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

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The comultiplication on $\mathbb{C} Q $ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=Q$ the ...
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Jacobson radical of the integral group ring

I am trying to prove that the Jacobson radical of the integral group ring $\mathbb{Z}G$ for a finite group is zero. Most of what I find on semisimplicity, Jacobson semisimplicity, has to do with ...
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15 views

PBW proof proposal

One version of the PBW theorem states: $\omega $:$\mathfrak {S} \mapsto \mathfrak {E} $ is an isomorphism of algebras. I am curious if this is a possible proof for the PBW theorem, part is taken ...
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Is the Transpose of a Representation an Equivalent Representation?

Suppose we are working over $\mathbf{Z}[G]$ where $G$ is finite. Suppose further we have two representations $\rho$ and $\rho^\prime$ such that $\rho^\prime=(\rho)^T$. Can we say that these two ...
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20 views

Explicit formula for invariant inner product of the standard representation of $S_3$

Let $V$ be a representation of a group $G$ over $\mathbb{C}$. Given the standard Hermmitian inner product $\langle\cdot,\cdot\rangle$ on $V$ we can always define a $G$-invariant inner product by ...
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The comultiplication on $\mathbb{C} S_3$ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=S_3$, ...
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1answer
22 views

Compute the isotropy representation

Suppose $SU(1,1)$ acts on the open unit disc $\mathbb{D}$ in the natural way, by linear fractional transformations. The isotropy group is $U(1),$ since it stabilizes the point $0.$ I am trying to ...
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One-dimensional representations of S5

The only one-dimensional representations of $S_5$ are the trivial representation and the sign representation. Why are these the only ones? Here's what I've got so far: the image of any ...
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Why does $\rho_{\mathbf{Z}^t\otimes P}=\rho_R$ imply the isomorphsim of $\mathbf{Z}^t\otimes P\cong R$?

so I have happened upon a thesis regarding the calculations of various $\mathbf{Z}D_6$ modules and their isomorphisms and came across a technique which is bothering me. Let me give an example. Let ...
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Is $B(w_1 w_2)B = (Bw_1B)(Bw_2B)$?

Let $B$ be a Borel subgroup of $GL_n$ and $W$ the Weyl group of $GL_n$. Let $w_1, w_2 \in W$. Is $B(w_1 w_2)B = (Bw_1B)(Bw_2B)$? If this is true. How to prove it? Thank you very much. Edit: If $l(w_1 ...
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Consider $\mathbf{Z}G$, $G$ finite. If the characters of two $\mathbf{Z}G$-modules are equal, does it follow that the modules are isomorphic?

So I have recently started to delve into integral representation theory and I was wondering if a particularly useful theorem survives the transition to integral rep theory. Basically, suppose we have ...
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13 views

Examples of the local Langlands correspondence

I'm trying to compile some examples of the local Langlands correspondence, with the aim of motivating the statement and also just giving some concrete traction on how it works. I would especially like ...
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23 views

matrix of the dual representation: inverse of the transpose

I have a doubt concerning the dual representation. Can someone check that what I wrote is correct please? Let $A: V \longrightarrow V$ be linear, the dual map $A^T : V^* \longrightarrow V^*$ is ...
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Multiplicity of irreducible $\mathbb{C}S_n$-modules

A known result in the representation theory of the symmetric group $S_n$ says: "Let $T_{\lambda}$ be a Young tableaux corresponding to a $\lambda \vdash n$, and let $M=M_{1} \oplus M_{2} \oplus ...
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1answer
51 views

Question concerning a list sorting problem

I have the following question: Let $a,b,c,d$ be four natural numbers with $a \leq b$ and $c\leq d$. I have written a program that produces a list, which has as entries all 2-tuples $(x,y)$ with ...
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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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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 ...
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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 ...
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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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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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27 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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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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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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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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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
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1answer
44 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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1answer
42 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 ...
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36 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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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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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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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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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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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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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 ...
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1answer
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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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28 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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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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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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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
28 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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1answer
43 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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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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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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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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1answer
31 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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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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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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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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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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132 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 ...