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

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What is an algebra and what is it's representation?

Heyho, i've kind of got an understanding problem what exactely an algebra and especially it's representation is. In my case it is said, that the relation $R_{12}(u-v) (L(u) \otimes \mathrm{I}) \; ...
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41 views

Understanding a step in this proof

I have it tagged as representation theory because it's out of my rep theory book but I'm really just misunderstanding a group theory aspect here. So the statement to prove is: if $G$ has odd order ...
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14 views

Abelian semi-simplification representation

Let $K$ be a number field with $G=\mathrm{Gal}(\bar K/K)$. Consider the representation of $G$ with value in a $\mathbb{F}_p$-vector space $V$ of dimension 2 $$ \rho : G \longrightarrow ...
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Let G be an abelian group. Let V be an irreducible faithful CG-module. Prove that dimV = 1 and G is cyclic.

I was wondering if I could get some help with the following problem. I know how to prove it with Schur's Lemma but I'm having problems without it. Let G be an abelian group. Let V be an irreducible ...
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90 views

How do modern algebraists think about diagonal matrices?

Let $\mathbb{K}$ denote a field and $A$ denote a $\mathbb{K}$-algebra. Then given a $\mathbb{K}$-subalgebra $\Delta$ of $A$, I suppose it make sense to declare that $m \in A$ is ...
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Do we have $\delta(ab)=\delta(a)\delta(b)$ implies $\Delta(cd)=\Delta(c)\Delta(d)$?

Assume that $B$ is an algebra which is also a coalgebras (we do not assume that $B$ is a bialgebra: we do not assume $\Delta(cd)=\Delta(c)\Delta(d)$). Assume that $A$ is $B$-comodule algebra. Then we ...
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30 views

Two questions about Schubert calculus and Schur functions.

I am reading the file. I have a question on pae 28. How to prove that $[X_{\{2,4\}}] = S_{(1)} = x_1 + x_2 + \cdots$ and $S_{(1)}^4 = 2 S_{(2,2)} + S_{(3,1)} + S_{(2,1,1)}$? I tried to verify ...
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Schubert calculus and number of lines satisfying some properties.

I am reading the file. I have a question on pae 18. It is said that: Given a line in $\mathbb{R}^3$, the family of lines intersecting it can be interpreted in $G(2, 4)$ as the Schubert variety $$ ...
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59 views

Non-linear equivariant maps between group representations

Given two representations $\pi_1$ and $\pi_2$ of a group $G$ (let's say it's a compact Lie group), a natural thing to study are linear equivariant maps A between them: $$ A \pi_1 = \pi_2 A $$ I'm ...
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44 views

Representations of cartesian product $G$

We know for both representations of a locally compact group $G$, their tensor product is a representation of $G \times G$ (cartesian product of $G$ with self). Is each representation of it of this ...
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48 views

Who are the mathematicians in US who are working on expander graphs right now?

I am familiar with only the "big" names doing this research like Gharan, Nikhil Srivastava, Dan Spielman, Jean Bourgain, Luca Trevisan, Elina Fuchs, Peter Sarnak , Amin Saberi and Terence Tao. I would ...
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37 views

Proving that this function is an Endomorphism?

Given that $\mathbb{C}G = W_1 \oplus W_2$ and $1 = e_1 + e_2$ where $e_1 \in W_1$ and $e_2 \in W_2$ Also Knowing that $$w_1e_1 = w_1, w_2e_1 = 0$$ $$w_1e_2 = 0 , w_2e_2 = w_2$$ Let $x \in G$ Now it ...
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55 views

What shall I learn in order to understand Auslander-Reiten theory and tilting theory?

I work on cluster algebras and quivers and hence I need to understand Auslander-Reiten theory and tilting theory as soon as possible. I have read some noncommutative algebra and homological algebra ...
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30 views

Minimal polynomial of endomorphism of permutation module

Let $G$ be a transitive permutation group on a set $\Omega$. If $n$ is the degree and $M\in\mathbb{Z}^{n\times n}$ is a symmetric matrix that is also contained in ...
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73 views

How to show that $\int_G f(t) dt = \int_G f(t^{-1}) dt$?

I am reading the lecture notes. On page 34, line 13, it is said that $\int_G f(t) dt = \int_G f(t^{-1}) dt$. How to prove this identity? I think that if we let $s=t^{-1}$, then ...
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Can Schroedinger equation be derived from the unitary representation of Galilean group? [migrated]

I have been trying to understand quantum mechanics as a unitary representation of spacetime symmetries. My first question is: Can Schroedinger equation be derived from the unitary representation of ...
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36 views

How to apply a double centralizer property on a faithful module of a self-injective Artin algebra?

Let all considered algebras be Artin algebras and let all considered modules be finitely generated. Let $A$ be left-QF-3 with minimal faithful left ideal $Ae$. Then the following are equivalent: ...
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When does a representation of $H\subset G$ on $V$ extend to a representation of $G$ on $V$?

Let $G$ be a finite group, $H$ a subgroup, and $\varphi:H\rightarrow GL(V)$ a finite-dimensional representation of $H$ over a characteristic zero, algebraically closed field. Let $\chi$ be the ...
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28 views

Unitary invariant positive definite form

Let $G$ be a finite group and $V$ a finite-dimensional compex vector space which is a $G$-module. Define $(u,v)=\sum_{x\in G} h(ux,vx)$, where $h(u,v)$ is a positive-definite hermitian form. By ...
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34 views

How to show that a certain module is injective over an endomorphism algebra?

Let $A$ be a self-injective Artin algebra and $M\in\ \mathfrak{mod}\ A$ with the property $\mathfrak{add}\ _AA = \mathfrak{add}\ M$. Let $I$ be a finitely generated injective $A$-module. Why is ...
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17 views

Projector method for tensor and double groups

I'm currently trying to understand a computation in my script. The setup is the following: We are looking at the double group of $C_{3v}$, i.e. $C^D_{3v}$. The character table is given by the ...
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39 views

How to find irreducible representations of $\mathbb{C}S_2$ and $\mathbb{C}S_3$

I just starting to learn representation, so still have a lots of thing that is unclear. And here is a question what I wish to attempt. Let $S_n$ be the symmetric group and I want to find all the ...
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63 views

Always exists a representation $\rho$ for an arbitrary group?

I am studying the representations of the fundamental group of a fixed surface into PSL$_2\mathbb{R}$, and a simple question aries in me. If $G$ is a group, and $V$ a vector space over a field ...
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30 views

* representation of an algebra

Suppose $G$ is a finite group. Let $A(G)$ be the algebra of functions from $G$ to $C$. Now we define the convolution of $a$ and $b$ in $A(G)$ as $(a*b)(x)=\sum\limits_{y\in G} a(xy^{-1})b(y)$. Let $U$ ...
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56 views

What does the notation $U\mathfrak{sl}_2$ mean, and why is the $U$ written in a different typeface to the $\mathfrak{sl}$?

A representation theory homework problem asks me to determine the finite dimensional irreducible representations and the finite dimensional indecomposable representations of $U\mathfrak{sl}_2$. I ...
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35 views

Induction and Compact induction of representations

Let $H \leq G$ be a subgroup of a finite group, $G.$ Suppose $(\sigma, W)$ is a representation of $H.$ Then we know that $Ind_H^G \sigma $ and $ind_H^G \sigma $ are isomorphic, where $$Ind_H^G ...
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23 views

Why aren't all elements of the $45_a$ representation of $SO(10)$ zero?

We can write elements of the $45_a$, where $a$ denotes antisymmetric, as $10 \times 10 $ matrices, because $$ 10 \otimes 10 = 1_s \oplus 54_s \oplus 45_a$$ Here $10$ denotes the fundamental ...
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24 views

For a root system, why does $\beta\in\Delta_+\setminus\{\alpha_i\}$ imply $(\beta+\mathbb{Z}\alpha_i)\cap\Delta\subset\Delta_+$?

Let $\mathfrak{g}(A)$ be a Kac-Moody algebra for a matrix $A$, with root basis $\{\alpha_1,\dots,\alpha_n\}$. There is a remark on the bottom of page 6 of Kac's Infinite Dimensional Lie Algebras ...
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52 views

Why is $\mathfrak{g}(A)=\mathfrak{g}'(A)$ iff $\det(A)\neq 0$?

In many sources (Victor Kac, Zhexian Wan, etc.), it's stated as a remark that if $\mathfrak{g}(A)$ is the Kac-Moody algebra of a generalized Cartan matrix $A$, then $\mathfrak{g}(A)=\mathfrak{g}'(A)$, ...
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Quadratic Casimir of SO(5)

In the article A Four Dimensional Generalization of the Quantum Hall Effect, arXiv:cond-mat/0110572, by Zhang and Hu Quadratic Casimir operator for $SO(5)$ is given as $$p^2/2+q^2/2+2p+q .$$ When ...
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Free product of two algebras and actions of algebras.

Let $A, B$ be two algebras. Suppose that $A$, $B$ acts on $V$. Then we have two maps $$ \delta_1: A \otimes V \to V, \\ \delta_2: B \otimes V \to V, $$ which satisfy the axioms of actions. Do we ...
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44 views

Short pedagogical introduction to Young-tableaux and weight diagrams?

I am looking for a short pedagogical introduction to Young-tableaux and weight diagrams and the relationship between them, which contains many detailled and worked out examples of how these methods ...
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1answer
25 views

Dual representation of $SL_n$ via Young diagram

Irreducible representations of $SL_n$ are encoded by Young diagrams with fixed number of rows not greater then $n-1$ (at least, I prefer this notation). There is an involution on these ...
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79 views

Decomposition of polynomial ring as $S_n$-module

I want to whether there is a containment relation between the $S_n$-modules $\mathbb{C}S_n$ and $\mathbb{C}[x_1,\ldots ,x_n]$. Is it true that $\mathbb{C}[x_1,\ldots ,x_n]$ contains an isomorphic copy ...
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What are the character functions of $\mathbb{Z}_N \times \mathbb{Z}_N$ ?

$\mathbb{Z}_N \times \mathbb{Z}_N$ is an Abelian group which I can think of to consist of all tuples of the form $(\omega ^a, \omega^b)$ where $0 \leq a,b \leq (N-1)$ and $\omega = e^{ \frac{2 \pi i ...
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32 views

Can two representations with different dimensions be isomorphic?

For a finite group G and two irreducible representations, with different dimensions. How would I show that they can not be isomorphic?
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Embedding of a finite group in a compact connected Lie group

How can one embed a finite group $G$ in a compact connected Lie group? I think if we take a faithful unitary representation of G , that will do the job.But if $G= Z/n$, then what should be the ...
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47 views

Are these functions characters?

In Ireland and Rosen are mentioned the functions $f,g:\mathbb{F}_{p^f}^*=G\to \mathbb{C}$, with $$f:x \mapsto \zeta_p^{x+x^2+\cdots + x^{p^{f-1}}}$$ $$g:x \mapsto \left( \frac{x}{P}\right)_m,$$ ...
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31 views

About irreducible group representations

Are the trivial representation, the alternating representation and the standard representation irreducible representations?! I can easily see that the trivial representation (which sends every ...
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Given the basis vectors of a 10-dimensional representation of $SO(10)$, how can I compute the basis vectors of the 54-dimensional representation?

Because $10 \otimes 10 = 1_s \oplus 54_s \oplus45_a$ we can write each element of $54$ as a $10×10$ matrix. The usual basis vectors of the 10-dim rep are $$ \begin{pmatrix}1 \\0 \\ \vdots ...
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Why doesn't the “naive” scalar product for $SO(n)$ yield something invariant?

By definition, for $SO(n)$ we have $g^T g=1$ for $g \in SO(n)$. Given some vector $v \in V$ and some representation $R: SO(N) \rightarrow \mathrm{Lin}(V)$, the defining condition above tells us ...
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50 views

Question about a passage in Fulton and Harris

So I was reading the first chapter of Fulton and Harris and they are determining the representations of $S_3$. I came along this passage and had some questions What do they mean when they say "the ...
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75 views

Where does the ambiguity in choosing a basis for a Lie algebra come from?

This is a follow-up to this question. For matrix Lie algebras, we can define the Lie algebra $g$ of a group $G$ as the set $T_a \in g$ that yield an element of $G$ when put into the exponential map: ...
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100 views

The generators of $SO(n)$ are antisymmetric, which means there are no diagonal generators and therefore rank zero for the Lie algebra?

Okay, this may be a silly question but I can't figure it out myself right now. By definition $O \in SO(n)$ fulfils $O^T O=1$ and $\det(O)=1$. For the generators of the group $ T_a \in so(n)$, this ...
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51 views

Recognizing action of semidirect product

I've been looking at some texts in representation theory and I see instances where the symmetric group $S_n$ and some other group, e.g., $GL(V_1) \times \ldots \times GL(V_n)$, act on a space. The ...
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39 views

A $\mathbb Z/p\mathbb Z[G]$ submodule with no complement

Let $G$ be a group acting on a set $X$ of size $n$. Suppose $G$ acts doubly transitively. If $p$ is a prime, this naturally gives a permutation representation on the vector space over $\mathbb ...
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42 views

Inducing representations from a subgroup of finite index.

Let $G$ be a group and $H$ a subgroup of finite index. Let $(\sigma , W) $ be a irreducible representation of $H$ (which need not be finite dimensional). 1) When can we extend this representation of ...
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43 views

In general, how do you construct a nontrivial representation of a group?

This is my first time studying representations. I'm not sure how to go about constructing a nontrivial representation of a group. Do I construct a function that satisfies the definition? Could you ...
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40 views

How do I construct a nontrivial linear representation of the group $G$=S' in $R^3$

$S'$ is defined as the unit circle. The product is defined as the sum of angles. How do I construct a linear representation in general? I don't know how to begin with this problem. Some ...
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40 views

GNS construction and representations

I am currently reading about C* from the following notes ( http://www.math.uvic.ca/faculty/putnam/ln/C%2A-algebras.pdf ). In the proof of GNS construction theorem 1.12.4 page 50 there is something I ...