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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95 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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36 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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13 views

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
18 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
23 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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27 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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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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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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20 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 ...