Tagged Questions

4
votes
1answer
83 views

How to understand $\frac{d}{dt}\{(\exp(tX))_*(Y)\}|_{t=0}=[X,Y]$?

Let $G$ be a Lie group on which $X$ and $Y$ are two vector fields. Let $G\xrightarrow{\exp(tX)} G$ be the (Lie theory) exponential map corresponding to $X$. Then of fundamental importance is ...
2
votes
2answers
38 views

On the eigenvalues of a linear transformation $\tau$ such that $\tau^3 = \mathrm{id}$

I am reading the book on representation theory by Fulton and Harris in GTM. I came across this paragraph. [..] we will start our analysis of an arbitrary representation $W$ of $S_3$ by looking ...
2
votes
2answers
40 views

On the proof of Schur's lemma in Fulton & Harris

I'm reading the book on representation theory by Fulton and Harris. I'm stuck with the proof of Schur's Lemma 1.7: Schur's lemma 1.7 If $V$ and $W$ are irreducible representation of $G$ and ...
1
vote
0answers
47 views

What to take from representation of $S_d$?

I am reading about group representations, and books I read all contain the representation theory for symmetric groups $S_d$. However none of them presents the material in a friendly way. After reading ...
10
votes
5answers
186 views

Applications of Character Theory

Some of the applications of character theory are the proofs of Burnside $p^aq^b$ theorem, , Frobenius theorem and factorization of the group determinant (the problem which led Frobenius to character ...
2
votes
2answers
101 views

Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$.

Suppose $G$ is an abelian group and $a\in G$ and $$f:\left<a \right>\to\Bbb T$$ is a homomorphism. Can $f$ be extended to a homomorphism on $G$: $$g:G\to \Bbb T$$ ? $\Bbb T$ is the circle ...
1
vote
0answers
32 views

Computing Invariant Subspaces of Matrix Groups

Does anyone have a program written in Mathematica (or SAGE or GAP) that computes the invariant subspace lattice of a matrix group?
7
votes
1answer
255 views

Do these two sets of matrices form groups?

Stimulated by some Physics backgrounds, consider the following two sets of matrices. Notations and definitions:Let $A,B$ be two complex $n\times n$ matrices, then $\left [ A,B \right ...
2
votes
1answer
32 views

properties of Sym^2 vector subspace/properties of tensor products

I have a problem: Let $V$ be an $n$-dimensional complex vector space and let $B=\{e_1,e_2,...,e_n\}$ denote the elements of a chosen basis. Let $\rho:G \to GL(V)$ be an irreducible representation. Let ...
1
vote
1answer
39 views

Representations - Tensor Product prove properties of tensor product

I have a problem: Let $V$ be an $n$-dimensional complex vector space and let $B=\{e_1,e_2,...,e_n\}$ denote the elements of a chosen basis. Let $\rho:G \to GL(V)$ be an irreducible representation. Let ...
3
votes
3answers
114 views

Are Clifford groups very *non-commutative*?

Clifford groups seem to be very non-commutative by the relation \begin{equation} \gamma_{i}\gamma_{j}=-\gamma_{j}\gamma_{i}. \end{equation} But is it really so? Can we put this degree of ...
3
votes
0answers
35 views

Duality of $Z(G)$ and $[G,G]$ in representation?

This question and its many wonderful answers illustrate many faces of the duality of $Z(G)$ and $[G,G]$, the centre/ commutator duality of a group. I was thinking about its manifestation in group ...
2
votes
0answers
33 views

Why does $d_{\alpha}$ divide $\#G$ for $\alpha\in\hat{G}$?

Let $\alpha$ be a unitary irreducible representation of a finite group $G$. Then we have \begin{equation} d_{\alpha}|\#G, \end{equation} where $d_\alpha$ is the degree of the representation and $\#G$ ...
2
votes
0answers
73 views

Representations of $\text{GL}_2(\mathbb{Q})$

Let's say that as a representation theorist I am naively interested in representations of $G(\mathbb{Q})$, where $G$ is an algebraic group defined over $\mathbb{Q}$. For the purposes of this question, ...
13
votes
1answer
197 views

Writing a group element as $ghg^{-1} h^{-1}$ and as $g^2 h^2$

I recently read the elegant paper Generalized Frobenius Schur Numbers, by Bump and Ginzburg, which I learned about here. The results in this paper imply the following: Let $G$ be a finite group ...
3
votes
0answers
55 views

Induction from trivial representation and number of irreducibles

Let $G$ be a finite group, $S$ a subgroup and $K$ a field, whose characteristic does not divide the group order. Let $\pi$ be the trivial representation of $S$. Are there criteria in how many ...
4
votes
2answers
79 views

Good book on representation theory after reading Rotman

I'm about to finish Rotman's "Introduction to the Theory of Groups" and I would like to continue my study of group theory with a book on representation theory. The book should give a broad overview ...
1
vote
1answer
37 views

Isomorphisms of the Lorentz group and algebra

I'm trying to read a few books on QFT and some seem to say the Lorentz algebra obeys $\mathfrak{so}(1,3)\otimes \mathbb{C} \cong \mathfrak{su}(2) \oplus \mathfrak{su}(2)$ while others say ...
1
vote
0answers
21 views

Tensor product of an irreducible $G$-representation and a one-dimensional representation [duplicate]

If $G$ is a finite group, $V$ is an irreducible $G$-representation and $W$ is any 1-dimensional $G$-representation (both over an algebraically closed field of characteristic zero), show that $V ...
0
votes
0answers
16 views

concerning coadjoint representation

Let $\xi $ be the vector field on $\frak{g}^*$ (dual of Lie algebra) which correspond to element $X$ of the Lie algebra $\frak{g}$. Then why have we $\xi(F)=K_*(X)F$ where here $K=Ad^*(g)$ is ...
3
votes
1answer
25 views

action of orthogonal group on the space of antisymmetric bilinear forms

What is the natural action of orthogonal group on the space of antisymmetric bilinear forms.
1
vote
0answers
46 views

irreducible representation of a group

Reduced group $C^\ast$-algebra of group $G$ is defined to be $G^*_{r}(G)=\overline{\lambda(L^1(G))}$ where $\lambda$ is left regular representation. My question is how to get a irreducible ...
7
votes
2answers
64 views

Complex finite dimensional irreducible representation of abelian group

I'm supposed to show that each Complex finite dimensional irreducible representation of an abelian group is one dimensional. For any map $\phi: V \rightarrow V$ it holds that $\phi(\rho(g)v) = ...
1
vote
2answers
69 views

irreducible representation of non-abelian p-group

Can someone help with the following problem? Let $G$ be a non-abelian group of prime-power order $p^n$ and $E$ be an irreducible $G$-space over $\mathbb{C}$ giving a faithful representation of $G$. ...
0
votes
0answers
60 views

Show group is isomorphic to finite Heisenberg group

Show that the group $\langle x,y,z$ $|$ $z = xyx^{-1}y^{-1}$, $zx=xz$, $zy = yz$, $x^n = \mathbb{I}, $ $y^n = \mathbb{I}$, $z^n = \mathbb{I} \rangle$, $(n \in \mathbb{Z_{>0}})$ is isomorphic to ...
3
votes
1answer
63 views

Exercise 2.15 M.Isaacs' Character theory of finite groups

I'm beggining to study character theory, and i'm doing some problems from Isaacs' Character theory book. I would need some help with this one: (2.15): Let $\chi\in \operatorname{Irr}(G)$ be ...
1
vote
1answer
68 views

Exercise 2.8 M.Isaacs' Character theory of finite groups

I'm a starter at character theory. I'm trying to do this exercise: (2.8) Let $\chi$ be a faithful character of a group $G$. Show that $H\subseteq G $ is abelian if and only if every irreducible ...
4
votes
1answer
81 views

Character theory exercises [closed]

I'm doing the exercises from chapter 2 of M.Isaacs' Character theory of finite groups, and I'm having problems with some of them. In particular, I would need help with these ones. Thank you very much ...
2
votes
1answer
42 views

Group representation scalar product

Let $\rho: G \rightarrow GL(V)$ be a finite dimensional complex representation of the group $G$. Show that there is an inner product on $V$ such that $G$ acts by unitary matrices. My approach so far ...
6
votes
2answers
103 views

Do group rings appear outside of representation theory?

I am particularly concerned with finite groups. I have seen group rings used in the fundamentals of representation theory as the dual notion to representations. I haven't ever seen them anywhere ...
3
votes
1answer
64 views

Character on conjugacy classes

Let $V_j$, $j = 1,2$ be finite dimensional representations of a group $G$. Show: $\chi_{V_j}$ is a constant on each conjugacy class of $G$, where $\chi_{V_j}$ is the character of the representation. ...
1
vote
0answers
39 views

Quote on the Littlewood-Richardson Rule

In Gordon James's paper "The representation Theory of the Symmetric Group" he says "The author was once told that the Littlewood–Richardson rule helped to get men on the moon but was not proved until ...
3
votes
1answer
93 views

Irreducible subgroups of $\text{GL}(2,p)$

I just wonder any sources that provides the list of all minimal irreducible subgroups of $\text{GL}(2,p)$. Here the term "minimal" means there is no proper subgroup that is irreducible. A list of ...
4
votes
2answers
68 views

Characters of irreducible representations

Suppose $G$ is a finite group of odd order and $\chi$ is the character corresponding to some 2-dimensional representation of $G$. Must $\chi(x)\neq 0$ for every $x\in G$?
3
votes
0answers
74 views

Irreducible representation of $S_4$

Could one please point out an irreducible representation of degree 2 of the group $S_4$. Thank you.
5
votes
0answers
51 views

Subgroup Structure of $\mathrm{SL}(2, p^2)$, and Its Irreducible Characters

I am taking a course in representation theory of finite groups,and somehow I ended up getting assigned to write a report on subgroup structure and irreducible characters of $\mathrm{SL}(2,7)$ and ...
1
vote
1answer
33 views

Is inversion of an irrep equivalent to inversion of the corresponding group element?

If $g\in G$ and $R:G\rightarrow GL\left(V\right)$ is the matrix form of an irreducible representation of $G$ then is the following statement true? $R^{-1}\left(g\right)=R\left(g^{-1}\right)$ Where ...
3
votes
1answer
53 views

Are the irreps of SO(n) orthogonal?

This seems like a trivial question, but I'm wondering if irreducible representations automatically inherit the properties of the group that they represent. Specifically, if I take an irrep of SO(4) ...
2
votes
0answers
38 views

Molecular vibrations and a generalisation of Wigner's rule for (non-finite) compact groups

years student of mathematics and write my script for my bachelor. The topic is "Representations of groups and applications in physics". I understand the representations very good but now i want to ...
3
votes
1answer
65 views

Are the irreps of SO(4) necessarily real?

I'm not a group theory buff, but I was confused when I used some code to generate the irreducible representatives of SO(4) and found the resulting matrix elements to be complex. Is this possible, or ...
1
vote
1answer
60 views

Relation between a representative of a conjugacy class and corresponding irreducible character value

Is there a relation between the representative order of a conjugacy class and the corresponding irreducible character value? Thanks in advance.
2
votes
1answer
24 views

quotient representation — show that it actually is a linear representation

Let $U \subset V$ be a $G$-invariant subspace and $\rho : G \to \mathrm{GL}(V)$ a linear representation. Show that $\rho_{U} : G \to \mathrm{GL}(V/U)$ with $\rho_U(g)(v+U) = \rho(g)(v) + U$ is a ...
2
votes
2answers
34 views

dual representation — show that it actually is a linear representation in the dual space

Let $\rho : G \to \mathrm{GL}(V)$ be a linear representation in $V$. Show that $\rho^* : G \to \mathrm{GL}(V^*)$ with $\rho^*(g)(f) = f \circ \rho(g^{-1})$ is a linear representation on $V^*$. So ...
1
vote
0answers
60 views

Questions about Fourier transform.

I am reading the notes Lecture notes on representation theory. I have some difficulty in proving a) in Exercise 1.2. We need to prove that $$ \mathcal{F}^2(f)(x) = qf^{-}. $$ I compute as follows: ...
1
vote
1answer
90 views

Irreducible Representations and Maschke's Theorem

$\mathscr L(V,W)^G = \mathscr L(V_1,W)^G \oplus...\oplus\mathscr L(V_k,W)^G$ and for each irreducible represtation of G on a space W, the number of $j\in (1,...,k)$ for which $V_j \cong W$ is ...
1
vote
1answer
58 views

If $f$ is an irreducible representation, what can we say about $g:x\mapsto f(x^{-1})$?

Let $G$ be a finite group, $K$ a field which characteristic does not divide the group order and $V$ a $K$ vector space. Suppose there is an irreducible representation $f: G \rightarrow GL(V)$, $x ...
3
votes
4answers
97 views

What is a minimal polynomial of a group element, and why would we care if it was quadratic?

EDIT: the $p$-stable definition I give below is incorrect. I have included the correct definition as an answer to this question. I am trying to understand the definition of a p-stable group. The ...
6
votes
0answers
75 views

Induced representation, Ind(Res(U))

I am reading a book of Fulton and Harris "Representation theory, a first course". Now it's all about representation theory of finite groups, and there is one exercise, which I can't solve: If $U$ is ...
0
votes
0answers
48 views

check subgroups using representation theory

I am not sure if this is the right place to ask such a question? But I will give it a try here. Given two groups $G,H$, there is a problem we encounter very often: whether $H$ is isomorphic to a ...
1
vote
1answer
65 views

the index in representation theory

I'm studying quantum field theory. In my text book, generators of compact groups are normalized by $\rm{Tr}$$(T^aT^b)=\frac{1}{2}\delta^{ab}$. However, the index $T(R)$ is defined by $\mathrm{Tr} ...

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