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

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Identifying the cotangent bundle of the flag variety

Suppose $G$ is a Lie group (or I guess a linear algebraic group), $P \subset G$ a Lie subgroup with Lie algebras $\mathfrak{g}$ and $\mathfrak{p}$ respectively. In Chriss and Ginzburg's book ...
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20 views

Why is 1/2+1/2 in the weight space for SO(5)

Let's consider $\mathfrak{so}(5)$ as the Lie algebra of $\mathrm{SO}(5)$, where the symmetric bilinear form is $x_1y_5+\cdots +y_1x_5$. Then the maximal torus is given by $$\left(\begin{array}{cccccc} ...
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What are the differences between the three editions of the book “The Structure of Compact Groups”?

meta pre-clarification: I looked into another question like this but the guy didn't mark any specific tags for this type of question. Here's a link to the amazon book: ...
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170 views

Geometric & Intuitive Meaning of $SL(2,R)$, $SU(2)$, etc… & Representation Theory of Special Functions

Many special functions of mathematical physics can be understood from the point of view of the representation theory of lie groups. An example of the power of this viewpoint is given in my question ...
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1answer
22 views

If $A$ is an abelian C*-algebra, and $\tau$ is pure then it is a character on $A$

If $A$ is an abelian C*-algebra,and positive linear functional $\tau$ is pure then it is a character on $A$. Murphy in his book(C*-algebras and operator theory) has below proof: While I think we can ...
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61 views

Invariants of $V^{\otimes N}$. [closed]

Let $V$ be a finite dimensional complex vector space, and $G = SL(V)$ be the group of linear transformations of $V$ with determinant $1$. (a) Show that $V^{\otimes N}$ contains a nonzero ...
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Another construction of Specht module.

Let $\lambda$ be a partition of $n$, and $\lambda^*$ the dual partition (i.e. having the transposed Young diagram). Let $z_i$ be vectors in $\mathbb{C}^{\lambda_i^*}$, and$$F_\lambda = \prod_i ...
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41 views

Artin Algebra Representation Chapter Resource Request

I am working through chapter 10 of Artin's Algebra 2ed which introduces Group Representations. However, I've found that the approach to introducing groups is unlike the usual method used by more ...
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1answer
24 views

Is the coaction $\delta: A \to H \otimes A$ injective?

Let $A$ be an algebra and let $H$ be a bialgebra. Suppose that $A$ is an $H$-comodule. Then we have a coaction $\delta: A \to H \otimes A$. Is the coaction $\delta: A \to H \otimes A$ always ...
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1answer
66 views

A Representation Theory Problem in Putnam Competition

The following was the B6 problem of 1985 Putnam Competition: Suppose $G$ is a finite group (under matrix multiplication) of real $n\times n$ matrices $\{M_i\}, 1\leq i\leq r$. Suppose that ...
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1answer
39 views

Embedding so(n) in su(n)

Is there any way of embedding $\mathfrak{so}(n)$ into $\mathfrak{su}(n)$ for any $n$ other than picking the antisymmetric matrices of $\mathfrak{su}(n)$? I know that for small $n$ one can use ...
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1answer
29 views

One Dimensional Representations of $GL_3(\mathbb{F}_p)$

I am currently trying to find the irreducible one dimensional representations for the subgroup $H$ of $GL_3(\mathbb{F}_p)$ consisting of the upper triangular matrices. I know the number of ...
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2answers
122 views

Ideal of nilpotent elements in non-commutative ring.

Let $R$ be a non-commutative ring such that every element is either invertible or nilpotent. I am trying to show that the set of nilpotent elements, denoted $I$, is a two sided ideal, but I am having ...
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1answer
50 views

Why mention the “self-conjugate” property in Tannaka duality?

Based on this Wikipedia section and this MathOverflow answer of Qiaochu, I believe I've understood Tannaka duality for finite groups. We wish to characterize a finite group $G$ as a subgroup of ${\rm ...
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23 views

Reducible Lie Algebra

I'm furthering my physics knowledge through a book called Lie Algebras in Particle Physics and am having trouble with one aspect of a problem. I believe because it's a question purely about ...
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26 views

Matrix representation of an operator

Murphy says : The pure states of $A=K(H)$ are precisely the states $\omega_x : A\to \Bbb C ~~;~~\omega_x(u) = \langle ux,x\rangle $ where $x$ is a unit vector of Hilbert space $H$ . Then he gives ...
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1answer
52 views

Questions about subrepresentations of a representation of a quiver.

Let $Q$ be the quiver $\cdot \to \cdot \to \cdot$. Then $$ \mathbb{C} \to^{f} \mathbb{C} \to^g \mathbb{C} \quad (1) \\ 0 \to^{0} \mathbb{C} \to^0 0 \quad (2) $$ are two representations of $Q$, ...
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1answer
33 views

Representations of the Lie algebra $\mathfrak{sl}(2,\mathbb{C})$

In Corollary 7.2 of http://math.uchicago.edu/~may/REU2012/REUPapers/Bosshardt.pdf, why is the set of weights an unbroken string? I understand we get a finite number of weights by looking at the ...
3
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1answer
73 views

Character theory - Exercise 5.14

I am trying to solve the exercise 5.14 from the Isaac Martins Character Theory of Finite Groups. Let $G$ be a nonabelian group and let $ f=min\{\chi(1) | \chi \in Irr(G), \chi(1)>1 \}. $ Show ...
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1answer
34 views

the span of a representation's action on a vector

Consider the image of the action of a group representation $\rho: G \to V$ on some vector $v \in V$: $$ \{ \rho(g) v : g \in G \} $$ It seems that the span of this set: $$ W_v \equiv ...
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1answer
21 views

matrix Lie group embedding as a manifold

Given a Lie group of matrices, and suppose for simplicity that it is globally generated through exponential map from its Lie algebra on a element. Is there a canonical way to embed it into ...
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1answer
35 views

symmetric group acting on torus

Let $S_k$ be symmetric group of order $k$. Let $T^k=S^1\times\cdots \times S^1$. Then $T^k$ is a Lie group. For each $\sigma\in S_k$, let $\sigma$ act on $T^k$ from right in the way $$ ...
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1answer
27 views

Prove that there exists an $m$th root of unity $λ ∈ C$ such that for all $g ∈ G$, $\chi(zg) =λ \chi(g)$

Suppose $\chi$ is an irreducible character of $G$. Suppose $z ∈ Z(G)$ and that $z$ has order $m$. Prove that there exists an $m$th root of unity $λ ∈ C$ such that for all $g ∈ G$, $\chi(zg) =λ ...
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7 views

Finding the eigenvalues of the matrix corresponding to an element of finite order in a group representation

Suppose that $g$ is an element of finite order (say $n$) in a group $G$ and $\rho:G\rightarrow GL(V)$ is a degree $n$ representation of $G$. If now I know $\chi_V$, how can I find the eigenvalues of ...
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1answer
15 views

A question involving condition for an element to be in the kernel of a representation ans the kernel of the coset representation

I came across the following question. Given a representation $\rho:G\rightarrow GL(V)$ with kernel $N$, let $\rho$ have character χ : $G$ → $\mathbb C$. Then for $g ∈ G$, it first asks to prove that ...
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13 views

Finding conditions for different representations to be faithful

I'm trying to find conditions under which each of the following representations are faithful: trivial, regular, coset, sign (for $G = S_n$), defining (for $G = S_n$) and degree 1 for $C_n$, the cyclic ...
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1answer
48 views

Young tableaux of $8\otimes 8$ in $SU(3)$

In Georgi's Lie Algebras in Particle Physics, one finds the following Young tableaux for $8\otimes 8$ in $SU(3)$: I am unsure of all the cancellations. Let us number the canceled tableaus increasing ...
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1answer
44 views

Finding normal subgroup from a character table

I have the following character table. Note I assume that $\chi_i$'s are all irreducible. $$ \begin{array}{|c|c|c|c|c|} \hline & C_1 & C_2 & C_3 & C_4 & C_5 \\ \hline \chi_0 & ...
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1answer
32 views

Finding irreducible subrepresenations of modular representation in GAP

Recently, I have been fiddling with modular representations in GAP. First from what I can tell, GAP does not have a good way built in to find things like Brauer characters of a given non-solvable ...
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1answer
71 views

Vector spaces over quaternions

Let $V$ be an $n$-dimensional vector space over the quaternions $\mathbb{H}$, and let $G$ be the multiplicative quaternion group. How would one show that $V$ would then be a $4n$-dimensional ...
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1answer
22 views

THE positive half-spin space of quaternion vector space

I have the following information: $T$ is the one-dimensional quaternion vector space with the canonical action of $\Gamma$, a finite subgroup of SU$(2)$. This makes sense as SU$(2)$ is the unit ...
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1answer
27 views

Question about using two associative algebras irreducible modules to prove the algebras are isomorphic

If two associative algebras $A_1$, $A_2$, over some field admit a bijection $f$ between irreducible modules such that dim $M_1$= dim $f(M_1)$ where $M_1$, ($f(M_1)$) are irreducible $A_1$ ...
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1answer
43 views

Complete the character table of group of order $21$

You are given the incomplete character table of a group $G$ with order $21$ which has $5$ conjugacy classes, $C_1,\dots,C_5$, which have sizes $1,7,7,3,3$. $$ \begin{array}{|c|c|c|c|c|} \hline & ...
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prove diagonalizablity using argument from representation theory

Let $\{X_1,X_2,...,X_n\} \subseteq GL_d$ be a subgroup of commutating matrices then show this matrices are simultaneously diagonalizable (using some argument from representation theory)
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2answers
51 views

Finding the character table of $Q_8$

Assume $G$ is a finite group. I am trying to construct the character table of $Q_8$, which is defined by $$Q_8=\{\pm 1,\pm i, \pm j,\pm k \}, \ i^2=j^2=k^2=-1, \ ij=k,jk=i,ki=j$$ By considering the ...
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1answer
47 views

What is the difference between the representation of a group and an algebra?

Sometimes, I come across this idea in physics -> the representation of Lorentz group: SO(3,1) and the representation of Lorentz algebra: so(3,1). At times, I mix them up. Is there a good intuitive way ...
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1answer
46 views

Let $\rho : G \rightarrow GL_n(\mathbb{C})$ be a representation show that $|\operatorname{tr} X| \leq \dim \rho$

Let $G$ be a finite group. Let $\rho : G \rightarrow GL_n(\mathbb{C})$ be a representation, pick $g \in G$ and write $X=\rho(g)$. Prove that all eigenvalues of $X$ are roots of unity, and deduce that ...
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How to rigorously show tensor identities using symmetry arguments?

I am wondering how to rigorously show tensor identities such as the following. Let $n$ denote the radial unit vector in $D$ dimensions. Then $\langle n_i n_j \rangle = \frac 1 D \delta_{ij}$ and ...
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1answer
30 views

What is the notion of “character” in the context of Cayley graphs?

I am looking at these notes, http://www.eecs.berkeley.edu/~luca/books/expanders.pdf On page 37, Lemma 5.16, the notion of "character" defined seems to be any map from the finite Abelian group to ...
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1answer
46 views

Example of $\det \rho(g)=\det \sigma(g)$ for all $g\in G$, but $\rho \not\simeq \sigma$

Give an example of a group $G$ and two representations $\rho$ and $\sigma$ of $G$ such that $\det \rho(g)=\det \sigma(g)$ for all $g\in G$, but $\rho \not\simeq \sigma$. At the moment but ...
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1answer
91 views

Ten dimensional representation of $S_6$

Let $S=\{1,2,3,4,5,6\}$. For every three-element subset $A\subset S$ and $B=S\setminus A$ consider the symbol $e_{(A|B)}$ for which we assume that $e_{(A|B)}=e_{(B|A)}$. Then the vector space $V$ ...
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1answer
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Reducibility of Cyclic groups

Let $G$ be the cyclic group $C_{4}$ and consider the 2-dimensional representations of G. Why does extending scalars to the complex numbers let this representation become reducible? I understand how it ...
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Decomposing some representations as a direct sum of irreducibles

I'm taking $V$ to be the standard representation of $S^3$. I'm looking for the decomposition of the following representations as a direct sum of irreducibles. (a) $V\bigotimes V$ (b) $Sym^2$ $V$ (c) ...
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1answer
28 views

Descending representation on the quotient group

I have this homework question in introductory representation theory: Let $ψ : G → GL(V)$ be a representation of $G$ and let $N$ be a normal subgroup of $G$. Define $ρ : G/N → GL(V)$ by $ρ(gN) = ...
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1answer
33 views

number of irreducible representations

Prove that if $H$ is an abelian subgroup of $G$ then each irreducible representation of $G$ has degree at most $\frac{|G|}{|H|}$ One proof is given in serre, but I would like to see some different ...
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22 views

Is each multiplicative linear functional on $L1(SL(2,R):SO(2,R))$ triviall?

We know that if $G=SL(2,R)$ and $H=SO(2,R)$ as a compact subgroup of $G$, then $\{ξ∈\hat{G};ξ|_H=1\}$ is triviall ($\hat{G}$ is the characters group). Can we conclude that each multiplicative linear ...
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Inner product doesn't matter for Schur orthogonality?

I'm reading Knapp's Basic Algebra, specifically the section about Schur orthogonality relations. Given a representation $R: G \to \text{End}(V)$, he defines $V_R$ to be the vector space of matrix ...
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4answers
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Quick question about formal definition of a representation

So I know that a linear representation is defined as $\rho : G \to GL(V)$ over some finite group $G$. So if we define the action of some group ring, say F[G] over some representation V, is this ...
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Iwasawa module: $O[[G]]$ is noetherian where $G$ is a compact $p$ adic Lie group

Let $G$ be a compact $p$ adic Lie group. Let $E$ be a finite extension of $\Bbb{Q}_p$ with ring of integers $O$. Then how to show that $O[[G]]$ is noetherian. I was reading the article here and on ...
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2answers
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Canonical Decompostion

This is regarding the proof of proposition 24(page 61) of Serre's Linear Representations of finite groups. Line 3 of the proof says that for $s\in G$, $\rho(s)$ permutes $V_i$? Can someone be kind ...