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

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Irreducible representations of a cyclic group over a field of prime order

Consider $G$ a cyclic group of order $n$ with prime $p\not|n$. How do I construct all the irreducible representations over $\mathbb F_p$? How many irreducible representations are there and what are ...
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206 views

Estimates on conjugacy classes of a finite group.

In Character Theory Of Finite Groups by I Martin Issacs as exercise 2.18, on page 32. Theorem: Let $A$ be a normal subgroup of $G$ such that $A$ is the centralizer of every non-trivial element ...
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188 views

Inducing a representation from a subgroup.

Find all the irreducible representations of the group given by: $<x,y,z|x^2=y^2=(xy)^2=z^6=1,zxz^{-1}=y, zyz^{-1}=xy>$. I have 8 conjugacy classes: $\{1\}, \{z^3\}, \{x,y,xy\}, ...
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310 views

Convolution of irreducible characters of a finite group

If $\chi^{\lambda}$ and $\chi^{\mu}$ are the characters of two irreducible representations $V^{\lambda}$ and $V^{\mu}$ of a finite group $G$, is there a simple way of proving that : $$ \chi^{\lambda} ...
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1answer
63 views

Is $m=n+1$ the largest $m$ such that $S_m$ has a faithful action on $\mathbb{Z}^n$?

I can write down a faithful action of $S_{n+1}$ on $\mathbb{Z}^n$. That is, I know of a way to explicitly give a homomorphism from $S_{n+1}$ to $GL(n,\mathbb{Z})$ that has a trivial kernel. An example ...
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60 views

Restriction representation to compact subgroup

Given $Sl(2, \mathbb{R})$, how can I define an unitary scalar product on the discrete series representation? Will the restriction to $SO(2)$ encounter a scaling for each irreducible ...
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78 views

Classifying continuous characters $\epsilon:\mathbf{C}^\times\to \mathbf{C}^\times$.

I recently saw the following claim: Let $\mathbf{C}$ denote the field of complex numbers together with its usual topology. If $\epsilon:\mathbf{C}^\times\to \mathbf{C}^\times$ is a continuous ...
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519 views

If every irreducible representation of a finite group has dimension $1$, why must the group be abelian?

Suppose $G$ is a group and that every irreducible representation of $G$ has dimension $1$. Why does this mean that $G$ is abelian? The number of $1$-dimensional representations of $G$ is given by ...
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126 views

One-parameter subgroup of SL(n) is diagonalizable

Given a group homomorphism $\rho\colon \mathbb{C}^\times \rightarrow SL(n,\mathbb{C})$. Why is it true that the subgroup $\rho(\mathbb{C}^\times)$ is diagonalizable, i.e. that there is a basis of ...
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90 views

How to bound the order of a finite group under the following hypotheses?

In the book Character Theory Of Finite Groups by I.Martin Issacs as exercise 2.14 Let $G$ be a finite group with commutator subgroup $G'$. Let $H \subset G' \cap Z(G)$ be cyclic of order n and ...
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285 views

A representation is semisimple if its restriction to a subgroup of index prime to Char(F) is semisimple

Let $G$ be a finite group and $H$ a subgroup whose index is prime to $p$. Suppose $V$ is a finite-dimensional representation of $G$ over $\mathbb{F}_p$ whose restriction to $H$ is semisimple. Prove ...
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Character formula for $S_n$ and $GL(V)$

In a set of lecture notes I'm reading, we consider representations of the symmetric group $S_n$ via treatment of Young tableaux, partitions of $n$ etc. (in what I believe is the standard approach) - ...
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45 views

Finite-dimensional representations of the Lie algebra of vector fields on a circle

I have just began to study infinite-dimensional Lie algebras and I am curious whether the Lie algebra $L$ spanned by the vector fields $z^n \partial/\partial z$, $n=0,1,2,3,\dots$ admits any ...
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333 views

Characters of a group with abelian subgroup of index 2

I was reading through a proof this morning which said that the characters of a group with abelian subgroup of index 2 are of degree at most 2. This feels like an easy result but I can't seem to work ...
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82 views

Representations of Central Products

What is a good reference for learning about representations/characters of central products of groups? By central product, I mean the following. If $G$ and $H$ are groups, containing isomorphic ...
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764 views

What is the idea of a monodromy?

Is there a connexion between : 1) The monodromy group of a topological space. 2) The $\ell$-adic monodromy theorem of Grothendieck. 3) The $p$-adic monodromy conjecture of Fontaine (which is now ...
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Minimal transitive subgroups of $GL(n)$

A subgroup $G$ of $\mathrm{GL}(n)$ acts on a vector space $V$ of dimension $n$, inducing an action of $G$ on the Grassmannian $\mathrm{Gr}(n,k)$ for each $k$. I'm interested in when this action is ...
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92 views

Cartan decomposition of measure of $GL(2, \mathbb{C})$

I am searching for a integration formula $$ \int\limits_{GL_2(\mathbb{C})} \phi(g) d g = \int\limits_{SU(2)} \int\limits_{SU(2)} \int\limits_{M} \phi(k_1mk_2) \omega(m) d m d k_1 d k_2 ,$$ where $M$ ...
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135 views

If $V$ is an irreducible group representation over a non-algebraically closed field $F$, what's $\dim_F\, (\operatorname{Hom}_G(V,V))$?

Let $F$ be a field and let $G$ be a group. Let $V$ be an irreducible $F$-linear representation of $G$. If $F$ is algebraically closed, then $\dim_F \,(\operatorname{Hom}_G(V,V)) = 1$ by Schur's ...
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Understanding the proof of Schur-Weyl Duality

I am teaching myself representation theory on $GL(V)$ and $S_n$ using my friend's lecture notes, and have reached a proof of the Schur-Weyl Duality theorem; on reading through I'm struggling to make ...
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91 views

Galois Extensions generated by Algebraic Representations

$\newcommand{\Q}{\mathbb Q}$ Let $F=\mathbb Q^{ab} \subset \mathbb C$, i.e. the algebraic numbers. Let $G$ be a finite group of order $n$ and let $\phi: G \rightarrow GL_m(F)$ be a representation. ...
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1answer
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Question about Tamafumi Kaneyama's Paper: “On Equivariant Vector Bundles On An Almost Homogeneous Variety”

My reference: http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.nmj/1118795362&page=record I have two question about Proposition 3.3.: Proposition3.3. ...
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About the intertwiners of a real representation and its complex conjugate

i am currently trying to understand a proof in Trautman's "The Spinorial Chessboard", namely theorem 4.2 on page 48. It states the following: If $\rho:\mathcal{A}\to\operatorname{End}_\mathbb{C} S$ ...
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161 views

(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 ...
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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. ...
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109 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
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186 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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51 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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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 ...
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224 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 ...
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103 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
135 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 ...
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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 ...
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1answer
72 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 ...
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258 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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150 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
370 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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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
249 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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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 ...
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1answer
105 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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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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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.
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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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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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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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592 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 $v = ...
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$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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167 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 ...
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524 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 ...