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

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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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1answer
58 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 ...
3
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1answer
121 views

Monodromy Representations

Let, be $V$ a connected smooth manifold and $q_1,q_2\in V$ and $F:U\to V$ a connected covering of degree $d$. This covering induces two monodromy representations $\rho_1:\pi_1(V,q_1)\to S_d $ ...
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87 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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2answers
671 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 ...
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Surprising but simple group theory result on conjugacy classes

I have read that for any group $G$ of order $2m+1$ (odd) with $n$ conjugacy classes, it is always the case that $16$ divides the value $(2m+1)-n = |G|-n$. This seems to me like an astonishing ...
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54 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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33 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
24 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
57 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 ...
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44 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 ...
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1answer
49 views

Reference request: what is the relation between classical r-matrices and quantum R-matrices?

I learned from a professor that $$ R=Id+(q-1)r+ o(q-1), $$ where $R$ is a quantum $R$-matrix and $r$ is the corresponding classical $r$-matrix. Here $o(q-1)$ denotes a term of the form $A(q-1)^2$, ...
2
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1answer
61 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 ...
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78 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 ...
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1answer
111 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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1answer
283 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& ...
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327 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 ...
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61 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. ...
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67 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 ...
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67 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 ...
4
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2answers
102 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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2answers
263 views

Why is $ U \otimes \operatorname{Ind}(W) = \operatorname{Ind}(\operatorname{Res}(U) \otimes W)$?

If $U$ is a representation of $G$ and $W$ is a representation of $H$, then why is $$ U \otimes \operatorname{Ind}(W) = \operatorname{Ind}(\operatorname{Res}(U) \otimes W)$$ I've tried to simply use ...
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155 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, ...
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1answer
40 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 ...
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30 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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53 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
55 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 ...
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67 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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1answer
57 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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152 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 ...
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Noncommutative Fourier Transform

The theory of Fourier transform for Euclidean spaces has analogues for locally compact abelian groups. In the noncommutative setting, representations can be used to define analogous transforms. My ...
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92 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 ...
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2answers
374 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 ...
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1answer
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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
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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 ...
4
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1answer
115 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 $p$. ...
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128 views

Does representation theory exists without Groups?

I need to know: is representation theory all about Groups? Is it necessary to be a finite group? Does representation theory exists without Groups? For example is there sample where representation is ...
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196 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 ...
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1answer
170 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 ...
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6answers
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Best books on Representation theory

What are some of the best books on Representation theory for a beginner? I would prefer a book which gives motivation behind definitions and theory.
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123 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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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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1answer
246 views

Generalizing Newton's identities: Trace formula for Schur functors

We work over $\mathbb C$. A general linear group ${\rm GL}(V)$ acts diagonally on the tensor power $V^{\otimes n}$ as $$(A^{\otimes n})(v_1\otimes\cdots\otimes v_n):=(Av_1)\otimes\cdots\otimes ...
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3answers
118 views

Representation theory in physics

0 down vote favorite I'm sorry if this is somewhat a dumb question. First: "Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements ...
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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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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) ...
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Dimensions of irreducible representations of finite groups over $\mathbb Q$

If $G$ is a finite group, then it is well known that there are finitely many inequivalent irreducible representations of $G$ over $\mathbb{C}$; moreover the sum of squares of dimensions of the ...
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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 ...
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1answer
110 views

Commutant Algebra of Matrix Representation

I am currently working on Bruce Sagan's The Symmetric Group. In the following example they show that for a representation that contains 2 different subrepresentations the commutative algebra Com$X$ ...
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2answers
139 views

Inner product in Maschke's Theorem

I am working through Maschke's Theorem on page 16 in Bruce Sagan's The Symmetric Group: In order to prove the theorem the author constructs an inner product $\langle v, w \rangle' = \sum_{g \in G} ...