# Tagged Questions

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### * 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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### 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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### 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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### 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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### 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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### 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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### 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 ...