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

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Tensor products and irreducible representations

Again something from Fulton and Harris I'm having trouble with: Exercise 2.33 (c). If $U$, $V$, and $W$ Are irreducible representations, show that $U$ appears in $V \otimes W$ if and only if $W$ ...
4
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1answer
310 views

Exercise 2.4 in Fulton and Harris

I'm trying to begin reading Fulton and Harris' Representation Theory and I'm having trouble with the following: Exercise. Show that if we know the character $\chi_{V}$ of a representation $V$, then ...
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1answer
44 views

Dimension of a representation and the order of an element in a group

Let $V$ be a representation of a finite group $G$ with $V$ being finite dimensional. Fix a $g \in G$. Is it necessarily true that $\dim V \geq \operatorname{ord} g$?
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347 views

Being isomorphic as representations of a group G

Let $G$ be a finite group. What is meant by two finite dimensional vector spaces (over $\mathbb{C}$) $V$ and $W$ being "isomorphic as representations of $G$"? To show that we have such an isomorphism, ...
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340 views

Does the $p$-torsion of an elliptic curve with good reduction over a local field always determine whether the reduction is ordinary or supersingular?

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $E/K$ an elliptic curve with good reduction. Does the $\mathbb{F}_p[\mathrm{Gal}(\overline{K})]$-module $E[p](\overline{K})$ determine whether the ...
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1answer
62 views

unitary representation and denseness.

I have the next unitary representation, $\pi : G\rightarrow \mathcal{U}(H)$, where G is a closed subgroup of $S_{\infty}$ (the group of bijective functions from $\mathbb{N}\rightarrow \mathbb{N}$), ...
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1answer
80 views

Isomorphism between $sl_{4}$ and the orthogonal group of $6$ variables

Let V be the irreducible $sl_{4}$-module with highest weight $\pi_{2}=\lambda_{1}+\lambda_{2}$ (i.e if $H=\left(\lambda_{1},\dots,\lambda_{4}\right)$ is a diagonal matrix in $sl_{4}$ with values ...
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2answers
123 views

Characters and permutation matrices

Suppose I represent the group $S_{\,n}$ using $n \times n$ permutation matrices. This is a valid group representation. Let $\chi$ be its character. Since $\chi(g)$ is complex and since ...
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1answer
49 views

What is the systematic way to convert any arbitrary finite dimensional representation into block diagonal form?

Given any arbitrary representation, how do I convert it into block diagonal form, or find its irreducible representation?
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1answer
82 views

Dimension of Lie algebra

Let $o(2l,F)$, with $l \ge2$ and $n=2l$ be the orthogonal lie algebra $\{L\in gl(n,F)|SL=-L^{t}S\}$ where $S=\begin{pmatrix} 0 &I_{l} \\ I_{l} & 0 \end{pmatrix}$. How can I show that ...
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1answer
72 views

Representation of $S_3$ on $\mathbb{C}^6$.

I am asked to decompose $S_3$ into its irreducible unitary representations, on $\mathbb{C}^6$. I wonder how does this differ from analysis the given in the next paper?: ...
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47 views

notation question $Aut_{Modk}(A)$

What is $Aut_{Modk}(A)$ if A is a finite generated algebra, k is a field? Is it a group of automorphisms of A as a vector space over k or what?
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58 views

Inner tensor and restriction

Let $\pi$ resp. $\chi$ be a finite dimensional representation resp. 1-diml. rep. of a finite group. We define $\chi \otimes \pi (g) = \chi(g) \pi(g)$ as a rep of $G$. For $N$ subgroup, does hold ...
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2answers
60 views

Notational query about representation theory and one-dimensional representations

Suppose $\theta$ is a one-dimensional representation of a group $G$, and $\rho : G \to \mathrm{GL}(V)$ is another representation. Define $\theta \otimes \rho : G \to \mathrm{GL}(V)$ given by ...
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51 views

What is the weight of an operator?

What does this mean: "The standard Cartan weight operators for SO(4) are $L_{12}$ and $L_{34}$. An SO(4) irrep is labeled by the highest weight defined by these operators, which is of the form ...
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2answers
184 views

A trivial question concerning $sl_{n}\mathbb{C}$ representations

The question is, does the fact $$ \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0& 0\\ 0 & 1 &0 \end{array}\right)^{2}=0, \left(\begin{array}{ccc} 0 & 0 & 0\\ 0 ...
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1answer
1k views

Question on fundamental weights and representations

I am a bit confused about the notion of "fundamental weights". In a complexified setting, I am thinking of my Lie algebra to be decomposed as, $\cal{g} = \cal{t} \oplus _\alpha \cal{g}_\alpha$ where ...
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2answers
193 views

Morita equivalence of acyclic categories

(Crossposted to MathOverflow.) Call a category acyclic if only the identity morphisms are invertible and the endomorphism monoid of every object is trivial. Let $C, D$ be two finite acyclic ...
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291 views

Topology on the tensor product of two topological vector spaces — how properties does it maintains?

Good morning, this is my first question in this website. If I have two topological vector spaces, say $A$ and $B$, I would like to know 1)how the topology on $A\otimes B$ is canonically defined? ...
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2answers
123 views

Why is dimension of irrep equal to number of copies of its associated submodule?

Let $G$ be a finite group. There exists a decomposition, as a $\mathbb{C}[G]$-module, $\mathbb{C}[G]=n_{1}V_{1}\oplus \cdots\oplus n_{r}V_{r}$, where the $V_{i}$ form a complete set of irreducible ...
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2answers
446 views

sl(2,C) and the harmonic oscillator

I've been studying the finite-dimensional representations of the lie algebra sl(2,C). I've read that these representations are related to the harmonic oscillator and the associated raising and ...
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1answer
57 views

Non-isomorphic representations of equal degree (namely, of $C_4$)

I'm learning about Representation Theory, and I've come across a statement I don't understand: "If $\rho$ and $\rho'$ are isomorphic representations, then they have the same dimension. However, ...
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1answer
90 views

Equivalent module categories

Let $A$ and $B$ be rings and let $A\text{-mod}$ and $B\text{-mod}$ be their abelian module categories. Let $F:A\text{-mod}\to M\text{-mod}$ and $F':B\text{-mod}\to A\text{-mod}$ be functors which ...
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1answer
107 views

Definition of $\mathrm{Hom}_G (V, V') $

The following appears in my notes: Suppose $V$ and $V'$ are vector spaces over a field $F$. Let $G$ be a group, and let $\rho : G \to GL(V)$ and $\rho' : G \to GL(V')$ be ...
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138 views

Representations and homogenous polynomials

$GL_{n}(\mathbb C) \times GL_{m}(\mathbb C)$ acts on the space $P$ of polynomial functions on the space of complex $n$ by $m$ matrices like this: $((g,h)f)(A)=f(g^{T}Ah) $ for $g \in GL_{n}(\mathbb ...
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1answer
169 views

dual representations and partitions

Suppose we have a partition $\mu$ of $n$. There is an associated polynomial irreducible representation $\phi_{\mu}$ of $GL_n(\mathbb C)$. How do I obtain a new representation of $GL_n(\mathbb C)$ ...
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1answer
168 views

$SO(3)$ acting on the space of $3 \times 3$ matrices

Let $SO(3)$ act on the space of $3 \times 3$ real matrices by conjugation. How can I decompose the space of matrices into the sum on minimal invariant subspaces and figure out what they are isomorphic ...
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2answers
338 views

An hermitian operator problem

It is possible to have two hermitian operators $A$ et $B$, with : $B^2 = \mathbb{I}d$ $[A,B] = i * \mathbb{I}d$ where $i$ is the usual (complex) square root of $(-1)$, and $\mathbb{I}d$ is the ...
3
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0answers
124 views

Integral forms of loop algebras.

The question following is about integral forms for semisimple Lie algebras and loop algebras constructed from them. Let $\frak g$ a finite-dimensional Lie algebra over $\mathbb C$ and $L(\frak ...
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0answers
79 views

Minimal embedding of exceptionnal Lie groups into special orthogonal groups

Let $G$ be a Lie group. The set of all $N$ such that $G$ is a subgroup of $SO(N)$ has a minimum $N_{\min}(G)$. (If I am not wrong, $N_{\min}(G)$ is supposed to be less or equal to $\dim(G)$) What is ...
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1answer
382 views

Product of $SU(2)$ representations

I am familiar with the interpretation of the irreducible representations (finite dimensional) of $SU(2)$ in terms of homogeneous polynomials of degree $n$. If I take two of these irreducible ...
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2answers
366 views

Isomorphism between group algebras

I am starting to study group algebras and I am stuck on the following problem. The first part is easy, but I copy it in case it helps to prove the second part. This exercise is taken from ...
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1answer
140 views

Question about induced representations

Let $G_1$ and $G_2$ be finite groups such that $G_1$ is a subgroup of $G_2$. Let $V$ be a representation of $G_1$ (over some field; I am not assuming that the characteristic of the field is $0$ or ...
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3answers
663 views

Character Table From Presentation

I've recently learned about character tables, and some of the tricks for computing them for finite groups (quals...) but I've been having problems actually doing it. Thus, my question is (A) how to ...
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1answer
192 views

Why is every $N$-invariant polynomial function on $n\times n$ matrices in the Plücker algebra?

Let $k$ be a field and $k[{\bf x}] = k[x_{ij}: 1 \leq i, j \leq n]$ be a polynomial algebra that I can think of as the algebra of functions on $n \times n$ matrices that are polynomial in each ...
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1answer
141 views

Right Regular Representation and Automorphic Forms

Let $R(g)$ denote the right regular representation of $SL_{2}(\mathbb{R})$ in $L^{2}(\Gamma\backslash SL_{2}(\mathbb{R}))$ with $\Gamma$ a congruence subgroup. Why is decomposing $R(g)$ equivalent to ...
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1answer
123 views

How to compute the Gel'fand Models for a (quantum) Lie Algebra

Given a lie algebra $g$, how does one approach finding the Gel'fand models? For clarity, by this I mean $\bigoplus_{\lambda\in P^+}V(\lambda)$ where $P^+$ are the dominant weights, and ...
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2answers
594 views

Categorical description of algebraic structures

There is a well-known description of a group as "a category with one object in which all morphisms are invertible." As I understand it, the Yoneda Lemma applied to such a category is simply a ...
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2answers
260 views

Reference for a “wild” problem

I am currently working on something related to the character theory of the group of unipotent upper triangular matrices with elements in a finite field. I have seen in many papers on the topic the ...
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1answer
127 views

Representation of finite group and Classifying finite simple group

My question is quite clear as the title of it. I had studied theory of representation of finite group and I also had known something about the program of classifying finite simple group, and in ...
4
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1answer
270 views

Representation of Finite Groups

Is it true that any finite group determined by representation over closed field? In other words, are there exists two different groups with the same representations? For example, any non-abelian group ...
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1answer
44 views

How to understand $2S4(6) = [2^3]S(3) = 2wrS(3)$?

For example in Kluener's data base of transitive subgroups of $S_n$ ( http://www.math.uni-duesseldorf.de/~klueners/minimum/minimum.html ), one can read their name like the one in the title. What ...
4
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1answer
199 views

Mackey and relatively projective modules

While reading over Alperin's Local Representation Theory and reminding myself how a module is relatively H-projective iff H contains some vertex of the module, I realized I could not prove a basic ...
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101 views

Question concerning semisimple Lie algebras

I'm currently solving a problem in Fulton's Representation Theory A first course and I'm not sure why a particular result is true. One part of the problem (exercise 14.15 if anyone is interested) ...
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1answer
69 views

Reducibility of an $\mathbb{R}[X]$-module

Let $q \in \mathbb{R}[X]$ be an irreducible polynomial of degree 2 and assume $n > 1$ is an intiger. My question is whether the $\mathbb{R}[X]$-module $\mathbb{R}[X] / (q^n)$ is reducible (i.e has ...
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4answers
539 views

How is $\operatorname{GL}(1,\mathbb{C})$ related to $\operatorname{GL}(2,\mathbb{R})$?

I am trying to get a grasp on what a representation is, and a professor gave me a simple example of representing the group $Z_{12}$ as the twelve roots of unity, or corresponding $2\times 2$ matrices. ...
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69 views

Looking for specific resource on the classification of complex irreducible representations of metacyclic groups

My apologies in advance if this question is in any way out of place. I'm currently need of a classification of complex irreducible representations of metacyclic groups. My current reference ...
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1answer
373 views

Question about representing the Dual Space

In Fulton and Harris' Representation Theory, right at the beginning when they introduce representations, they note The dual $V^{\ast} = \mbox{Hom}(V,{\mathbb C})$ of $V$ is also a representation, ...
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1answer
147 views

Hereditary Algebras

I recently began to study representation theory of algebras and I found this problem: Suppose $\Lambda$ is a finite dimensional algebra over an arbitrary field. If $\Lambda$ is hereditary, basic ...
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1answer
131 views

$2×2=3+1$ for $\operatorname{GL}_2$

If $V$ is the natural representation for $\operatorname{GL}(2,q)$, then $V⊗V$ appears to decompose into the direct sum of a (strange?) one-dimensional module and a three dimensional module. I've ...