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

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Extensions and Kazhdan's Property (T)

Is Kazhdan's property (T) stable under extensions? i.e. if $G$ is an extension of a group with property (T) by a group with property (T), does it follow that $G$ has property (T)?
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1answer
147 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 ...
1
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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 ...
7
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2answers
185 views

Reference for densities and pseudoforms and non-tensorial representations of $\operatorname{GL}(n)$ and associated vector bundles

I'm looking for a reference that will set me straight on a few things. It started out with densities. In John Lee's book, "Introduction to Smooth Manifolds", densities on vector spaces are functions ...
3
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0answers
107 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) ...
1
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2answers
199 views

Tensor product representations of $\text{sl}(2;\mathbb{C})$

Define $V_m$ as the space of all homogeneous polynomials in two complex variables of degree $n$. Then we can define a representation of $SU(2)$ on the space $V_m$ by the formula $$[\Pi_m(U)f](z) = ...
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0answers
73 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 ...
6
votes
1answer
152 views

Historical reference request: Young tableaux

I am writing up an article on the RSK correspondence. To this end, I want to understand the history behind the invention of the Young tableaux and how it was introduced into the study of the symmetry ...
2
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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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0answers
79 views

2-Brauer characters of the symmetric group $\mathfrak{S}_3$

In a previous question, I asked how to compute Brauer characters of the alternating group $\mathfrak{A}_3$; the answer to this question provided a solution for all cyclic groups. I would now like to ...
2
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0answers
90 views

symplectic representations: when could the center act trivially?

I'm considering a problem about symplectic representation of real reductive group, which fits more or less into the setting of symplectic representations discussed in Milne's survey ''Shimura ...
2
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0answers
82 views

radical layers equal socle layers

I've read that the radical layers of the group algebra $kP$ of a $p$-group $P$ coincides with the its socle layers (char $k = p$). What does this tell me about the structure of the group algebra or ...
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1answer
44 views

A question about bounding character ratios

The following question arose in a research project, and I'm sure it must be well known. I even know a very indirect proof, and would love to know if anybody knows a simple one. Here is the question. ...
4
votes
1answer
181 views

2-dimensional $\ell$-adic representations [closed]

In an assignment, I have to give an example of a 2-dimensional $\ell$-adic representation of the absolute Galois group of $\mathbb{Q}$, bu I am faced with the problem that I do not a lot of these. Or ...
3
votes
2answers
314 views

Irreducible representations of $\mathbb{Z}$

I'm wondering what are the irreducible representations of the group ($\mathbb{Z}$,+). Knowing that for $\mathbb{Z}_n$ the 1-dimensional representations are the nth roots of unity, I considered taking ...
2
votes
1answer
271 views

The symmetric algebra as a g-module

I'm quite sure that this question is not difficult, but I think that my understanding of the definitions is just not deep enough yet: Given a lie algebra $g$, and a $g$-module $V$ (or equivalently a ...
11
votes
2answers
671 views

What is the standard definition of an ordinary (local) $p$-adic Galois representation?

Let $V$ be a $n$-dimensional $\mathbf{Q}_p$-vector space with a continuous action of $\operatorname{Gal}(\bar{L}/L)$, where $L$ is a complete discretely valued field of characteristic zero with ...
32
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5answers
2k views

Can every group be represented by a group of matrices?

Can every group be represented by a group of matrices? Or are there any counterexamples? Is it possible to prove this from the group axioms?
5
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1answer
119 views

representation of a group by an integral matrix

A representation of a group $G$ on a vector space $V$ over a field $K$ is a group homomorphism from $G$ to $GL(V)$, the general linear group on $V$. That is, a representation is a map $ \rho ...
8
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2answers
711 views

Characters of Symmetric and Antisymmetric Powers

Let $V$ be a representation with character $\chi$. I would like to have a formula for the characters of the representations $\mathrm{Sym}^m[V]$ and $\wedge ^m[V]$ in terms of $\chi$. Fulton and ...
4
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2answers
500 views

Classification of unitary irreducible representation

I recently learnt that one can explicitly classify the unitary irreducible representations of $\mathrm{SL}(2,\mathbb R)$. In the end one has a list of all these representations given by explicit ...
4
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2answers
985 views

a visual route to learning Galois theory

I really like the ideas of Galois theory: that you can think about all the algebraic numbers you can make starting with some set of them that there is some structure to this set of "algebraically ...
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2answers
672 views

one dimensional representation is irreducible

let $G$ be a finite group. let $V$ be an $F$-vector space. and $\rho:G\rightarrow GL(V)$ be a one dimensional representation. I don't see why it is automatically irreducible. My guess: $V=\langle ...
2
votes
1answer
131 views

Complete reducibility of finite-dimensional representations of $\mathfrak{sl}_2(\mathbb{C})$

By Weyl's theorem every finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{C})$ is completely reducible, because $\mathfrak{sl}_2(\mathbb{C})$ is a (semi) simple Lie algebra. It seems there ...
5
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1answer
476 views

Step by step procedure to obtain irreducible representations and construct character table of a group

I am studying group theory and character table of $S_2$ is given in the book. But how to obtain this table is not given. Can someone explain how exactly to construct this table?
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1answer
99 views

How to calculate the orders of these groups: $T_n(F_p) , SL_3(F_p) , O_3(F_2) , SO_3(F_3) , O_2(F_7)$

where $T_n(F_p)$ denotes the the group of $n \times n$ invertible upper triangular matrices with entries in the field $F_p$ and $p$ is a prime. $O_3(F_2)$ is the orthogonal group. $SO_3(F_3)$ stands ...
4
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1answer
168 views

Valuations on number fields

I'm trying to explicitly compute modular representations of some finite groups -- the easiest example to discuss is the cyclic group $C_3$ when $p=3$. The three ordinary irreducible modules for $C_3$, ...
7
votes
3answers
168 views

Is $\mathbb{Z}[G]$ integral over $\mathbb{Z}$?

Here $G$ is a finite group(not neccessarily abelian),then there is a statement in some representation book that $\mathbb{Z}[G]$ is integral over $\mathbb{Z}$.That is, every element in $\mathbb{Z}[G]$ ...
4
votes
2answers
229 views

Isomorphism between $\operatorname{Hom}_{\operatorname{End}(V)}(V,W) \otimes V$ and $W$

Let $V$ be a vector space over $\mathbb C$ and $W$ a $\operatorname{End}(V)$-module. I'm having difficulty seeing why the map $$ \operatorname{Hom}_{\operatorname{End}(V)}(V,W) \otimes V \to W $$ ...
3
votes
1answer
62 views

Eigenvalues of the matrix $(-1)^{i_1+i_2+\cdots+i_k+j_1+j_2+\cdots+j_k}$

$M_{[i],[j]}=(-1)^{i_1+i_2+\cdots+i_k+j_1+j_2+\cdots+j_k}$, where $1\le i_1<i_2<\cdots<i_k\le n$ and $1\le j_1<j_2<\cdots<j_k\le n$, can be taken to be an $\left(n\atop ...
5
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1answer
881 views

Spherical harmonics give all the irreducible representations of $SO(3)$?

It is mentioned in Wiki that the spaces $\mathcal{H}_{k}$ of spherical harmonics of degree $k$ give ALL the irreducible representations of $SO(3)$. Could anyone tell me where can I find the proof? ...
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483 views

Subgroups as isotropy subgroups and regular orbits on tuples

Is there some natural or character-theoretic description of the minimum value of d such that G has a regular orbit on Ωd, where G is a finite group acting faithfully on a set Ω? Motivation: In ...
2
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1answer
158 views

How to find a representation with given order?

This term I have to study "Representation theory of finite group", and my professor chooses the book "Representations and characters of groups" by Gordon James and Martin Liebeck, which was published ...
4
votes
2answers
189 views

Projective modules over $k[X,Y]/(X^3,Y^5)$

I'm searching for an example of a module, that is not projective for $k[X,Y]/(X^3,Y^5)$, but projective for the two subalgebras $k[X]/(X^3)$ and $k[Y]/(Y^5)$. (I don't think it is relevant, but in ...
4
votes
2answers
206 views

More on the versions of the Peter-Weyl theorem

The following three statements appear analogous: For a finite group $G$, the group algebra $\mathbb C[G]$ decomposes as $\bigoplus_{V {\rm\ irred}} V^* \otimes V$. (Peter-Weyl) For a compact group ...
5
votes
1answer
197 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 ...
3
votes
2answers
455 views

Irreducible representations of a cyclic group of order p over a field of q elements when p and q are distinct primes

What is a sufficient condition for the existence of an irreducible representation of degree $n$ of the cyclic group of order $p$ over the field of $q$ elements when $p$ and $q$ are distinct primes? ...
5
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3answers
463 views

Reference request for algebraic Peter-Weyl theorem?

It seems that, for $GL_n$, and possibly for something like complex reductive groups $G$ in general, there's an algebraic version of the Peter-Weyl theorem, which might say that the coordinate ring of ...
2
votes
1answer
113 views

Subspaces of Representations of Lie Groups

Question 8.17 from Fulton's Representation Theory reads as follows: Let $V$ be a representation of a connected Lie group $G$ and $\rho: \frak{g} \to$ $ \operatorname{End}(V)$ the corresponding map of ...
12
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1answer
248 views

Do all algebraic integers in some $\mathbb{Z}[\zeta_n]$ occur among the character tables of finite groups?

The values of irreducible characters of a finite groups are always sums of roots of unity; do all sums of roots of unity (i.e. algebraic integers in the maximal abelian extension of $\mathbb{Q}$) ...
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0answers
158 views

Geometric Interpretation of Maschke's Theorem

I was wondering if anyone here could pitch a plausible geometric interpretation of Maschke's Theorem for $FG$-modules (or at least for a particular instance of its conclusions.) It seems reasonable ...
4
votes
1answer
289 views

Local Langlands correspondence: Weil-Deligne group

While reading the book 'Langlands correspondence for loop groups', I came across the definition of the Weil group $W_F$ and the Weil-Deligne group $W'_F = W_F \ltimes \mathbb{C}$ with action ...
7
votes
1answer
556 views

Centers of quotients of Lie Groups

Exercise 7.11 in Fulton's Representation Theory asks to prove that: (a) Show that any discrete normal subgroup of a connected Lie group $G$ is in the center $Z(G)$ (b) If $Z(G)$ is discrete, show ...
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1answer
55 views

An Embedding of $PGL_n \Bbb C$

I have a question about the projective general linear group. How does one realize it as a matrix group? Specifically, what is an embedding of $PGL_n \Bbb C \to GL_k \Bbb C$ for some $k$? In this case, ...
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2answers
56 views

Spanning the Lie Algebra of $SL_{2}(\mathbf{R})$

What is a sufficient criteria for testing whether or not a set of matrices span the Lie algebra of $SL_{2}(\mathbf{R})$?
2
votes
1answer
142 views

Opposite and connected quivers problem

Here are two problems from Elements of the Representation Theory of Associative Algebras by D. Simson, et. al (Page $65$). $1$. Let $Q=(Q_{0},Q_{1},s,t)$ be a quiver. Prove $(KQ)^{op} \cong KQ^{op}$ ...
3
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1answer
325 views

Complete reducibility of sl(3,F) as an sl(2,F)-module

I was reading the Weyl's theorem on the complete reducibility of a finite dimensional representation of semi-simple Lie algerba and wanted to apply the theorem to the following problem which was ...
3
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0answers
176 views

Behaviour of group representations under extension of field

When studying behaviour of linear representations of finite groups under extension of fields, I came up across two natural questions, which I couldn't solve (Reference: Representations of finite ...
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0answers
184 views

How to proof that the irreducible representations of the upper triangular $n$ by $n$ matrices are $V_1,…,V_n$?

Here the ground field $k$ is algebriacally closed.$A$ is the algebra of upper triangular $n$ by $n$ matrices. I already know that $V_i$ which is 1-dimensional, and any matrix $x$ acts by ...
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1answer
608 views

Importance of Group Representation theory

I was reading about the group representation, but couldn't really why is it important or interesting. Can you someone explain about some of the important mathematical applications (not from physics, ...