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

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3
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1answer
286 views

Normal subgroups of $S_N$

Is there a list of all normal subgroups for $S_N$? What is a criteria for a finite group to be a normal subgroup of $S_N$? Which of them are kernels of irreducible representation? From a partition ...
3
votes
1answer
82 views

Construction of representations

Is there an example, where given a conjugacy class in a finite group, can we construct an irreducible representation from it?
7
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1answer
104 views

Computing the dimension of representations in a reducible induced representation

I'm looking at the induction of representations of a parabolic subgroup of $Sp_4$ into the whole group. There are some cases that the result is reducible, and I need to compute the dimensions of the ...
0
votes
1answer
33 views

generalized eigenspaces for many operators III

Let $\phi_1, \cdots, \phi_n$ be commutative linear operators on a vector space $V.\,\,$ Then we have $$V=\oplus V_{(a_i)}, \text{ where } V_{(a_i)} = \{x\in V \mid \exists p \,\,\text{ such that }\,\, ...
0
votes
1answer
107 views

generalized eigenspaces for many operators II

Let $\phi_1, \cdots, \phi_n$ be commutative linear operators on a vector space $V.\,\,$ Then we have $$V=\oplus V_{(a_i)}, \text{ where }\, V_{(a_i)} = \{x\in V \mid \exists p \,\,\text{ such that ...
2
votes
2answers
257 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 ...
1
vote
1answer
178 views

generalized eigenspaces for many operators

Let $\phi_1, \cdots, \phi_n$ be commutative linear operators on a vector space $V.\,\,$ Then we have $$V=\oplus V_{(a_i)}, \text{ where }\, V_{(a_i)} = \{x\in V \mid \exists p \,\,\text{ such that ...
11
votes
2answers
800 views

Restriction to a normal subgroup

More exam preparation. Let $A$ be a normal subgroup of a finite group $G$ and $V$ an irreducible representation of $G$. Show that either $\text{Res}_A^G V$ is isotypic (a sum of copies of one ...
2
votes
1answer
175 views

Finding double coset representatives in finite groups of Lie type

Is there a standard algorithm for finding the double coset representatives of $H_1$ \ $G/H_2$, where the groups are finite of Lie type? Specifically, I need to compute the representatives when ...
2
votes
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 ...
14
votes
2answers
597 views

finite subgroups of PGL(3,C)

The enumeration of finite subgroups of $\operatorname{PGL}_2(\mathbb{C})$ is one of the classic classification problems: mathematicians in many fields know well that the answer is cyclic groups, ...
7
votes
5answers
695 views

Shortest way of proving that the Galois conjugate of a character is still a character

Let $G$ be a finite group and $\chi$ a character of $G$. The values of $\chi$ generate an abelian Galois extension $K$ of $\mathbb{Q}$, and so one can consider the conjugate $\sigma(\chi)$ of $\chi$ ...
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2answers
79 views

Is $\mathbb{Q}[[v^{-1}]] \cap \mathbb{Q}(v)=\mathbb{Q}[[v^{-1}]]$?

I have a question about the some rings and fields. Is $\mathbb{Q}[[v^{-1}]] \cap \mathbb{Q}(v)=\mathbb{Q}[[v^{-1}]]$?
6
votes
2answers
362 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 ...
2
votes
1answer
688 views

Why does the trivial representation have degree 1?

If you have a representation from $G \to Aut(V)$, it has degree $1$ if $V$ is a 1-dimensional vector space over $F$. The trivial representation sends any element of $G$ to the trivial automorphism $v ...
1
vote
1answer
58 views

How do you find the matrices of a representation given the matrices of subrepresentations?

Specifically, there is a passage in Dummit and Foote that says Suppose $V$ is a finite-dimensional $FG$-module and $V$ is reducible. Let $U$ be a $G$-invariant subspace. Form a basis of $V$ by ...
0
votes
0answers
96 views

Prehomogeneous vector spaces

How can I prove that the following 2 prehomogeneous vector spaces are not isomorphic? 1) $(GL_1^2 \times SL_2,2\Lambda_1\oplus\Lambda_1, \mathbb{C}^3\oplus\mathbb{C}^2)$ 2) $(GL_1^2 \times ...
5
votes
2answers
979 views

Irreducible representations of a semidirect product

I have two finite groups. The irreducible representations of their product are given by tensor products of the irreducible of representations of the groups. Is there a way to build the irreducible ...
1
vote
1answer
92 views

questions about the paper: Affine quivers and canonical bases

I am reading the paper Affine quivers and canonical bases. I have a question on page 114 of the paper. In the proof of property (b), line 6 of page 114, why "for each $\gamma \neq 1$, $tr(\gamma, ...
2
votes
1answer
649 views

relations between root lattice and weight lattice

Let $Q$ and $P$ denote the $\mathbb{Z}$-span of the simple roots and fundamental weights respectively. What are the relations between $Q$ and $P$? Does $P$ contain $Q$? Thank you.
46
votes
3answers
930 views

How to think of the group ring as a Hopf algebra?

Given a finite group $G$ and a field $K$, one can form the group ring $K[G]$ as the free vector space on $G$ with the obvious multiplication. This is very useful when studying the representation ...
2
votes
0answers
176 views

clarification on the definition of a group C*-algebra

I've been trying to understand the definition of a group C*-algebra. Given a topological group $G$ and a C*-algebra $A$, let $u: G \to A$ define a unitary representation $U(G)$ of $G$ on $U(A)$, the ...
3
votes
3answers
417 views

How can there be multiple irreducible representations of a group each having distinct dimension?

The matrix elements for the $(2l+1)$-dimensional irreducible representation of SO(3) are given by: $D_{m',m}^l(\phi_1,\Phi,\phi_2)=[i^{m'-m}\sqrt{(l+m')!(l-m')!(l+m)!(l-m)!}$ ...
4
votes
2answers
305 views

How to generalise $(\wedge^2 \chi)(g) = \frac{1}{2}(\chi(g)^2-\chi(g^2))$?

One can decompose $\bigotimes^2 V = \bigvee^2 V \oplus \bigwedge^2 V$, getting a corresponding decomposition for representations, say when $V$ is a module for some finite group $G$. One then has the ...
9
votes
1answer
340 views

Field of definition of representations of symmetric groups

Can one show in an elementary way, without recourse to Young tableaux etc., that the complex representations of symmetric groups are realisable over $\mathbb{R}$? It is easy to show that they are all ...
6
votes
2answers
4k views

What is the meaning of an “irreducible representation”?

What does it mean to talk about the "irreducible representatives of SO(3)"? I'm struggling to understand the concept of irreducible representations. Could someone give a concrete example for someone ...
12
votes
3answers
367 views

Complex Galois Representations are Finite

In A First Course in Modular Forms, Diamond and Shurman leave as an exercise ($9.3.3$) that every complex Galois representation is finite. While I think I have worked through this exercise here, this ...
22
votes
2answers
767 views

Surprising but simple group theory result on conjugacy classes

I have read that for any group $G$ of order $2m+1$ (odd) with $n$ conjugacy classes, it is always the case that $16$ divides the value $(2m+1)-n = |G|-n$. This seems to me like an astonishing ...
0
votes
1answer
64 views

A compact group with a $G$-finite function

Suppose we have a compact group $G$ with continuous $f$ on $G$ that is also $G$-finite. I am told that then, out of all the irreducible representations $\pi$ of $G$ we must have $\pi(f)\neq 0$ only ...
7
votes
3answers
689 views

Classifying the irreducible representations of $\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}$

I need help with the following problem: Suppose $n$ is some positive integer, and $n|p-1$. Classify all irreducible representations of $$\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}.$$ ...
2
votes
1answer
582 views

Irreducible representations and simple lie algebras

Could you give me a hint how to prove the following? A representation $R$ of a complex Lie algebra $\mathfrak{g}$ is irreducible iff the image $R(\mathfrak{g})$ is a simple Lie algebra.
18
votes
2answers
583 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 ...
8
votes
1answer
305 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, ...
3
votes
2answers
315 views

Minimal embedding of a group into the group $S_n$

I am aware this has come up recently (Embedding of finite groups for example) but after searching I haven't found the particular answer I'm looking for. Suppose I know the character table and can ...
1
vote
1answer
317 views

Irreducible representation decomposition

Any faithful finite group representation can be written as a sum of irreducible representations $\rho = \oplus_{i} a_i \rho_i$ such that $Ker(\rho)=0=\bigcap_i Ker(\rho_i)$ - is this sufficient to ...
0
votes
1answer
122 views

Sum of non-principal representation over the group is zero?

I am stuck on an exercise. (a) Let $\mathfrak{X}$ be an irreducible $F$-representation of $G$ over an arbitrary field. Show that $\sum_{g \in G} \mathfrak{X}(g) = 0$ unless $\mathfrak{X}$ is the ...
8
votes
3answers
534 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 ...
4
votes
2answers
175 views

Character theory questions

I am following the text by Isaacs on character theory and I have a few questions. From p. 10, it seems like an reducible representation is one whose matrix at each group element can be written in a ...
1
vote
1answer
297 views

the volume of 3-sphere

My question is why the volume of 3-sphere that lies between a rotation angle of $\theta$ and $\theta+d\theta$ is $$2\pi \sin^2(\theta/2) d\theta$$
1
vote
1answer
424 views

Dimension of irreducible representations of finite p-groups

Let $G$ be a finite $p$-group. Is it true that every irreducible representation of $G$ over an algebraically closed field of characteristic zero ($\mathbb{C}$, for example) must have dimension a ...
3
votes
1answer
116 views

What is the dimension of a representation

What is meant by dimension of a representation in the following excersize: "Prove that any irreducible representation of an abelian group has the dimension of 1"? I looked at the solution, and it ...
7
votes
1answer
237 views

What is a description for the following number theoretic object?

The title couldn't quite contain the question, so I didn't attempt to make it precise. I should note that this is the third or fourth question I've asked these past two days about problems I've been ...
3
votes
1answer
161 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 ...
6
votes
3answers
322 views

Basic Representation Theory

I'm reading a paper that uses representation theory, and I'm stuck on something simple. Say $\Delta$ is an abelian group, and $\hat{\Delta}$ its group of irreducible characters. Say $M$ is a ...
2
votes
2answers
323 views

An 'opposite' or 'dual' group?

Let $(G, \cdot)$ be a group. Define $(G, *)$ as a group with the same underlying set and an operation $$a * b := b \cdot a.$$ What do you call such a group? What is the usual notation for it? I tried ...
4
votes
1answer
167 views

Questions about p-adic representations

In a paper I'm currently reading, they have the following situation: $k$ is some number field that doesn't have a primitive $p^{th}$ root of unity, and $k(\zeta_p)$ a field above it with Galois group ...
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1answer
104 views

Question about notation / terminology

I'm given the following in a homework question: Let $G$ be a group and $k$ an algebraically closed field. (a) Show that the action of $G \times G$ on $C_k (G)$ defined by $$ (g_1, g_2) \varphi (x) ...
1
vote
2answers
300 views

What are the differences between classical Yang-Baxter Equation and quantum Yang-Baxter Equation?

what are the differences between classical Yang-Baxter Equation and quantum Yang-Baxter Equation? Thank you very much.
12
votes
3answers
680 views

Dimensions of irreducible representations of finite groups over $\mathbb Q$

If $G$ is a finite group, then it is well known that there are finitely many inequivalent irreducible representations of $G$ over $\mathbb{C}$; moreover the sum of squares of dimensions of the ...
8
votes
3answers
480 views

Does every irreducible representation of a compact group occur in tensor products of a faithful representation (and its dual)?

Let $G$ be a compact (Hausdorff) group and $V$ a faithful (complex, continuous, finite-dimensional) representation of it. (Hence $G$ is a Lie group.) Is it true that every irreducible representation ...