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

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Are homomorphisms into PGL related to the Schur multiplier?

I've been trying to understand homomorphisms from a finite group $G$ into $\operatorname{PGL}(n,R)$ for $n$ a positive integer, and $R$ a commutative ring with 1, usually a field. I had been under ...
4
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3answers
297 views

Matrix which commutes with permutation matrix

I'm trying to show that if $A$ commutes with all $3\times 3$ permutation matrices, then $A$ has to be of the following form: $ A = \begin{pmatrix} a & b & b \\ b & a & b \\ b & b ...
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320 views

Applications of representation theory in physics

The notes of a lecture on basic group and representation theory I attended last semester begin with a bit of motivation for the argument. They give the following examples for applications in physics: ...
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63 views

A finite-type quiver has no self-loops

I am reading through Etingof et al's notes on representation theory, and they assert in Exercise 5.4(c) on page 80 that a finite-type quiver has no self-loops. I think the way to show this is to ...
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35 views

Why is arbitrary linear representation of $S_3$ spanned by action of $\tau \in A_3$?

Quoted this question. I am reading the book on representation theory by Fulton and Harris in GTM. I came across this paragraph. [..] we will start our analysis of an arbitrary representation ...
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81 views

Splitting fields of symmetric groups

Is it true that $k$ is a splitting field of $S_n$ if and only if the characteristic $p$ of $k$ is zero or larger than $n$? The fact that the character table (over $\mathbb C$) has only integer entries ...
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1answer
40 views

What defines the dimension of a representation?

For example, if I have trivial representation of $S_3$, why does it have dimension 1? Why can't I take a vector space of dimension 2 and map all the vectors identically so I would have a ...
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45 views

faithful representation related to the center [duplicate]

Let $H$ and $K$ be two finite groups, $G = H \times K$. $\phi$ is an irreducible representation of $H$, and $\psi$ is an irreducible representation of $K$ (both representations are ...
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169 views

A representation of a finite group which is not completely reducible

Maschke's theorem says that every finite-dimensional representation of a finite group is completely reducible. Is there a simple example of an infinite-dimensional representation of a finite group ...
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1answer
47 views

Why is $\sum_{g \in G} \rho(g) =0$ for any nontrivial irreducible representation

Let $F$ be an arbitrary field, and $(\rho, V)$ be an irreducible representation of $G$. Then $$\sum_{g \in G} \rho(g) = \begin{cases} 0 & \text{ if } \rho \neq 1_G, \\ |G|1_V & \text{ if } ...
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24 views

complete irreducibility of projection representation

Let $G$ be a finite group, and $F$ be a field. $V$ is a finite-dimensional vector space over $F$. Then a group homomorphism $\rho: G \rightarrow PGL(V)$ is called a projection representation of $G$. ...
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82 views

If $\chi\in\operatorname{Irr}(G)$, $N\unlhd G$, and $\langle\chi_{N},1_{N}\rangle\ne 0$, then $N\subset \operatorname{Ker}(\chi)$.

Let $N \unlhd G$ and $\chi \in \operatorname{Irr}(G)$. Suppose that $\langle\chi_{N},1_{N}\rangle\ne 0$. Show that $N\subset \operatorname{Ker}(\chi)$. Hint: Use that, for any character ...
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118 views

Dimension of the space of algebraic Riemann curvature tensors

Given $n\in \mathbb N$, consider the vector space $\mathbb R^{n^4}$ whose elements I will denote by $(R_{abcd})$ with indices $a,b,c,d \in \{1, \dots, n\}$. This vector space is $n^4$-dimensional. The ...
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1answer
49 views

further decomposing the canonical decomposition of a representation

I am trying to learn representation theory by myself,so please pardon me if this is dumb. let V be a representation of a finite group G, $W_1,...,W_h$ be the distinct irreducible representations of G, ...
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53 views

character of finite groups

Suppose that G is a finite group. Let $\theta_n: G\rightarrow\mathbb{N}$, such that $\theta_n(g) = |\{h\in G\mid h^n=g\}|$ for all $g\in G$. Let $\chi_i$ be the distinct irreducible character of G. ...
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29 views

Existence of weights of a finite dimensional representation of a semisimple Lie algebra

Let $\mathfrak{g}$ be a semisimple complex Lie algebra. I want to show that every finite dimensional irreducible representation of $\mathfrak{g}$ is a weight module, and I need the existence of at ...
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1answer
38 views

Proving that $Hom_G (V,W)$ is 1-dimensional when $V,W$ are irreducible

Question: Let $G$ be a group. For any two representations $V,V'$ of $G$ over $\mathbb C$, let $Hom_G (V,V')$ denote the space of all linear maps $h: V\rightarrow V'$ such that $h\rho'_g = \rho_g ...
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99 views

Problem 5.15, I. Martin Isaacs' Character Theory

Isaac's Character theory of finite groups book, Problem 5.15: Let $H \subseteq G$ and suppose $\phi$ is a character of $H$ with $det(\phi)=1_{H}$. Let $\chi={\phi}^{G}$ and show ...
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1answer
105 views

bests book of representation theory for algebraic number theorists

I am looking for some of the best books on representation theory for an algebraic number theorists> I would prefer a book that is more number theoretical (e.g, galois representations, p adic ...
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96 views

Usefulness of the concept of equivalent representations

Definition: Let $G$ be a group, $\rho : G\rightarrow GL(V)$ and $\rho' : G\rightarrow GL(V')$ be two representations of G. We say that $\rho$ and $\rho'$ are $equivalent$ (or isomorphic) if $\exists ...
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207 views

Question regarding the definition of direct sum decomposition of a representation

Please bear with me. I am trying to learn representation theory of finite groups from J.P. Serre's book by myself. Here, the author has used the word 'representation' for the homomorphism $\rho : ...
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65 views

Confusion regarding direct sum decomposition of representations from Serre's book

Sorry if the question is dumb. I am trying to learn representation theory of finite groups from J.P.Serre's book by myself. In section 2.6 on canonical decomposition, he says that let V be a ...
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1answer
60 views

Eigenspace decomposition for semisimple module

Let's start with a prime $p$ and a group $\Delta$ of order prime to $p$. Let $M$ be a finite $\mathbb{F}_p[\Delta]$-module of order a power of $p$. I want to find a decomposition into eigenspaces ...
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1answer
85 views

Questions about the Space of Matrix Coefficients

Apologies in advance for the basic question: In reading up on representation theory, I came across a confusing definition for the $M(\rho)$, the space of matrix coefficients of a representation $(G, ...
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151 views

Representation problem from Serre's book

I asked this question yesterday on the setting of an exercise problem (Ex 2.8) from Serre's book "Linear representations of Finite Groups" (I'm teaching myself representation theory...) Now that that ...
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31 views

$O(3)$ and $SO(3)$ with resticted representations

I know the following facts: $O_3(\Bbb{R})\cong SO_3(\Bbb{R})\times C_2$ The irreducible representations of $SO_3(\mathbb{R})$ are only of odd dimensions, call the representations: ...
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1answer
35 views

Trace of the action of the Hecke algebra

Let $G$ be any finite group, $H$ a subgroup of $G$, and $\mathcal{R}$ the Hecke algebra associated to this data (i.e. the space of $H$-bi-invariant maps $G \longrightarrow \mathbb{C}$ with the ...
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83 views

Representation problem: I don't understand the setting of the question! (From Serre's book)

Ex 2.8 of Serre's book "Linear Representations of Finite Groups" says: Let $\rho:G\to V$ be a representation ($G$ finite and $V$ is complex, finite dimensional) and $V=W_1\oplus W_1 \oplus \dotsb ...
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1answer
110 views

Question on representation theory

This is a question from J.P.Serre's book 'Linear representation of finite groups',section 2.4 The question: Let $G$ be a finite group. Show that each character of $G$ which is zero for all $g \ne 1$ ...
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90 views

Irreducible characters form orthonormal basis of set of class functions

I am reading Serre's book (Linear Representations of Finite Groups). Theorem 6 in chapter 2 says that the irreducible characters $\chi_1,\dotsc,\chi_h$ of a finite group $G$ form an orthonormal basis ...
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Representations of $SO_3(\mathbb{R})$ from $SU_2(\mathbb{C})$

Define $V_n$ as the linear space of all homogeneous polynomials of degree $n$ in two variables $x$ and $y$. Define also the representation $\rho_n$ of $SL_2(\Bbb{C})$ on $V_n$ by: ...
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101 views

When is $R(G\times H) = R(G) \otimes R(H)$?

Suppose $G$ and $H$ are discrete groups. If $\rho_G$ and $\rho_H$ are reps of $G$ and $H$ on $V_G$ and $V_H$, respectively, then we get a rep of $G\times H$ on $V_G\otimes V_H$ by sending $(g,h)$ to ...
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1answer
106 views

Clarifications on the faithful irreducible representations of the dihedral groups over finite fields.

I would like to clarify a few things in the answer to this question: Faithful irreducible representations of cyclic and dihedral groups over finite fields 1) When a representation extends, what does ...
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98 views

When a group algebra (semigroup algebra) is an Artinian algebra?

When a group algebra (semigroup algebra) is an Artinian algebra? We know that an Artinian algebra is an algebra that satisfies the descending chain condition on ideals. I think that a group ...
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1answer
70 views

Standard representation of $O_h$ in $\mathbb{R}^3$

I want to give the standard representation of the complete octaedergroup $O_h$ in $\mathbb{R}^3$. To which group is $O_h$ isomorphic, and how to obtain a standard representation of the group? What ...
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1answer
30 views

Show the $\mathbb{C} S_3$-module of dimension 2 has $S(V \otimes V)$ is not irreducible

Consider the $\mathbb{C} S_3$-module of dimension 2, call it $V$. I want to concretely show that $S(V \otimes V)$ is not irreducible. I found a representation for $S_3$ over $\mathbb{C}$ of degree ...
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1answer
273 views

Learning Roadmap for Borel - Weil - Bott Theorem

Next semester I may study a course where the ultimate goal is to get to the Borel - Weil - Bott (BWB) Theorem, if not at least try to understand it in the case that we have $G = \text{SL}_n$. I have ...
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3answers
93 views

Extending and inducing irreducible representations

I sense this may be a simple question, but it is one I haven't been able to find an answer for, possibly due to the use of different terminology. Referring to this question: Faithful irreducible ...
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1answer
38 views

What is the smallest dimension possible for a representation of $D_8 \times Q_8$ which is faithful over $F$?

Consider $D_8 \times Q_8$, where $D_8$ is the dihedral group of order 8; $Q_8$ the quaternions. Let $F$ be a field of characteristic not equal to 2. What is the smallest dimension possible for a ...
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1answer
77 views

Bounds on Young Tableau Element locations

I'm having trouble finding some elementary results on the following. Let $Y$ be a standard Young Tableau of shape $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_n)$ with $N:=\sum_{i=1}^n\lambda_i$. My ...
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1answer
80 views

If $M \simeq N$ in ${\tt stmod}(G)$ will $M \oplus \text{(proj)} \simeq N \oplus \text{(proj)}$ in ${\tt mod}(G)$?

Let $G$ be a finite group and ${\tt stmod}(G)$ the stable module category for $G$, i.e., the category whose objects are $G$-modules and whose morphisms are $G$-module homomorphisms modulo those that ...
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Representation of Complexification of Lie Algebra

Is the following obvious? I think it is, but wanted to make sure before an exam tomorrow! "There is a bijection between the complex representations of a real Lie algebra and the complex ...
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Definition of a group representation

A representation of a group on a vector space $V$, is a group homomorphism $f: G \to GL(V)$, where $GL(V)$ is the general linear group. However, Wikipedia defines it as a map $G \times V \to V$ such ...
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89 views

The regular representation for affine group schemes

I want to understand the regular representation of an affine algebraic group. An affine algebraic group as I know it, is a functor from the category of $k $ -algebras to groups that is representable ...
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1answer
140 views

Relationship between number of conjugacy classes and number of irreducible representations of a group

For a finite group G the number of irreducible representations over an algebraically closed field F is at most the number of conjugacy classes whose sizes are coprime to the characteristic of F. What ...
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183 views

Degrees of faithful irreducible representations of $\mathbb{Z}_n$ over finite fields

I came across the following theorem while studying representation theory over finite fields. A cyclic group $\mathbb{Z}_n$ has a faithful irreducible representation of degree $d$ over $\mathbb{F}_p$ ...
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Can one reformulate tensor methods and young tableaux to account for spinor representations on $\operatorname{SO}(n)$?

Standard tensor methods and Young tableaux methods don't give you the spinor reps of $\operatorname{SO}(n)$. Is this because spinor representation are projective representations? If so, where does ...
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135 views

Show that a p-group has a faithful irreducible representation over $\mathbb{C}$ if it has a cyclic center

A p-group is a group of order $p^d$ where p is a prime. If the center has order $p^m$ (since its order must divide the order of the group) then we have a one dimensional faithful irreducible ...
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23 views

Split 3D representation of S3 in irreducible components

I saw from this post that you can prove that the 3d representation of S3 is reducible. What if I want to split this representation in a sum of irreducible representation?
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184 views

Division algebra over a an algebraically closed field

I am reading my notes and I stumbled upon a proof that I dont fully understand, and I was hoping maybe someone could clear the details. The main goal was to show that if $k$ is algebraically closed, ...