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

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(Non-)Formality of A-infinity algebra implies derived (non-)equivalence?

Take an unital differential graded (dg) $k$-algebra $A$, we can regard it as $A_\infty$-algebra with $m_1$ as differential and $m_2$ as algebra multiplication, and $m_n=0$ or all $n\geq 0$. Take a dg ...
3
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0answers
102 views

Space of functions on the vertices of a cube; Representations

$1.$ The problem statement, all variables and given/known data (Sorry, don't know how to get TeX to work...) Consider the space of functions $V_{\nu}$ defined on the vertices of a cube. ...
1
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1answer
125 views

Continuous representation restricts to homomorphism $G \to O(n)$

As part of a problem I've been set, I'm required to show that if $G$ is a compact group then there is a continuous group homomorphism $G \to O(n)$ if and only if $G$ has an $n$-dimensional ...
5
votes
1answer
202 views

Complex and Real Representations, their differences by decomposition

1. The problem statement, all variables and given/known data Decompose $\mathbb{C}^{5}$, the 5 dimensional complex Euclidean space) into invariant subspaces irreducible with respect to the group ...
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66 views

How to characterize the image of $PGL(V)$ in $\mathbb{P}(W)$ for an irreducible $GL(V)$-representation $W$

I'd like to ask two versions of my question, one a very specific case that I suspect may have a fairly easy and classical answer, the other the general case which I would like to find references on. ...
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103 views

Reference Request - Spaces of Smooth Vectors

I was recently looking for examples of non-nuclear spaces of smooth vectors of representations of Lie groups. I'll recall the basic definitions. Let $\pi$ be a unitary irreducible representation of a ...
2
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347 views

All irreducible representations of Pauli group

I'm supposed to find out all irreducible representations of Pauli group, that is, the group generated by Pauli matrices $\sigma_k(k=1,2,3)$. It has 16 elements: $\pm 1, \pm i, \pm \sigma_k, \pm i ...
2
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0answers
111 views

Existence of a 1-dimensional invariant subspace

Show the existence of a 1-dimensional invariant subspace for any 5-dimensional complex representation of the group $A_4$, where $A_4$ is the alternating group of degree 4. Any hints?
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1answer
170 views

A problem regarding to Schur's lemma

Let $\rho: G \to GL_n(\mathbb{C})$ be an irreducible representation, and $g\in Z(G)$. Show $\rho(g)$ is a scalar multiple of the identity matrix $I$. I think I have it, here is my solution: Since ...
2
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2answers
465 views

The Ext functor in the quiver representation

First take a question as an example: Let $f:L\to M$ be an irreducible morphism in $\mathrm{mod}-A$ and $X$ be a right $A$-module. Show that ...
1
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1answer
84 views

Infinitely many nilpotent elements in $\mathbb{C}[G]$

Suppose $G$ is a finite group and $F$ is a field such that $\mathrm{char}\;F$ doesn't divide $|G|$. Suppose that $F$ is algebraically closed and $G$ is not abelian. How can I prove that $F[G]$ has ...
6
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376 views

Two non-isomorphic groups with the same complex character table

Could you give me an example of two non-isomorphic groups with the same complex character table?
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1answer
228 views

indecomposable module which is not cyclic

In Etingof's notes entitled "Introduction to Representation Theory," he asks the reader to produce an example of an indecomposable module which is not cyclic (Problem 1.25(c)). The exercise even comes ...
1
vote
1answer
402 views

Groups of order 21

Let $G$ be a group of order 21: 1) Determine all possible values of $n$, where $n$ is the number of conjugacy classes of $G$. 2) Determine all the possible decomposition of $\mathbb{C}[G]$ as a ...
3
votes
1answer
75 views

a function as a character

I meet difficulty in Problem 4.5 in the book "Representation theory of finite group, an introductory approach" of Benjamin Steinberg : For $v=(c_1,\cdots,c_m)\in(\mathbb{Z}/2\mathbb{Z})^m$, let ...
0
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1answer
352 views

Regarding Schur's lemma, that $T = \lambda I$, the uniqueness of $\lambda$.

I'm reading through the proof in Artin's "Algebra" of Schur's lemma (second statement): if $T:V \to V$ is a $G$-invariant linear operator with respect to $\rho$ an irreducible representation, then $T ...
2
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0answers
67 views

Why is the $\mathbb{Z}$-span of a set of representations an ideal of the representation ring?

I am studying a proof of Brauer's theorem. The proof makes use of the following claim, which I haven't been able to convince myself of: Let $G$ be a finite group and let $R[G]$ be the representation ...
8
votes
1answer
128 views

Flatness of residual representations associated to modular forms

Let $f\in S_k(\Gamma_1(N),\chi)$ be a Hecke eigenform of weight $k\geq 2$, $p$ an odd prime not dividing $N$, and $K_f$ the number field generated by the Hecke eigenvalues of $f$. Fixing a prime ...
2
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44 views

Is $\sigma \operatorname{Res}_H^G(U) \cong U$ if $\sigma \in G/H$? [duplicate]

Possible Duplicate: Why is $ U \otimes \operatorname{Ind}(W) = \operatorname{Ind}(\operatorname{Res}(U) \otimes W)$? I am working on a problem out of Fulton and Harris: Show that $U ...
8
votes
3answers
2k views

Symmetric and exterior power of representation

Is there exists some simple formula for characters $$\chi_{\Lambda^{k}V}~~~~\text{and}~~~\chi_{\text{Sym}^{k}V}$$ for some representation $V$ of finite group? Thanks.
6
votes
1answer
80 views

r-th transvectants and $\mathbb{C}G$-module maps

Suppose $V=\mathbb{C}^2$ and $G=SL(V)=SL_2(\mathbb{C})$. We define $C_n = H_{\mathbb{C},n}(V,\mathbb{C}) \cong S^n(V^*)$, the n-th symmetric power of the dual of $V$, i.e. the homogeneous polynomials ...
3
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2answers
55 views

A 1-1 homomorphism from $\operatorname{Iso}(\mathbb{R}^2)$ to $GL(3,\mathbb{R})$

In class we saw A 1-1 homomorphism from $\operatorname{Iso}(\mathbb{R})$ to $GL(2,\mathbb{R})$ $$\operatorname{Iso}(\mathbb{R})\cong \left\{ \begin{pmatrix}\pm1 & x\\ 0 & 1 ...
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0answers
50 views

Are tempered representations unitarizabile?

Let $G$ be a locally compact, unimodular group and $Z$ be its center Clearly, square integrable representations with central unitary character is unitarizabile, since their matrix coeffecient imbed ...
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2answers
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Irreducibility of the standard representation of $S_n$.

The permutation representation of $S_n$ is $\mathbb C^n$ with elements of $S_n$ permuting the basis vectors $\{e_1, e_2, \ldots, e_n\}$. It has a trivial subrepresentation spanned by the vector $v = ...
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51 views

$GL_2$-Invariants of $\mathbb{C}[X,Y]$

One of the problems in some work I'm doing tells me to consider $GL_2$ acting on $\mathbb{C}[X,Y]$, induced by the natural representation of $GL_2$ on $\mathbb{C}^2$. I just wanted to check 2 things: ...
2
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2answers
243 views

Two questions about $KG$-modules

Let $K$ be a field of $\operatorname{char}= p>0$ , let $G$ be finite group of order $p$, and $V$ is non zero $KG$-module. How do I show that there exist non-zero $v\in V$ such that $gv=v $ for ...
2
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2answers
775 views

How does one decompose the regular representation of $S_3$?

I need to decompose the regular representation of $S_3$ into irreducible ones. What I know so far is this: $S_3$ is generated by $\tau = (12)$ and $\sigma = (123)$. If $v$ is an eigenvector of ...
4
votes
1answer
137 views

Weil's proof of a theorem on finite irreducible representations of products of compact groups

Theorem Let $G$ and $H$ be compact groups. Let $ρ$ be a finite dimensional irreducible continuous representation of $G×H$ over the field of complex numbers. Then $ρ$ is a tensor product of irreducible ...
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2answers
146 views

What is the smallest degree of a homogenous polynomial invariant under the action of $D_{2n}$ in the plane.

If we put a regular polygon centered on the origin in $\mathbb{R}^2$ then we can think of $D_{2n}$ as isometries of the plane. What is the degree of the smallest polynomial invariant under these ...
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2answers
145 views

How does the standard representation restrict to the cyclic group generated by (1234).

So we have the group $S_4$ which has the standard representation. We also have the subgroup generated by permutation (1234). This is isomorphic to $C_4$ which has four irreducible representation. How ...
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2answers
112 views

Existence of Hermitian form .

I came across this question in my notes : If $G$ is finite group and $V$ is finite dimensional $CG$ module ( $C$ complex ) , how do i show that there exist a positive definite Hermitian Form $(.,.)$ ...
2
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0answers
236 views

Examples of decomposition representation

Here is a question in the book "Representation theory of finite group, an introductory approach" of Benjamin Steinberg. (Question 3.8(2), page 25) that I need some hints from you : Give an example of ...
2
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1answer
86 views

Explanation of $|G|=d_1 n_1^2 + \cdots +d_s n_s^2$ in representation theory

Can anyone help me to understand the equation $$|G|=\sum^s_{i=1} d_i n_i^2$$ please? The context is representation theory of finite groups over a field $\mathbb{K}$ of characteristic zero. I know ...
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1answer
199 views

Induced and Restricted Representation Manipulation

$\newcommand{\Ind}{{\text{Ind}}}$ $\newcommand{\Res}{{\text{Res}}}$ $\newcommand{\ds}{{\displaystyle}}$ $\newcommand{\inv}{{^{-1}}}$ I am doing Exercise 3.16 from Fulton Harris. http://bit.ly/JeTz1J ...
6
votes
3answers
300 views

Understanding the representations of group and Modules.

I am trying to understand and have a good grasp on Representation theory . I was asking to myself " what essentially is the difference between representation of some group $G$ and a $KG$ module, how ...
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1answer
287 views

does the trivial representation always induce the permutation representation?

Does the trivial representation always induce the permutation representation? Is this true for each field $\mathbb{K}$ or just for representations over $\mathbb{C}$?
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2answers
293 views

Few questions on Character of representation .

a) What does it mean to say that the Character of a representation is irreducible on its own? b) If Char($K$) is $0$ then kernel of character is a normal subgroup of G , why ?? c) Over a field of ...
4
votes
1answer
424 views

induced representation, dihedral group

I am trying to construct the induced representations of the dihedral group $G=D_p$, $|D_p|=2p$, if I take the subgroup $H=\langle r \rangle \cong C_p$, which is generated by the rotations. I have ...
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0answers
47 views

Type 1 group $\leftrightarrow$ every irreducible representation has a trace

Is it equivalent for a separable, locally compact group: the group is type $1$, every unitary representation $\pi$ has a trace $\mathrm{tr} \; \pi: C_c^\infty(G) \rightarrow \mathbb{C}$?
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1answer
81 views

Stationary paths

Let $Q$ be a finite quiver and denote the stationary parts of $Q$ by $e_{i}$. Suppose we have two arrows $f,g$ such that their composition $f \circ g$ is equal to $e_{i}$. Does this always implies ...
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1answer
107 views

When are two Representations equivalent?

Let $K$ be a field, $G$ group, $V$ and $W$ be finite dimensional $K[G]$ modules and $X$ and $Y$ be the representations afforded by $V$ and $W$ respectively.I need to show that $X$ and $Y$ are ...
1
vote
1answer
136 views

Why are there $|G/G'|$ 1-dimensional representations of $G$?

Let $G'$ be the derived subgroup of a finite group $G$. We have a correspondence $\{\mathrm{reps \ of \ G/G'}\} \longleftrightarrow \{\mathrm{reps \ of \ G \ with \ kernel \ containing \ G' }\} $ ...
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3answers
130 views

A simple question about representation of a group

I am taking a course that mentions that sometime we would like to look at a group $G$ as a group of matrices. From another course I took a while ago I remember that this is called a representation. I ...
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213 views

Generating the partners in a multi-dimensional irreducible representation.

I am trying to block diagonalize a Hermitian matrix using the irreducible representations of its symmetry group. Using the group's character table, it is straightforward to generate a set of ...
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1answer
78 views

algebraic group to the lie algebra and hom

Let $G$ be a linear algebraic group and let $\rho:G \rightarrow GL(V_{1})$ and $\psi:G \rightarrow GL(V_{2})$ be finite representations. Why is $Hom_{G}(V_{1},V_{2}) \subset Hom_{\mathfrak ...
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1answer
152 views

Dimension of subspace fixed by subgroup representation.

If $G$ is an abelian group with cyclic subgroup $H$ and $(\rho,V)$ is a (permutation) representation of $G$. Then I can form a representation of $H$ by considering the composition ...
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1answer
267 views

Equivalent matrix representations

Let $K$ be a field, $G$ a group and $G'=[G,G]$ the commutator subgroup of $G$. Show Two matrix representations of $G$ over $K$ of degree $1$ are equivalent only if they are identical. The group $G$ ...
4
votes
1answer
450 views

Representation theory of $GL(2,\mathbb{C})$

I want to classify all unitary representations of $GL(2,\mathbb{C})$ from the representation theory of $SL(2, \mathbb{C})$. Is this possible? Knapp claims that one obtains all irreducible ...
2
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0answers
64 views

Representations of $U(d)$. Calculation of Gelfand-Zeitlin patterns for particular vectors.

Following structure is given $\left(\mathbb{C}^d\right)^{\otimes n}$. Consider irreducible representations of $U(d)$. And consider the fully symmetric subspace $T_{\alpha}$ in ...
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277 views

Group of Hermitian and Unitary matrices

This question is continuation of an earlier question asked in Matrices which are both unitary and Hermitian Consider the unitary group $U(n^2)$ and consider the subset $R$ of Hermitian Unitary ...