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

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Matrices over a finite field with given Jordan normal form over the algebraic closure

Can one describe the (conjugacy classes of) square matrices over a finite field such that over the algebraic closure of this finite field their Jordan normal form consists of one Jordan block? (Such ...
1
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2answers
48 views

Confusion about Lie groups in Fulton & Harris

Near the beginning of chapter 8 (titled Lie groups and Lie algebras) authors motivate the definition of Lie algebra. I'm confused by two things in just one sentence: ($G$ is a Lie group) The ...
3
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0answers
53 views

How much we know about the Group from its Complex character table?

Suppose $G$ is a finite group and suppose that complex character table of $G$ is given.It is well known that from character table we cannot determine the Group uniquely (For example $Q_8$ and $D_8$ ...
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43 views

Considering a permutation representation of a transitive $G$-set

Suppose $X$ is a transitive $G$-set, where the size is greater than $1$, and $\pi=\pi_X$ the associated permutation representation. What is its character $\chi$? I thought that the permutation ...
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1answer
27 views

global dimension of bounded path algebra

Can someone give me some example : how to calculate de global dimension of some bounded path algebra. 1-My problem is that I do not know how to find the projective resolution of a simple module. ...
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1answer
83 views

Proving that $\pi_{X \times Y} \simeq \pi_X \otimes \pi_Y$

If $X$ and $Y$ are $G$-sets and $X \times Y$ is a G-set by $g \cdot (x,y)=(g \cdot x , g \cdot y)$. \pi is the corresponding permutation representation. Prove that $\pi_{X \times Y} \simeq \pi_X ...
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0answers
8 views

Dimension of a finite union of locally closed subsets

Let $X$ be an irreducible variety and $\{X_i\}_1^m$ be a finite collection of locally closed subsets, which are not necessarily disjoint. I have trouble convincing myself of the following result: ...
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0answers
30 views

Decomposition of vector spaces.

Let $V$, $W$ be finite dimensional vector spaces (over a characteristic zero field $\mathbb{K}$) and $\lambda=(\lambda_1, \cdots, \lambda_n)$ a partition of an integer $m$. Let $L_{\lambda}V$ and ...
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15 views

Questions about the indivisible imaginary root in affine root system.

I am reading the paper. On page 5, $\delta$ is defined as the indivisible imaginary root in $\widehat{\Delta_+}$. $\Lambda_0 \in \widehat{\mathfrak{h}^*}$ is the unique element satisfying $\langle K, ...
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0answers
21 views

if $\chi$ is faithful then exist $\phi$ which constitutes $\chi$

Let $\chi$ be a faithful representation of a group $G$ then prove exist a representation $\phi$ of $G$ such that $<\chi^n,\phi>=0$ I was trying to do it like, let $N=ker(\chi)$ and consider the ...
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2answers
53 views

Abelian groups cannot have characters of degree 2

I was attempting the following exercise: Assume that $G$ is a simple group. Let $\chi$ be an irreducible character of degree $2$, and $g \in G$ be an element of order $2$. Prove that $\chi (G) ...
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Irreducible character of degree greater than one takes value zero on some conjugacy class

It is a standard fact that irreducible character of a finite group of degree $>1$ takes value $0$ on some conjugacy class. A proof for example can be found here. I would like to know whether there ...
4
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1answer
52 views

Orbits of algebraic groups (dimension of connected components)

Let $X$ be an algebraic variety with algebraic group $G$ acting on it. Let $x\in X$. I am trying to prove that all connected components of the orbit $Gx$ are of dimension $\dim G - \dim G_x$, where ...
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1answer
32 views

Artin-Wedderburn theorem and square dimension

Let $A$ be a finite-dimensional simple algebra over $\mathbb{C}$ of dimension $n$. By Wedderburn's theorem, we have that $A$ is isomorphic to a matrix ring $M_r(\mathbb{C})$, which is of dimension ...
2
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1answer
38 views

Representations of a product of Lie groups

Let $G=G_1\times G_2$ be a product of two compact Lie groups. Is every finite dimensional irreducible representation of $G$ a tensor product of irreducible representations of $G_1$ and $G_2$? This ...
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1answer
58 views

Thm 5.19 (Brauer) in Isaacs Martin - Character Theory of Finite Groups

I would please like some help to understand the proof of thm 5.19 (Brauer) in Isaacs Martin - Character Theory of Finite Groups. It states: Let $\chi$ be a character of G with $[\chi,1_G]=0$. Let ...
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0answers
28 views

Decomposition of regular representation [closed]

Let $G$ be a group and consider its regular representation. We may uniquely decompose this representation into sums of irreducible components. What does it mean to find a basis for each component?
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2answers
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Etingof Problem 5.1, “Field embeddings”

Recall that $k(y_1, \dots, y_m)$ denotes the field of rational functions of $y_1, \dots, y_m$ over a field $k$. Let $f : k[x_1, \dots, x_n] \to k(y_1, \dots, y_m)$ be an injective $k$-algebra ...
2
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34 views

$GL(V)$ representations and Schur modules.

Let $W$ be a fine dimensional complex vector space of dimension $n$ and $L_{\lambda}W$ the Schur module associate to the partition $\lambda=(\lambda_1, \cdots,\lambda_{n-1})$, where $\sum_i ...
1
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1answer
35 views

Size of conjugacy classes in SL(2,3)

I've been given the representations of the conjugacy classes for a group presentation $G = <x,y,z | x^2 = y^3 = z^3 = xyz>$ which is isomorphic to $SL(2,\mathbb{F}_3)$ which are: ...
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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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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
23 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 ...
4
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1answer
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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25 views

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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42 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 ...
1
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1answer
25 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
68 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 ...
0
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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 ...
2
votes
2answers
124 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 ...
0
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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
votes
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 ...
2
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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 ...
-1
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1answer
37 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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0answers
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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0answers
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
49 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
45 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 ...
1
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1answer
72 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 ...