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

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Defining a ring homomorphism in proving $|C| \cdot \frac{\chi(C)}{\dim V} $

Lemma irreducible representation $\rho: G \rightarrow GL(V)$, $C$ conjugacy class, then $$|C| \cdot \frac{\chi(C)}{\dim V} $$ is an algebraic integer. In the start of this proof we have: ...
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Isomorphism from $\mathbb{C}[G]$ to $\prod_{i=1}^h M_{n_i}(\mathbb{C})$.

What I want to ask is the proof of the Proposition 10. in "Linear Representations of Finite Groups" by Jean-Pierre Serre. Let $\rho_i : G \rightarrow GL(W_i)$ be the distinct irreducible ...
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2nd half of proof of $\dim V^G=\frac{1}{|G|}\sum_{g \in G}\chi(g) $

Lemma. Let $\rho: G \rightarrow GL(V)$ be a representation, character $\chi$. Then $$\dim V^G=\frac{1}{|G|}\sum_{g \in G}\chi(g) $$ Proof. RHS: $$\frac{1}{|G|}\sum_{g \in G}tr ...
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Spanning $\mathbb{I}$ in $V$

I have the following definition, Definition If $\rho$ : $G \rightarrow GL(V)$ is a representation we call $v \in V$ $G$-invariant if $$g \cdot v =v \ \ \forall g \in G $$ Then I have the statement ...
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Using $\displaystyle \mathbb{C}[G]\cong \bigoplus_{irreducible \ \rho}\rho^{\dim \rho}$ for $S_3$

Let $G=S_3$. $$\chi_{\mathbb{C}[G]}=(6,0,0)=1 (1,1,1)+1(1,1,-1)+2(2,-1,0)=1\chi_{\mathbb{I}}+1 \chi_{\xi}+2\chi_{\triangle} $$ since $\displaystyle \mathbb{C}[G]\cong ...
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47 views

Irreducibility of a certain polynomial associated to an irreducible representation of a finite group

Let $k$ be an agebraically closed field of characteristic 0. Let $G$ be a finite group of order $n$. A representation of $G$ is a homomorphism $\psi: G \rightarrow GL(V)$ where $GL(V)$ is the general ...
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62 views

Question about Poincare duality and homology of a cylinder.

I am reading the paper. I have some questions about Poincare duality and homology of a cylinder. On page 9, example 2.6. Let $X = \mathbb{R} \times S^1$ be a cylinder and $Y = X/(0 \times S^1 )$, ...
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117 views

Centralizer of $\mathbb{C}[G]$ in $\mathbb{C}[H]$

I found this result, but can't understand how to prove. Let $H$ be a subgroup of $G$. Then prove $Z(\mathbb{C}[G],\mathbb{C}[H])$ is commutative iff every irreducible $G$ module when restricted to ...
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110 views

Dual of a matrix lie algebra

In fact I already calculate the dual space with a formula, but I did'd understand some steps of the formula. So, I want to calculate the dual space of The lie algebra of $SL(2,R)$. Knowing that ...
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66 views

Can two different characters of $S_n$ have the same _multiset_ of values?

As I was going through various representation-theory posts in the site, I stumbled upon this one: Characters of the symmetric group corresponding to partitions into two parts. Now, that question ...
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98 views

Semisimple implies complete reducibility

Why does a semisimple Lie algebra imply complete reducibility? I have that a semisimple Lie algebra is a Lie algebra with no non-zero solvable ideals. Complete reducibility means that every invariant ...
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27 views

References request: are there some references about simple modules of group algebras?

Are there some references about constructing the simples, determining the dimensions of simple modules and describing decompositions of tensor products of simple modules of group algebras? Thank you ...
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33 views

How to show that the ordinary quiver of a semisimple algebra is a quiver consisting of isolated points?

It is said that the ordinary quiver of a semisimple algebra is a quiver consisting of isolated points? How to prove this result? Thank you very much. Edit: the ordinary quiver is the quiver defined ...
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32 views

Irreducible representation of $G=\mathbb{R}$

How can one prove that the irreducible representation of $G=\mathbb{R}$ is $e^{kx}$? ($k\in\mathbb{C}$) Thank you.
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90 views

Bourbaki's proof of normal basis theorem Part 2

Let $K/k$ be a finite Galois extension of a field $k$, $G$ its Galois group. The normal basis theorem states as follows. There exists an element $\alpha$ of $K$ such that $\{\sigma(\alpha)\ |\ ...
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Projective representation of braid group

The representation theory of braid group $B_n$ is tough, not to mention the projective representation. But my problem is simpler: how to find out all the one-dimensional projective representations of ...
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67 views

Schurs Lemma for endomorphisms

Schur's lemma states that for an irrep $(\varphi,V)$ any endomorphism $\phi: V \mapsto V$ is given by a scalar mapping. Lets say we are in the complex case, then this would mean: $\phi = \lambda I$ ...
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133 views

Irreducible Representation of a $p$-group over field of characteristic $p$ is trivial (Dummit and Foote 18.1 #22)

I'm working on question 22 of Dummit and Foote 18.1. I know this question has been answered in other posts, but I'm confused about the method this text recommends using: Let $p$ be a prime, let $P$ ...
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24 views

Which representation of $ su(N) = A_l $ is $\Gamma(1,0,0,\dots,0,0,1)$?

I was wondering which representation of the Lie algebra $ su(N) $ is $\Gamma(1,0,0,\dots,0,0,1)$? where $ (1,0,\dots,0,1)$ are the Dynkin labels of the representation. My guess is that $ \Gamma $ is ...
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How to compute the pointwise stabilizer subgroup of a fixed-point subspace?

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$. Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$. Let ...
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Basic Representation Theorey: Bijective Correspondence Between Representations (Dummit and Foote 18.1 #3)

I am working on the following question from Dummit and Foote: Prove that the degree 1 representations of $G$ are in bijective correspondence with the degree 1 representations of $G/G'$ (where $G'$ is ...
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Basic Representation Theory: One Dimensional Representation (Dummit and Foote 18.1 #2)

I'm working on question 2 in 18.1 of Dummit and Foote. The question states: Let $\phi : G \to GL_n(F)$ be a matrix representation. Prove that the map $g \to det(\phi(g))$ is a degree 1 ...
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Prove that $Q_8 \not < \text{GL}_2(\mathbb{R})$

Problem 18.1.10 in Dummit and Foote's Abstract Algebra, third edition: Prove that $\text{GL}_2(\mathbb{R})$ has no subgroup isomorphic to $Q_8$. [EA: The quaternion group]. [This may be done by ...
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106 views

Checking the definition of absolutely irreducible representations

The definition of an irreducible representation $(\rho, V) $ is one with no subrepresentations. Am I correct in saying that a absolutely irreducible means "it is irreducible over the algebraic ...
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32 views

Show that $ (φ^G )_K = (φ_{H∩K})^K $ with Mackey's theorem

Suppose H,K ≤ G e θ $ ϵ $ Char(H). Show that Z(θ)≤H. Suppose H,K ≤ G and HK = G. Se $ φ $ ϵ Char(H) show that $ (φ^G )_K = (φ_{H∩K})^K $. For the proof I have to use the Mackey's theorem. How do I ...
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188 views

Differences between primitive central idempotents and primitive orthogonal idempotents

I asked this question in mathoverflow. But it was closed. So I ask it here. If we have a complete set of primitive orthogonal idempotents of an algebra $A$, then we can obtain simple modules, ...
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54 views

Counterexample to exactness of functor from group representations to fixed points

I recently asked this question. Now, the answer there claimed that the functor $()^G:Rep_G\to Vect_{\mathbb{C}}$, where $Rep_G$ are complex representations of a group $G$, and $V^G=\{v\in V: ...
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72 views

Functor from category of group representations to space of $G$ invariants

For a representation $(V,\rho)$ of a group $G$, define the subspace of $G$-invariants by $$ V^G=\{v\in V: \rho(g)v=v\quad \forall g\in G\} $$ and want to prove the following: $V\mapsto V^G$ ...
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45 views

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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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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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 ...
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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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49 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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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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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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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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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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78 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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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
50 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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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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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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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 ...
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135 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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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
39 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.
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53 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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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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130 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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51 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 ...