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

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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 ...
2
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0answers
82 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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69 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. ...
6
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2answers
369 views

On “complexifying” vector spaces

I think this question should be quite trivial. For some reason I did not really get the author's argument. I shall use the symbols from the book to avoid ambiguity. In the book "Lectures on Lie ...
3
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2answers
245 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
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1answer
106 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 ...
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2answers
417 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 ...
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2answers
529 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 ...
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106 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 ...
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1answer
401 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?
4
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1answer
162 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 $C_3$, ...
6
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1answer
477 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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147 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
2answers
211 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 $$ \operatorname{Hom}_{\operatorname{End}(V)}(V,W) \otimes V \to W $$ ...
2
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1answer
59 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 ...
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391 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 ...
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2answers
186 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 ...
2
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1answer
148 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 ...
3
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2answers
426 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? ...
11
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1answer
217 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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1answer
259 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 ...
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2answers
54 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 $SL_{2}(\mathbf{R})$?
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1answer
46 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, ...
2
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1answer
116 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 $(KQ)^{op} \cong KQ^{op}$ ...
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3answers
542 views

Representations of a cyclic group of order p over a field of characteristic p?

Let $p$ be a prime. My eventual goal is to prove that the only irreducible representation of a $p$-group over a field of characteristic $p$ is the trivial representation. At the moment, I'm trying ...
2
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1answer
262 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 ...
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147 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 ...
3
votes
1answer
213 views

fundamental representation of $\ sl(l+1,F)$

This problem concerns the topic representation theory of Lie Algebras. The main purpose of the exercise is to study the form of the fundamental dominant weights of a Lie Algebra. I would be very ...
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1answer
100 views

Simple modules and blocks

Two questions. Q1. Let $A$ be a finite-dimensional algebra over a field with block decomposition $A=A_1\oplus A_2\oplus\cdots\oplus A_r$ and $M$ an $A$-module. If every composition factor of $M$ lies ...
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209 views

Question about Real Representations

I am stuck on a problem in Fulton's Representation Theory: A First Course. Exercise 3.39 states: Let $V_0$ be a real vector space on which $G$ acts irreducibly, $V=V_0 \otimes \Bbb C$ the ...
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273 views

A question on partitions of n

Let $P$ be the set of partitions of n. Let $\lambda$ denote the shape of a particular partition. Let $f_\lambda(i)$ be the frequency of $i$ in $\lambda$ and let $a_\lambda(i) := \# \lbrace j : ...
2
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1answer
105 views

Form of isomorphisms between subgroups of GLn

Suppose that I've been given two subgroups of the general linear group (over the real numbers), which are isomorphic. From this information alone, can I deduce the form of the isomorphism? I suspect ...
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1answer
69 views

What is the correct seminormal representation of (23) corresponding to the [22] partition of S4?

I am studying an undergraduate thesis, called A Fast Fourier Transform for the Symmetric Group, by Tristan Brand, one time a student at Harvey Mudd College. Source: ...
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1answer
330 views

Decomposing left regular representation of cyclic group over $\mathbb{Q}$

Let $G$ be a cyclic group of order $p$, where $p$ is prime. Let $V = \mathbb{Q}(G)$ be the rational group ring of $G$. How do you explicitly decompose $V$ as a direct sum of irreducible ...
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1answer
477 views

On Frobenius reciprocity theorem

The classical Frobenius reciprocity theorem asserts the following: If $W$ is a representation of $H$, and $U$ a representation of $G$, then $$(\chi_{Ind W},\chi_{U})_{G}=(\chi_{W},\chi_{Res ...
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374 views

Class function as a character

Suppose $G$ is a finite group, and $\{ \chi_1,\chi_2,\cdots,\chi_k \}$ be the complete set of irreducible complex characters of $G$. If $\theta$ is a class function on $G$, i.e. function from $G$ to ...
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1answer
314 views

Proof that Lie group with finite centre is compact if and only if its Killing form is negative definite

I am gathering material for an exposition on Lie theory and I am after proofs that a Lie group with finite centre is compact if and only if its Killing form is negative definite. I know of one, ...
2
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1answer
140 views

Projective Tetrahedral Representation

I can embed $A_4$ as a subgroup into $PSL_2(\mathbb{F}_{13})$ (in two different ways in fact). I also have a reduction mod 13 map $$PGL_2(\mathbb{Z}_{13}) \to PGL_2(\mathbb{F}_{13}).$$ My question is: ...
3
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0answers
108 views

Blocks and covering modules

I have a question relating to p109 of Local representation theory by JL Alperin. Let $G$ be a finite group and let $N$ be a normal subgroup. If $B$ is a block of $G$, why must $B$ be a summand of ...
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1answer
100 views

Matrices and dimension question

Let $k$ be an algebraically closed field and consider the $k$-algebra $M_{n}(k)$ (the set of all $n \times n$ matrices with entries in $k$). Let $T$ be an indecomposable $M_{n}(k)$ module, why must ...
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1answer
95 views

Sum of all simple submodules

Let $M$ be an $A$-module where $A$ is a $k$-algebra ($k$ an algebraically closed field). If $\operatorname{Soc}{(M)}$ denotes the socle of $M$, i.e the sum of all simple submodules of $M$ why is that ...
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1answer
86 views

subrepresentations of the product of nonisomorphic irreducible representations

Let G be a finite group, and let V and W be two irreducible representations of G over a field k. How do I show that if V and W are not isomorphic, then the G-subrepresentations of the product V x W ...
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1answer
52 views

Blocks and simple modules

I have a (probably very straightforward) question about blocks and simple modules. The problem I'm having is on p103 of Local representation theory by JL Alperin. Let $G$ be a finite group. Let $B$ ...
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1answer
83 views

Defect groups and subgroups

I would like to prove the following statement from Alperin's Local representation theory, p101: Lemma Let $b$ be a block of the subgroup $H$ of $G$ and let $D$ be a defect group of $b$. If $b^G$ is ...
5
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1answer
207 views

Decomposability and irreducibility for representations

I am trying to understand the analogies between linear representations and permutation representations: A representation $\rho:G \to GL(V)$ is irreducible if it has no invariant subspace apart from ...
2
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1answer
402 views

character tables for groups of order $pq^2$

What is the character table for groups or order $pq^2$? The classification of order $pq^2$ groups has already been discussed in relation to Sylow theory. For the Abelian groups, $\mathbb{Z}_p ...
3
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1answer
162 views

Endomorphisms of a representation

Let $G$ be a group acting continuously on a free $\mathbb{Z}_p$-module of finite rank. Assume that $End_{G}(T)$ and $End_G(T/p)$ are the homotheties. Is it possible that $End_{G}(T/p^n)$ contains ...
8
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1answer
141 views

Representations of SL(2) as an algebraic group

I understand the finite-dimensional representations of $\text{SL}(2,\mathbb C)$ as a Lie group and their correspondence with Lie algebra representations of $\mathfrak{sl}(2,\mathbb C)$. Does anyone ...
2
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1answer
294 views

Representation Decomposition of $ \operatorname{Sym}^{k+6} V$

In Fulton and Harris's Text Representation Theory: A First Course, exercise 1.12(b) asks to show that $\operatorname{Sym}^{k+6}V \cong \operatorname{Sym}^kV \oplus R$ as representations of $\frak ...