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

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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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239 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 ...
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1answer
404 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 ...
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1answer
76 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 ...
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1answer
362 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 ...
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68 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
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1answer
135 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 ...
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45 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 ...
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3answers
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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
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1answer
81 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 ...
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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
51 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 ...
3
votes
3answers
1k views

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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0answers
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: ...
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2answers
255 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
811 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
138 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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148 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
158 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
113 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 $(.,.)$ ...
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0answers
240 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
208 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 ...
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3answers
324 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
304 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
310 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
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1answer
437 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
51 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
108 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 ...
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1answer
146 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 ...
8
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1answer
248 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
158 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
272 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
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1answer
474 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
67 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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0answers
296 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 ...
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1answer
243 views

Representation theory of $U(2)$

Where can I find a complete description of all irreducible representation of $U(2)$?
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1answer
913 views

Irreducible representations of a tensor product

Let $A, B$ be finitely generated (noncommutative) algebras over a field $k$ (say, algebraically closed). Can we get all irreducible representations of $A \otimes_k B$ from tensoring representations of ...
2
votes
1answer
73 views

connected algebraic groups

Let $G$ be a linear algebraic group over $\mathbb C$. Let $\psi$ be a finite dimensional regular representation of $G$ into $GL(V)$. Suppose $G$ is connected. I would like to show for $v$ in $V$ the ...
8
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2answers
793 views

Universal Cover of $SL_{2}(\mathbb{R})$

Why does the universal cover of $SL_{2}(\mathbb{R})$ have no finite dimensional representations?
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80 views

simple QG-modules, if G is the dihedral group with 2p elements

I know, if $G$ ist the cyclic group of order $p$ ($p$ odd), that the simple $\mathbb{Q}G$ modules are $\mathbb{Q}$ and $\mathbb{Q}(\zeta_p)$. In that way I get the irreducible representations of the ...
4
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1answer
119 views

An analogy between group actions and group represenations

I was trying to make a 'dictionary' between group action and group representation terms using the $\mathbb{C}[-]$ functor. I immediately found that if the set $Y \subset X$ is invariant under the ...
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2answers
269 views

Is there some kind of character theory for representations of finite dimensional algebras?

We know that for a representation $V$ of a Lie algebra or a quantum group, we can define character of $V$ as $ch(V)=\sum_{\mu} dim(V_{\mu})e^{\mu}$, where $V_{\mu}$ is the weight space of $V$ with ...
2
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2answers
159 views

Exercise at the Beginning of Part II in Fulton's Book on Young Tableaux

In Fulton's Book Young Tableaux, there's an Exercise at the beginning of part II for which I cannot find a solution (there doesn't seem to be one for this exercise in my copy of the book). It reads: ...
7
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1answer
195 views

How can I determine in practice whether two elliptic curves over $\mathbb{Q}$ have isomorphic $p$-torsion?

Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{Q}$ with good, ordinary reduction at an odd prime $p$. I'm wondering how to determine whether $E_1[p]$ and $E_2[p]$ are isomorphic ...
4
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1answer
177 views

Free groups and Kazhdan's property (T)

Showing non-amenability of a (non-abelian) free group is somewhat easy and one can do this immediately after the definition of amenability. Is there an easy proof of the fact that free groups do not ...
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2answers
290 views

Reference request: GL(n)

There are many places, which describe the unitary irreducible representations of $GL(n, F)$ with $F = \mathbb{C}$ or $F =\mathbb{R}$. Basically, we obtain a bunch of parabolically induced ...