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

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Prove that $\mathrm{Ind}_{\mathbb{I}}^G \cong \mathbb{C}[G]$

Prove that $\mathrm{Ind}_{\mathbb{I}}^G \cong \mathbb{C}[G]$. Apparently: $$\langle \mathrm{Ind}_{\mathbb{I}}^G \mathbb{I}, \chi \rangle_G \overset{Frob.Rep.}= \langle \mathbb{I}, ...
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2answers
35 views

Action of universal R-matrix of U_q(sl_2)

My question is really simple but requires a few definitions. No special knowledge of quantum groups should be needed, it is more about tensor algebra. Let $q \in \mathbb{C}$ with $q \neq 0, \pm 1$. ...
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0answers
29 views

Schur and Weyl modules.

Let $m$ be a non-negative integer and $\lambda=(\lambda_1, \cdots, \lambda_s)$ a partition of $m$. If $V$ is a vector space of dimension $n$ (over a field $\mathbb{K}$), we can consider the Schur ...
2
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2answers
49 views

Replacing entries of dice by average of it neighbours

I am interested in Representation Theory. I came across the following answer while reading this question on Mathoverflow. An example from Kirillov's book on representation theory: write numbers ...
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0answers
10 views

Inverting the the decomposition of tensor product representation into irreps

Suppose I have two unitary representations $U_V, U_W$ of a group $G$ on finite-dimensional vector spaces $V$ and $W$. I know that the tensor product representation $U_V\otimes U_W$ need not be ...
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28 views

Invariant subspace vs. irreducible subspace (terminology)

In a course in representation theory I was presented the following proposition: Let $(\pi,V)$ be a finite dimensional irreducible representation with a cyclic vector. $V$ has a unique max. proper ...
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1answer
25 views

explicit components of regular representation of $S_4$

Consider (left) regular complex representation of $S_4$. It has two 2-dimensional irreducible components. I need exact form of elements in those components (probably, having one element I may get ...
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1answer
21 views

Need definition of symmetric and antisymmetric tensor representations of a Lie algebra

I couldn't find a definitive answer online. Suppose we have a representation of a Lie algebra $(\pi,V)$. Consider the symmetric and antisymmetric vector subspaces of the $k$-th tensor product of ...
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9 views

Reordering indexed expressions (combinatorics)

To me, it appears always as a little 'magic' when people reorder expressions, indexed by highly complex combinations of permutations and I would like to know in deep and formally what really is going ...
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69 views

Why Jacobson, but not the left (right) maximals individually?

When we are working with Path Algebras, it does not need very sophisticated tools to prove that for a finite, connected, acyclic quiver $Q$, the Jacobson Radical of $KQ$ is nothing but the arrow ...
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79 views

Regge symmetry and outer automorphisms of Dynkin diagrams

Quantum $6j$-symbols are the coefficients of the change of basis matrix in the central extension of Temperley-Lieb algebra(see the book by Kauffman and Lins). It is my understanding that Ocneanu has ...
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2answers
44 views

The set of non-conjugate elements

I have $H \leq G$ where $G$ is a group. Now for any $t \notin H$ we have $H \cap tHt^{-1} = e$ Now $N$ is a set of all elements of $G$ which are not conjugate to any element of $H$ I want to ...
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2answers
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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37 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 ...
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1answer
37 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
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1answer
127 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^{*} ...
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1answer
41 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 ...
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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
50 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
49 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 ...
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1answer
78 views

Understanding the proof of Jordan-Hölder Theorem.

I need some help to understand the proof of this theorem which can be found in the book Introduction to Representation Theory by Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex ...
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30 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 ...
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1answer
39 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
57 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 ...
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1answer
64 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 ...
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1answer
21 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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1answer
26 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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39 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
36 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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30 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
146 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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25 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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35 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
48 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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70 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
86 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 ...
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2answers
49 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
57 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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45 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
30 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. ...
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1answer
90 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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36 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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17 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 ...
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2answers
55 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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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 ...