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

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37 views

Is the trivial representation a subrepresentation of a tensor power of any irreducible complex representation of a finite group?

Let $G$ be a finite group, $V$ an irreducible complex representation and $\mathbb{1}$ the trivial representation. Question: $\exists n >0$ such that $\mathbb{1} \le V^{\otimes n}$?
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31 views

Characters of Linear Algebraic Groups

Reading about the semi-invariants of quivers, I see a fact which is frequently referred to in the literature, and is assumed to be trivial. However, I don't see that very easily. So, I was wondering ...
2
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1answer
35 views

Characters of (distinct) irreducible finite-dimensional representations of $A$

I need help to understand the proof of this theorem. The theorem can be found in the book Introduction to representation theory by Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex ...
3
votes
1answer
125 views

Computing character of a representation and irreduciblity

for a finite field $k$ I have $G = SL_2(k)$ a group. $H \leq G $ and $H = \lbrace $ $\begin{bmatrix} a & b \\ 0 & d\\ \end{bmatrix} \vert a,b,d \in k \rbrace $ Now $\omega : k^{*} ...
3
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1answer
40 views

To show that $A_4$ is solvable

I need to show that $A_4$ is solvable. From what i know the definition of solvable expects to give some chain of subgroups such that each subgroup in the chain is normal to the one in which it is ...
1
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1answer
29 views

Irreducible representation of $S_3$ on $\mathbb C^3$

Does there exists an irreducible representation of the group $S_3$ on $\mathbb C^3$? The representations that I can think of all have a $1$ dimensional subspaces that are fixed.
2
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1answer
49 views

Product of chracter

From Isaac's character theory book; $3.12$ Let $x\in Irr(G)$ and $g,h\in G$. Show that $$\chi(g)\chi(h)=\dfrac{\chi(1)}{|G|}\sum_{z\in G}\chi(gh^z)$$ I had thought that it was related to $3.9)$; ...
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0answers
46 views

Any continuous group homomorphism from $\mathbb{R}$ to $GL(n,\mathbb{C})$

Any continuous group homomorphism $\phi$ from $\mathbb{R}$ to $GL(n,\mathbb{C})$ is of the form $\phi (t)=exp(tX)$ for some $X\in M(n,\mathbb{C})$. Can anyone give hints for the proof of this fact? I ...
4
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1answer
61 views

Understanding the proof of the Jordan-Hölder Theorem.

I need some help to understand the proof of this theorem which can be found in the book of Introduction to representation theory by Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex ...
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0answers
27 views

Regular representation, representability of the fiber functor, and hom-distributivity for Hilbert spaces

I've culled together a slick proof of $\Bbb C[G]\cong\bigoplus_{V\in\widehat{G}}{\rm End}(V)$ (Peter-Weyl decomposition) for finite groups using the fact that the fiber functor (that is, the forgetful ...
1
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1answer
35 views

Dihedral group is supersolvable

I need to show that Dihedral group $D_n$ is supersolvable. My Approach : I think the existence of a normal chain $\{e\} = G_0 \leqslant G_1 \leqslant ... \leqslant G_n = G$ satisfying following ...
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2answers
55 views

Dimension of irreducible module divides the dimension of the algebra?

Fact: $\chi(1)$ divides order of $|G|$ where $\chi$ is an irreducible character of $G$. Above fact is equivalent to say that if $V$ is an irreducible $A=\mathbb C [G]$ module then $\dim(V)$ divides ...
2
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1answer
60 views

Is there a natural permutation representation of a wreath product of groups?

Is there a "natural" embedding of a $G \wr H$ into the group of permutation matrices? Like an element of $G\wr H$ looks like, $g=((g_1,g_2,..,g_{\vert H \vert}),h), \forall g_i \in G, h \in H$. Now ...
1
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1answer
20 views

When does a faithful representation remain faithful on a quotient representation?

Suppose I have a faithful complex representation of some finite group $(V,\pi)$. I can show that whenever this representation contains the trivial representation $(\mathbb{C}v,1)$, so that as a module ...
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0answers
10 views

If $k_{\lambda\mu}>0$ then $\lambda$ dominates $\mu$

If $K_{\lambda\mu}>0$ then $\mu ≤ \lambda$, I need to prove this by using representation theory and also using combinatorics? Although the result is obvious I have no clue how should I start the ...
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1answer
23 views

Why is the $\mu_n$ representation rational?

In their paper "On the irregularity of cyclic coverings of algebraic surfaces" by F. Catanese and C. Ciliberto, the authors consider the following situation. Let $A = V/\Lambda$ be a $g$-dimensional ...
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36 views

Problem from Serre's book

Here is the Q. from serre's representation book I tried solving this the following way let $\Phi : W \rightarrow W_0$ be the given map which takes $w$ to $f_w$ now in $W_0$ we have $f_w(h) =0 ...
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1answer
30 views

Isomorphism between an group and its double dual

I wanted to prove that for an abelian group $G$ , $\phi : G \rightarrow \hat{\hat{G}}$ is an isomorphism where $\hat{G}$ is a set of all irreducible characters of $G$ for $x \in G$, ...
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29 views

Proof of Wigner-Mackey in Serre.

The question is regarding the proof of Wigner-Mackey given as Proposition 25 of Linear Representation of finite groups by Serre. It is on page 23. The fifth line of the proof of $(b)$ on page 63 ...
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1answer
15 views

Proof that the space of morphisms between equivalent irreps has dimension 1.

Schur's lemma says that for finite group representations, this space between non-equivalent irreps has dimension 0, and that the morphisms between identical irreps are homothety. Yet I forgot how to ...
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0answers
143 views

Matrix Elements of Real Represententations

Suppose that $G$ is a finite group and we have a unitary irreducible representation $\rho:G\rightarrow \hom V$. Suppose we fix a basis $\{e_i\}_{i\geq 1}$ of $V$ and with respect to this basis we have ...
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1answer
24 views

Algebraic Indepence of Functions over Infinite Field

Can someone point in the right direction to a reference or give me an idea of the proof of the following fact. My field theory is rusty: Let $K$ be an infinite field of arbitrary characteristic. ...
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0answers
20 views

Gauge transformation laws, proof in Kobayashi & Nomizu Foundations of Differential geometry

I have two questions about this proof found in K&N's Foundations of Differential Geometry. 1) Can someone please explain how they deduce ...
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34 views

Finite matrix power over $\Bbb Z$

$p=\text{prime}$. $p[\Bbb N_{T_1\leq T_2}]=\{0\}\cup \{p^t:t\in\Bbb Z, T_1\leq t\leq T_2\}$. Given $T\in\{0\}\cup\Bbb N$, what is largest $s\in\Bbb N$ such that there is a partition $$0=T_0\leq ...
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1answer
47 views

Finite matrix power over $\Bbb F_q$

What is largest $s\in\Bbb N$ such that a matrix $M\in\Bbb F_q^{n\times n}=\Bbb F_{p^r}^{n\times n}$ could satisfy $$M^i\neq I,\quad\forall i\in\Bbb Z_+:0<i<s$$ $$M^0=M^s=I?$$
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2answers
60 views

Conceptual description of the isotypical component

This is probably rather simple but I have not found it in the literature. Consider the category $C$ of representations of a finite group $G,$ over a field $k$ of characteristic not dividing the order ...
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0answers
15 views

Normal form over $\mathbb{Z}$ of matrices of order $2$

Suppose $M \in GL_k(\mathbb{Z})$ is of order $2$. That is, $M^2 = 1$ and $M \ne 1$. Then is it true that upto a change of $\mathbb{Z}$ basis, $M$ has the form $$\begin{pmatrix}J \\ & J \\ & ...
3
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3answers
71 views

Matrices over a finite field with given Jordan normal form over the algebraic closure

Can one describe the (conjugacy classes of) square matrices over a finite field such that over the algebraic closure of this finite field their Jordan normal form consists of one Jordan block? (Such ...
1
vote
2answers
47 views

Confusion about Lie groups in Fulton & Harris

Near the beginning of chapter 8 (titled Lie groups and Lie algebras) authors motivate the definition of Lie algebra. I'm confused by two things in just one sentence: ($G$ is a Lie group) The ...
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0answers
53 views

How much we know about the Group from its Complex character table?

Suppose $G$ is a finite group and suppose that complex character table of $G$ is given.It is well known that from character table we cannot determine the Group uniquely (For example $Q_8$ and $D_8$ ...
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43 views

Considering a permutation representation of a transitive $G$-set

Suppose $X$ is a transitive $G$-set, where the size is greater than $1$, and $\pi=\pi_X$ the associated permutation representation. What is its character $\chi$? I thought that the permutation ...
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1answer
26 views

global dimension of bounded path algebra

Can someone give me some example : how to calculate de global dimension of some bounded path algebra. 1-My problem is that I do not know how to find the projective resolution of a simple module. ...
0
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1answer
80 views

Proving that $\pi_{X \times Y} \simeq \pi_X \otimes \pi_Y$

If $X$ and $Y$ are $G$-sets and $X \times Y$ is a G-set by $g \cdot (x,y)=(g \cdot x , g \cdot y)$. \pi is the corresponding permutation representation. Prove that $\pi_{X \times Y} \simeq \pi_X ...
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8 views

Dimension of a finite union of locally closed subsets

Let $X$ be an irreducible variety and $\{X_i\}_1^m$ be a finite collection of locally closed subsets, which are not necessarily disjoint. I have trouble convincing myself of the following result: ...
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0answers
29 views

Decomposition of vector spaces.

Let $V$, $W$ be finite dimensional vector spaces (over a characteristic zero field $\mathbb{K}$) and $\lambda=(\lambda_1, \cdots, \lambda_n)$ a partition of an integer $m$. Let $L_{\lambda}V$ and ...
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0answers
15 views

Questions about the indivisible imaginary root in affine root system.

I am reading the paper. On page 5, $\delta$ is defined as the indivisible imaginary root in $\widehat{\Delta_+}$. $\Lambda_0 \in \widehat{\mathfrak{h}^*}$ is the unique element satisfying $\langle K, ...
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0answers
21 views

if $\chi$ is faithful then exist $\phi$ which constitutes $\chi$

Let $\chi$ be a faithful representation of a group $G$ then prove exist a representation $\phi$ of $G$ such that $<\chi^n,\phi>=0$ I was trying to do it like, let $N=ker(\chi)$ and consider the ...
0
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2answers
52 views

Abelian groups cannot have characters of degree 2

I was attempting the following exercise: Assume that $G$ is a simple group. Let $\chi$ be an irreducible character of degree $2$, and $g \in G$ be an element of order $2$. Prove that $\chi (G) ...
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0answers
15 views

Irreducible character of degree greater than one takes value zero on some conjugacy class

It is a standard fact that irreducible character of a finite group of degree $>1$ takes value $0$ on some conjugacy class. A proof for example can be found here. I would like to know whether there ...
4
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1answer
48 views

Orbits of algebraic groups (dimension of connected components)

Let $X$ be an algebraic variety with algebraic group $G$ acting on it. Let $x\in X$. I am trying to prove that all connected components of the orbit $Gx$ are of dimension $\dim G - \dim G_x$, where ...
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0answers
19 views

How to write down the associator matrix for a given tensor product?

I've been working on a problem related to Wigner 6j-symbols for the last few months, and one thing that's bugged me is the definition of the 6j-symbol as giving the coefficients of the associator map ...
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1answer
32 views

Artin-Wedderburn theorem and square dimension

Let $A$ be a finite-dimensional simple algebra over $\mathbb{C}$ of dimension $n$. By Wedderburn's theorem, we have that $A$ is isomorphic to a matrix ring $M_r(\mathbb{C})$, which is of dimension ...
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1answer
36 views

Representations of a product of Lie groups

Let $G=G_1\times G_2$ be a product of two compact Lie groups. Is every finite dimensional irreducible representation of $G$ a tensor product of irreducible representations of $G_1$ and $G_2$? This ...
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1answer
57 views

Thm 5.19 (Brauer) in Isaacs Martin - Character Theory of Finite Groups

I would please like some help to understand the proof of thm 5.19 (Brauer) in Isaacs Martin - Character Theory of Finite Groups. It states: Let $\chi$ be a character of G with $[\chi,1_G]=0$. Let ...
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0answers
28 views

Decomposition of regular representation [closed]

Let $G$ be a group and consider its regular representation. We may uniquely decompose this representation into sums of irreducible components. What does it mean to find a basis for each component?
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2answers
72 views

Etingof Problem 5.1, “Field embeddings”

Recall that $k(y_1, \dots, y_m)$ denotes the field of rational functions of $y_1, \dots, y_m$ over a field $k$. Let $f : k[x_1, \dots, x_n] \to k(y_1, \dots, y_m)$ be an injective $k$-algebra ...
2
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0answers
33 views

$GL(V)$ representations and Schur modules.

Let $W$ be a fine dimensional complex vector space of dimension $n$ and $L_{\lambda}W$ the Schur module associate to the partition $\lambda=(\lambda_1, \cdots,\lambda_{n-1})$, where $\sum_i ...
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1answer
34 views

Size of conjugacy classes in SL(2,3)

I've been given the representations of the conjugacy classes for a group presentation $G = <x,y,z | x^2 = y^3 = z^3 = xyz>$ which is isomorphic to $SL(2,\mathbb{F}_3)$ which are: ...
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0answers
95 views

Identifying the cotangent bundle of the flag variety

Suppose $G$ is a Lie group (or I guess a linear algebraic group), $P \subset G$ a Lie subgroup with Lie algebras $\mathfrak{g}$ and $\mathfrak{p}$ respectively. In Chriss and Ginzburg's book ...
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20 views

Why is 1/2+1/2 in the weight space for SO(5)

Let's consider $\mathfrak{so}(5)$ as the Lie algebra of $\mathrm{SO}(5)$, where the symmetric bilinear form is $x_1y_5+\cdots +y_1x_5$. Then the maximal torus is given by $$\left(\begin{array}{cccccc} ...