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

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finite subgroups of SO(3)

As is well-known, all finite subgroups of $SO(3)$, except for cyclic and dihedral groups, are isomorphic to $A_4$, $S_4$, or $S_5$. The classical proof of this fact uses the geometry of regular ...
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192 views

The unique closed orbit in GIT quotient fibers for polynomial actions of Gl

The following reasoning must contain a flaw somewhere because I end up with something absurd, and I cannot figure out where the mistake is. I hope that someone can point it out to me. Let $M$ be the ...
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1answer
70 views

How to show that the restriction of $\pi$ to the subrepresentation $W$ factor through $G/N$?

I am reading the lecture notes. On Page 16, Line 1, it is said that the restriction of $\pi$ to the subrepresentation $W$ factor through $G/N$. What does "factor through" mean? How to show that the ...
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78 views

Why $G/N$ is discrete?

I am reading the lecture notes. On page 15, Line -5, why $G/N$ is discrete? Thank you very much.
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132 views

How to show that two representations are equivalent?

I am reading the lecture notes. On page 14, example of $C_{c}^{\infty}(G)$. I am trying to show that the map $A$ takes $f$ to $g\mapsto f(g^{-1})$ is an invertible element of ...
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64 views

How to show that $\pi^*(g)=\chi(\det g)^{-1}$?

I am reading the lecture notes. On page 14, how could we show that $\pi^*(g)=\chi(\det g)^{-1}$? I think that $\langle \pi^*(g)\lambda, v \rangle = \langle \lambda, \chi(\det g)^{-1} v \rangle = ...
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Simple components and the irreducible characters of the group ring $K[G]$

Let $K$ be an algebraically closed field of characteristic $0$ and $G$ a finite group. I know that the group ring $K[G]$ is a semisimple and so decomposes as a direct sum of $m$ simple components ...
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60 views

Questions about reductive groups.

I am reading the lecture notes. Let $G$ be a reductive group and $(\pi, V)$ a representation of $G$. For $v \in V$, define $\operatorname{Stab}(v)=\{g\in G \mid \pi(g)v=v\}$ and $V^{\infty}=\{v\in V ...
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character of induced representation and the lifting representation

$A:H\rightarrow GL_n(\mathbb C)$ is a representation. $H$ is a subgroup of $G$. We define the lifting $\overline A(\sigma)=A(\sigma)$ if $\sigma\in H$, $\overline A(\sigma)=0$ if $\sigma\in G-H$. I ...
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408 views

Trinonions, Quaternions, Quinonions, Sextonions, Septonions, Octonions

There are quaternions and octonions and even sextonions but what about trinonions, quinonions and septonions. Are there 3, 5, and 7 dimensional algebras which could be called trinonions, quinonions ...
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38 views

Can I find a finite increasing filtration for every $V\in\mathfrak{O}$?

Let $V\in{O}$, I want to proof there exists a finite increasing filtration by submodules $0=V_0\subset V_1\subset\cdots\subset V_n=V$ such that $V_{i+1}/V_i$ is a highest weight module. Since $V\in ...
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166 views

What is the relationship between diagonalizability and complete reducibility?

I've been wrestling with a certain paragraph in Dummit & Foote (pg 849) and would appreciate clarification. Let $G = \langle g \rangle$ be a cyclic group of order $n$ and assume the field $F$ ...
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2answers
83 views

linear character of a finite group

I am reviewing my notes of algebra. It's not a long proposition so I tried to prove it by myself but failed. We have a finite group $G$ and a linear character $\chi$ of $G$. I need to show ...
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174 views

Computing ext over graded rings

This question came up as I was reading Beilinson, Ginzburg, Soergel paper Koszul Duality Patterns in Representation Theory. Suppose that $A$ is a Koszul ring (for the definition of Koszul ring ...
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42 views

explicit realization of irreducible representations of simple lie algebras

I know explicit realization of irreducible representations of simple lie algebra $sl_n$ when the highest weight of that representation is a fundamental weight.Is there any explicit realization of any ...
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reference request: proof that group characters are a basis for $L^2$

I know the following must be very standard, but I haven't found it in any of the functional analysis books to which I have access. Do you know where I can find a self-contained proof?: If $G$ is a ...
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202 views

Polynomials invariant under the action of $S_m \times S_n$

The polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ has a maximal subring invariant under the action of $S_n$ on the variables. This is the ring of symmetric polynomials. Suppose we have ...
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34 views

Norms of Primitive Idempotent

I'm working on a proof on Algebraic Graph Theory. I'm almost done, except that I'm not quite sure on this step. Is this equation true? $ \Vert E_j \hat{x} - u \cdot E_j \hat{y} \Vert ^2 = \Vert E_j ...
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434 views

Problem 1.24, Introduction to representation theory, Etingof

Let $k$ be a field and $n$ and $N$ be two nonnegative integers. Let $A = k[x_1, \ldots, x_n]$, and let $I \neq A$ be any ideal in $A$ containing all homogeneous polynomials of degree $\geq N$. Show ...
4
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193 views

How to write $SO(2n)$ characters in terms of rotation angles?

Say one is working in a representation of $SO(2n)$ such that it has the highest weights $(h_1,...,h_n)$. And let $\{H_i\}_{i=1}^{n}$ be a basis in the Cartan of $so(2n) = Lie(SO(2n))$. Now one says ...
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128 views

Does representation theory exists without Groups?

I need to know: is representation theory all about Groups? Is it necessary to be a finite group? Does representation theory exists without Groups? For example is there sample where representation is ...
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155 views

Projections in group $C^*$-algebras

Let $G$ be an amenable, discrete and infinite group. Cosinder its group C*-algebra $C^*(G)$ canonically represented on $B(\ell_2(G))$ by the left-regular representation $x\mapsto \delta_x$. Take the ...
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192 views

Is the group of rational numbers linear over $\mathbb{Z}$

Is the additive group of rational numbers isomorphic to a subgroup of $\mathrm{GL}(n,\mathbb{Z})$ for some $n$?
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66 views

Identification of dual space

Let $Q$ be a finite quiver and let $M$ denote the $k$-vector space generated by all arrows. Let $Q^{\ast}$ denote the opposite quiver of $Q$, i.e reverse all arrows. Now let $M^{\ast}$ denote the ...
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The matrix form of a representation of $S_3$.

I am going through some notes on group theory, and one problem states: Consider the three-dimensional representation of $S_3$ constructed as follows: Choose a basis $v_1,v_2,v_3$ of ...
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1answer
120 views

Research paper in harmonic analysis that can be read in parallel to studying the subject.

In my idle hours I started to learn some math I touched only superficially in academia in former times. Among others I am working through the books of A. Deitmar on harmonic analysis. I've almost ...
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95 views

Number of Cocharge Tableaux Summing to Fixed Numbers

First, some background: I will assume the anglophone conventions for Young tableaux in what follows. Given a standard Young tableau $T$ of shape $\lambda$, we can define the cocharge tableau $C(T)$ as ...
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116 views

Finding correct equivalence matrix for all group representations

This question is linked to a previous one that I asked: Attempting to find a specific similarity (equivalence) matrix I have a group of 24 elements, with two generators. I need to find an ...
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65 views

Definite positive measure and GNS representation

Let $G$ be a locally compact group. Let $\mu$ be a positive definite complex measure ([D, p295]): we have $\mu(f*f^*)\geq 0$ for any compact support continuons function $f \in C_c(G)$. In [D, p ...
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312 views

When are path algebras of quivers hereditary.

Suppose we have a finite quiver with relations, possibly with oriented cycles. Is it known when the path algebra of this quiver (with relations) is hereditary?
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Definition of the Young Symmetrizer

I have a question regarding the equivalence of two definitions of the young symmetrizer. First, some notation: let $\lambda$ be a partition of $n$. Given a $\lambda$-tableau $T$ (that is, a tableau of ...
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Number of Standard Young Tableaux of n cells

I know there is a 1-1 correspondance between the number of standard young tableaux of $n$ cells and the number of involutions in $S_n$. Number of involutions in $S_n$ satisfies the recurrence relation ...
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53 views

question on group representations

Here is a problem I faced in algebra. $\rho: A_4 \rightarrow End_{\mathbb C}\mathbb C^{10}$ is a representation of $A_4$. Then show that there is a vector $v \in \mathbb C^{10}$ such that $v$ is an ...
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1answer
130 views

Quotient vector space and representations

Supppose $A$ is an $k$-algebra, and $V$ is a finite dimensional representation of $A$ with subrepresentation $W\subset V$. So in the following short exact sequence, everything is a representation of ...
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The monodromy and cut planes.

I am trying to understand the following example from the book Riemann surface by Donaldson (p.48). This was given after introducing the notion of monodromy of the covering. Consider the Riemann ...
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1answer
117 views

Equivalence of tensor reps & tensor products of reps

Let a finite-dimensional vector space $V$ over $\mathbb R$ or $\mathbb C$ with dual $V^*$ and a group $G$ be given. Let $\rho:G\to\mathrm{GL}(V)$ be a representation, and let $T_kV$ and $V^{\otimes ...
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Two dimensional complex group representations

Michael Artin's Algebra, chapter 10 (both unstarred, and complex representations) M.8 Prove that a finite simple group that is not of prime order has no nontrivial representation of dimension 2. ...
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1answer
243 views

Clifford Algebra for understanding Atiyah Singer Index Theorem Reference Request

I am interested in studying Atiyah Singer Index Theorem and Spin Geometry and would like to study Clifford Algebras and their representations for this purpose. I have a book 'Clifford Algebras : An ...
4
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1answer
110 views

Proof that ideal of Plücker relations is a prime ideal

I am reading section 8.4 of Fulton's Young tableaux where he defines a certain ring as follows. Fix a complex vector space $E$ of dimension $m$ and integers $d_1,\ldots d_s$ such that $m \geq d_1 > ...
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57 views

Irreducible component of representation of symmetric group modulo $p$

Let $p$ be a prime, $n$ an integer divisible by $p$. Consider the natural representation of the symmetric group $S_n$ with coefficients in $\bar{\mathbb{F}}_p$ (namely to each permutation we associate ...
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If $N \unlhd G$, $G/N$ solvable, $\chi \in Irr(G)$, $\psi \in Irr(N)$ and $<\chi_N , \psi>_{_N} \neq 0$, then $(\chi(1)/\psi(1)) | [G:N]$.

If $N \unlhd G$, $G/N$ solvable, $\chi \in Irr(G)$, $\psi \in Irr(N)$ and $\langle\chi_N , \psi\rangle_{_N} \neq 0$, then $(\chi(1)/\psi(1)) | [G:N]$. Can anybody help me to prove that?
2
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3answers
328 views

Constructing the character table of a group

I am aware that, given a group, there is no simple general procedure to construct the character table of the group (over complex numbers). However, for specific groups, we could use helpful additional ...
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1answer
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Computing Brauer characters of a finite group

I am studying character theory from the book "Character Theory of Finite Groups" by Martin Isaac. (I am not too familiar with valuations and algebraic number theory.) In the last chapter on modular ...
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Learning representation theory of Lie groups for someone who knows Lie algebras

I'd like to learn the representation theory of Lie groups. I have a good knowledge of semisimple Lie algebras and their representation theory as well as the basics of Lie groups. To what extent are ...
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1answer
84 views

Representation Theory - Example of a not-G-stable V

I'm studying linear representations for algebraic groups for the moment. And I kind of got stuck on some theorem. The existence of a finite linear representation makes use of the fact that $V$ is ...
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62 views

How to calculate different matrix representation from the same permutation group?

How to calculate 2 dimension representation, unitary representation, 3 dimension representation etc tips: my book teach to use schmidt to calculate
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127 views

Number of Inequivalent Difference Sets In Elementary Abelian 2-groups

I have reason to believe that there is only one$(2^{2s+2},2^{2s+1}-2^s,2^{2s}-2^s)$- difference set (based on experimentation in GAP), up to equivalence/complementation, in any elementary 2-group of ...
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67 views

convolution inequality on R

Let $\nu$ be a complex Radon measure on $\mathbb{R}$ such that $$ \int_{\mathbb{R}} \check{\overline{f}}*f\ d\nu\geq 0 $$ for any complex continuous function $f$ with compact support, where ...
3
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1answer
88 views

Irreducible $\mathbb{F}G$ -module

There are two independent questions. Q1: $G$ is a finite group. $\mathbb{F}$ is a field such that Char$\mathbb{F} \nmid |G|$. Let $V$ be an irreducible $\mathbb{F} G$-submodule of $\mathbb{F} ...
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Odd representations over absolute Galois Group of totally real field

(This might be an easy quesiton but I'm new to representations and could use a helpful pointer or two) Let $G_{\mathbb{Q}}= \text{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$ be the absolute Galois ...