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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68 views

Why all irreducible representations appear 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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51 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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69 views

Decomposition of a representation into a direct sum of irreducible ones

I'm studying representation theory and in the book (Fulton and Harris) the author makes the following proposition with the following proof: Proposition: For any representation $V$ of a finite ...
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19 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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46 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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33 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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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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30 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 $G$-...
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113 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 $\mathfrak{g}_\...
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30 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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33 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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79 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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24 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: $$\varrho\left(e\right)=\left(\begin{...
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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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46 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 $2 ...
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$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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30 views

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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35 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 Q(\...
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47 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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23 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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75 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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75 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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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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51 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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38 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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99 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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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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20 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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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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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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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 $h(g,v)=f(g)v,\...
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28 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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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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29 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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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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21 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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10 views

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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22 views

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 representations ...
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33 views

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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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 \end{array}\right),\,\,\rho\left(a_{1}\right)=\frac{1}{2}\left(\...
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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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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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66 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) \times\...
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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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24 views

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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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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53 views

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 $x \in k^n, x\neq 0$...