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

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2
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What are the main differences among representations of $GL(n, \mathbb{R})$, $GL(n, \mathbb{C})$, and $GL(n,k)$?

What are the main differences among representations of $GL(n, \mathbb{R})$, $GL(n, \mathbb{C})$, and $GL(n,k)$? Here $k$ is a non-archimedean local field. Thank you very much.
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0answers
56 views

Prove the Weyl's complete reducibility Theorem on finite-dimensional $\mathfrak{g}-modules$ by Kostant's $\mathfrak{n}$-cohomology result

I've met an exercise in Kumar's book ("Kac-Moody Groups, their Flag Varieties and Representation Theory", Chapter III, page 89, Ex. 3.2. E, (1) & (2)). But I have no idea about its proof. Any ...
9
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2answers
223 views

History of the matrix representation of complex numbers

It is well-known to many that $\mathbb{C}$ can be represented by matrices of the form $\left[ \begin{array}{cc} a & b \\ -b & a \end{array} \right]$. For example, see this question or this ...
3
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1answer
28 views

Direct Sum Decomposition of Semisimple Algebra by Schur's Lemma

In Remark $3.1.2$ of Etingof's Introduction to Representation Theory, it says that a semisimple representation $V$ of an algebra $A$ is isomorphic to $\oplus_X Hom_A(X,V)\otimes X$ running over all ...
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0answers
55 views

Positively graded k-algebra

Suppose we have a positively graded $k$-algebra $A=\bigoplus_{i\ge 0}A_i$, such that $A_0$ has finite global dimension. Furthermore, all $A_i$ are finite dimensional and $A$ is generated in degree $0$ ...
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50 views

Isomorphic Dual and Conjugate Representations of a Lie Algebra

Let $\frak{g}$ be a complex Lie algebra $\frak{g}$, and $R:\frak{g} \to $End$(V)$, a representation for some finite dimensional complex vector space $V$. As is well-known, we can construct from $R$ ...
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18 views

Irreducible representations of the Lorentz Lie algebra

I am reading about the irreducible representations of the Lorentz Lie algebra. The author states that the general irreducible representation is given by $d^{(j_1,j_2)}(J_i) = d^{(j_1)}(T_i) \otimes ...
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0answers
47 views

Positive definite functions generated by irreducible representations — what do people call them?

Let $G$ be a group and $\pi:G\to B(H)$ be its irreducible unitary representation (one can endow $G$ with topology and claim that $\pi$ is continuous in some sense, this doesn't matter). For a given ...
2
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0answers
34 views

What does it mean for a representation to contain a character?

I'm trying to understand the statement "The representation $\pi$ contains the trivial character of $N$ if and only if it contains an irreducible representation $\sigma$ of $B$ containing the trivial ...
0
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0answers
21 views

Irreducible representations of group

I'm basically interested in $C^*$-algebras $A$, where the following conditions for a $^*$-representation $\pi$ on Hilbert space $H$ are all equivalent: 1. $\pi$ is irreducible i.e. there are no ...
2
votes
1answer
50 views

Is the coefficient ring $R$ of a group ring $RG$ necessarily projective as an $RG$-module?

So this may be a trivial question but I am new to the idea of group rings. Suppose we have a ring $R$ and a group $G$, I was wondering if the trivial $RG$-module $R$ is projective? In which case, how ...
0
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1answer
45 views

Irreducible module

I've met the following definition of irreducible module: an $R$ module $M$ is said to be irreducible if it contains no proper submodules: in other words, if $N \subset M$ is a submodule than either ...
1
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0answers
38 views

Show that isotropic function S(A) and A have same eigenvectors

Given $\boldsymbol{A}$ is a positive definite, symmetric second order tensor and $\boldsymbol{Q}\boldsymbol{S}(\boldsymbol{A})\boldsymbol{Q}^T = \boldsymbol{S}(\boldsymbol{QAQ}^T)$ $\forall ...
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0answers
27 views

Is a representation of a Lie group determined by its weight diagram?

I am reading about representations of $\mathfrak{su}(3)$. The author claims that $\mathbf{3}\otimes\bar{\mathbf{3}} = \mathbf{8}\oplus\mathbf{1}$, where $\mathbf{3}$ is the fundamental ...
0
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1answer
54 views

Group Theory- S3 table

$\begin{matrix} & e& a& b& c& d&f \\ e& e& a & b& c& d&f \\ a& a& b& e& d& f&c \\ b& b& e& ...
10
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2answers
185 views

Applications of Algebra in Physics

Often I have heard about the link between Algebra (in particular Representations of Groups and Algebras) and some "indefinite" field of Physics. I have a good preparation in Algebra and ...
6
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1answer
88 views

Representations of $\mathfrak{su}(3)$

I am confused about the notation for representations of $\mathfrak{su}(3)$. Often a bold number is used to denote a particular representation e.g. $\mathbf{3}$ is used to denote the fundamental ...
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47 views

Family of equivalent unitary representations is not a set.

I have recently come across a statement in the book: Kazhdan's property (T) by B. Bekka, P. de la Harpe, A. Valette at the beginning Appendix F.2. Fell topology on sets of unitary representations. ...
4
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0answers
43 views

When does $n$-dimensional algebra have $m$-dimensional faithful representation?

Suppose we have an $n$-dimensional associative unital algebra $A$ over a field $k$ (assume $\operatorname{char}(k)=0$ and maybe even $k$ is closed). I would like to know what is the minimal ...
1
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1answer
22 views

Can any $\theta \in \text{Hom}(S^\lambda,M^\mu)$ be written as $\theta = \kappa_t$?

I am currently working on Bruce Sagan's The Symmetric Group. I am struggling to understand why the following proposition should be true. Suppose that the field of scalars is $\mathbb{C}$ and ...
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0answers
34 views

Dimension of the Image of Young Projectors corresponding to Tensor factors.

Suppose I define the action of the symmetric group on abstract tensors as shuffling indices. I know this is very naive. I apologise, I am a physicist and working on a problem that involves tensors ...
4
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2answers
97 views

Identifying the algebra

In order to solve an obscure (physics) problem I have been considering whose details are not important, I am looking for elements (I am thinking in terms of matrices and their products but this may ...
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0answers
27 views

I want to decompose a tensor product using Littlewood-Richardson rule, How do I find the component of this in each irreducible space?

Let me set up the notation I am using. $(abc,de)$ denotes the standard Young tableau where the first row is $abc$ and the second row is $de$. Each young tableau corresponds to the young symmetriser, ...
2
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0answers
29 views

Why do we have $\{s\} = \pi\{t\}$ for some $\pi \in C_t$?

I am currently working on Bruce Sagan's The Symmetric Group. In the proof of Corollary 2.4.2, the book says Let $t = t^\lambda$ be a $\lambda$-tableau and $s = s^{\mu}$ be a $\mu$-tableau, where ...
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1answer
30 views

Repeated Irreducible Representations in a representation

I'm reading through Serre's - Linear representations of finite groups. He has the following theorem (theorem 4 of chapter 2): Let $V$ be a linear representation of $G$, with character $\phi$ and ...
0
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1answer
46 views

Finding a lie group structure on $\mathbb R^n\setminus\{0\}$

I want to find all maps $g: \mathbb R^n\setminus \{0\} \rightarrow GL_n(\mathbb R)$ which satisfy the properties $g$ is differentiable and injective $g(g(a)b) = g(a)g(b)$ for all $a,b\in\mathbb ...
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0answers
24 views

Presentation for Kernel of Induction map from $\oplus R(H)$ to $R(G)$

This is an exercise in Serre's representation theory book: Suppose that $X$ is a family of subgroups of $G$ stable under conjugation and taking subgroups Let $N$ be the kernel of map $\mathbb{Q} ...
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0answers
27 views

Signed column sum for Young tableau $t$

I am currently working on the book The Symmetric Group by Bruce Sagan. The following passage comes before introducing Specht Modules: Suppose that the tableau $t$ has rows $R_1, R_2, ..., R_l$ ...
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1answer
35 views

Irreducible Representations of $<X,Y>/\{[X,Y]=Y\}$

I was doing exercises from Etingof's Introduction to Representation Theory and came across this problem. $2.16.2$ Find all irreducible representations of the Lie algebra $L$ with generators $X$ and ...
1
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1answer
33 views

Basic Manipulation of Adams operations in R(G)

This is part of an exercise in Serre's representation theory book I am self-studying, but mostly it is about manipulation of symmetric polynomials. Let $\rho$ be a representation of a finite group ...
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25 views

Dimension of a weight space which is of weight $0$.

Let $V$ be a module of a Lie algebra $\mathfrak{g}$ and $V_{0}$ be the weight space of $V$ of weight $0$. $$ V_0 = \{ v\in V: h.v = 0, h \in \mathfrak{h} \}, $$ $\mathfrak{h}$ is a Cartan subalgebra ...
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0answers
27 views

Matrix representation of Boolean algebra?

Is there such a thing as matrix representations of Boolean algebra? Give a boolean algebra with finite elements {a,b,c...} and operations $\cap, \neg$, we can regard $\cap$ as matrix multiplication ...
0
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0answers
21 views

How to compute $\lambda(h_i)$?

Let $\lambda$ be a weight and $h_i = h_{\alpha_i} \in \mathfrak{h}$, $\alpha_i$ is a simple root. $\mathfrak{h}$ is a Cartan subalgebra of a Lie algebra $\mathfrak{g}$. How to compute $\lambda(h_i)$? ...
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1answer
53 views

Classification of separable algebras up to Morita equivalence

Is there a simple classification of separable algebras up to Morita equivalence, working over a particular field $k$? For example, over $\mathbb{C}$, every separable algebra is Morita equivalent to ...
3
votes
2answers
156 views

The definition of the right regular representation

I'm having difficulties understanding the definition of the right regular representation as it appears in Dummit & Foote's Abstract Algebra text. On page 132 it says Let $\pi:G \to S_G$ be the ...
0
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0answers
28 views

Measure in dual group - Kirillov theory

Let $G$ be a nilpotent connected, simply connected lie group. With the orbit method Kirillov describes the classes of equivalence of all irreducible unitary representations. Hence one identifies the ...
3
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1answer
52 views

Question on unitary representation of non-compact simple Lie groups

The following is an exercise appearing page 148 in Knapp's book, representation theory of semisimple groups. Let $G$ be a connected linear non-compact Lie group with simple Lie algebra $\mathfrak g$. ...
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1answer
94 views

Elements whose orders are multiple of $p$ [closed]

Let $G$ be a non-solvable group, $N$ an abelian minimal normal $p$-subgroup of order $p^r$ with $p\notin \pi(G/N)$, $N=C_G(N)$ and $K=G/N\cong A_5$. By these assumption we can conclude that $G$ has ...
5
votes
1answer
78 views

Decomposing a matrix representation

I am currently working on the following problem: Assume that $X$ is a reducible matrix representation of the form \begin{equation} X(g)=\left( \begin{array}{c|c} A(g) & B(g)\\ \hline ...
3
votes
1answer
106 views

Computing values of centralizers in a non-solvable group with a given property

A finite group G satisfies property $P_n$ if for every prime integer $p$, $G$ has at most $(n−1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...
2
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1answer
25 views

precise definition of “irreducible representation” (of associative algebras with unit)

Let $K$ be a field and $A$ an associative $K$-Algebra with unit. By a representation of $A$ I mean a homomorphism of $K$-Algebras with unit $f\colon V\rightarrow{End}_K(V)$ where $V$ is a finite ...
3
votes
2answers
86 views

Is the tensor product of two representations a representation?

I am a little bit uncertain about an argumentation showing that a given map of a topological group is somehow obviously continuous. In the following I will rely on the book of Anthony W. Knapp „Lie ...
3
votes
1answer
41 views

Simultaneous diagonalisable matrices

I am well aware that there are already several questions and posts regarding the following topic. However, I could not find any answer to the following problem in Bruce Sagan's book The Symmetric ...
0
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1answer
31 views

Unitary matrix for matrix representation

In the book The Symmetric Group the author says: Let $\chi$ and $\psi$ be characters of the $G$-module $V$. By picking an orthonormal basis for $V$, we obtain a matrix representation $Y$ for ...
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5answers
118 views

Getting an intuitive feel for induced representations

I'm reading about induced representations for research. Particularly, I'm trying to get a firm grasp on the finite group case before venturing on to the locally compact case. I've been looking at ...
3
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0answers
77 views

The order of the representative elements of conjugacy classes

Suppose $A$ is an arbitrary subset of group $G$ and $K_G(A)$ be the number of $G$-conjugacy classes contained in $A$. A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ ...
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2answers
28 views

Multiplicity of G-module

I am currently working on Bruce Sagan's The Symmetric Group. The following proposition is given without proof: Let $V$ and $W$ be $G$-modules with $V$ irreducible. Then dim Hom($V$,$W$) is the ...
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0answers
32 views

Lie algebra: symmetric and exterior power of representation

If $\mathfrak{g}$ is a Lie algebra, $V$ and $W$ are representation of $\mathfrak{g}$ we define the action of $\mathfrak{g}$ on $V \otimes W$ in the following way: $X \cdot (v \otimes w)=(X \cdot v) ...
2
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1answer
31 views

Why $\rho(t)^{-1}(H-\frac{\partial}{\partial h_{\rho^{\vee}}}) \rho(t) = H - \frac{1}{2}(\rho^{\vee}, \rho^{\vee})$?

I am reading the paper. On page 17, line 15, why $$ \rho(t)^{-1}(H-\frac{\partial}{\partial h_{\rho^{\vee}}}) \rho(t) = H - \frac{1}{2}(\rho^{\vee}, \rho^{\vee}) $$? Here $$ H = \frac{1}{2} \sum_{i\in ...
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0answers
20 views

Invariants of the symmetric group

Let $V_\lambda$ be an irreducible representation of the symmetric group $S_n$ as usual labeled by parition $\lambda$ of $n.$ Question. Is there any general information about the algebra of ...