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

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When are all irreps left ideals in the group algebra and generated by idempotents?

Let $A$ be an associative algebra. I am wondering under what conditions we can get all irreducible representations of $A$ as left ideals $A\cdot e$ with $e\in A$ an idempotent. This is certainly the ...
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74 views

A non-integrable representation of the Heisenberg Algebra

Let $\mathfrak h$ be the Heisenberg algebra in dimension 1, generated by vectors $P$, $Q$ and $I$ satisfying $[P,Q] = I$, $[P,I] = [Q,I] = 0$. A representation of $\mathfrak h$ on a Hilbert space $X$ ...
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105 views

If $K[Q_8]\cong K[D_8]$, char $K=p$ odd, $p=?$

Denote $Q_8$ to be the quaternion group, and $D_8$ to be the dihedral group with order 8, then we know that the group algebra $\mathbb{C}[Q_8]\cong \mathbb{C}[D_8]$ since $Q_8$ and $D_8$ have the same ...
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66 views

A group of linear isomorphisms of $\mathbb C^n$ must have an invariant subspace

Let $G$ be a finite group acting linearly on $\mathbb C^n$, and suppose that $|G| < n^2$. I am trying to show that there is a nonzero invariant subspace $W\subset\mathbb C^n$, i.e. $g(w) \in W$ ...
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175 views

Projective indecomposables versus general indecomposables

Given a finite dimensional algebra, what is the exact relation between the indecomposable projective modules, and a general indecomposable module? In the case of an oriented quiver without cycles for ...
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103 views

Do field automorphisms of a character imply outer automorphisms of the group?

Apologies for the imprecise wording of the title. In studying the basic representation theory of finite groups, I've been struck by a pair of phenomena present in every example I've worked with but ...
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132 views

Young tableaux: evaluate action of permutation

Consider the irreducible representation $V$ in the symmetric group $S_5$ corresponding to the Young diagram (these are meant to be boxes): $$[\;\;][\;\;] \\ [\;\;][\;\;] \\ [\;\;]\;\;\;\;$$ (a) List ...
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118 views

Invariant inner products on infite-dimensional representations

Let $G$ be a compact group and let $V$ be it's continuous representation. It is well known that if $V$ is finite-dimensional, then there is an $G$-invariant inner product on $V$. I haven't found a ...
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92 views

Semisimple objects in abelian categories

Let $\mathcal A$ be any Grothendieck abelian category and $0 \neq M \in \cal A$ an object. It is true that $M$ admits a simple subquotient? It is certainly true for $\mathcal A=R-Mod$ since $M$ ...
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189 views

Is there formula name and proof for this theorem ? ( guess it's called Burnside character formula)

The formula answers: how many tuples $(\sigma_1,\sigma_2,\dots,\sigma_n)$ of elements of a given group $G$ such that (1) $\sigma_i\in C_i$ , where $C_i$ stands for conjugacy class. (2) ...
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56 views

Getting a derivation from a path of automorphisms of an algebra

These representation theory notes leave the following claim to the reader: Recall that a derivation on an algebra is a map $d$ such that $d(ab)=d(a)b+ad(b)$. If $A$ is a finite-dimensional algebra ...
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112 views

DFT shift theorem generalizations?

The DFT shift theorem implies that any circular shift in the input space is equivalent to a phase change in the frequency domain, while the absolute values are preserved. $$ ...
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1k views

Understanding induced representations

Let $G$ be a group and $H$ be a subgroup. Let $\phi:H\rightarrow GL(V)$ be a representation of $H$. There are three constructions in Wikipedia, but I am not really convinced by these. My question is: ...
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2answers
118 views

1 dimensional representations of $S_n$

I want to show that $S_n$ has only two 1 dimensional represnetations. mainly the trivial and sign represnetations. Where I assumed that our Field we're working on is with characteristic $\neq 2$. ...
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135 views

Is there some relation between characters in representation theory and multiplicative characters?

A character of a group representation is obtained by taking trace of each matrix in this representation. The word character is often used in the sense that it is a homomorphism from a group to ...
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317 views

Finding All Irreducible Representations of $SO(3)$

I've read that one may prove that all irreducible representations of $SO(3)$ are tensor product representations of the fundamental representation (or tensor product representations of the spin 1/2 ...
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412 views

$SU(2)$ Representation of $SO(3)$

I've often seen it written that $SU(2)$ is a "two-valued representation" of $SO(3)$ (in theoretical physics books mainly). I have a major conceptual issue with this however. I know there is a Lie ...
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328 views

Looking for texts in representation theory

I recently finished a course in representation theory, and while I learned a lot from it, I know that there's a lot more in the subject that I missed. For the course we used Fulton and Harris as a ...
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242 views

Young Tableaux as Matrices

These questions are motivated only by curiosity. Take a Young tableau of shape $(\lambda_1,\lambda_2,\ldots,\lambda_n)$, where $\lambda_1\geq\lambda_2\geq\ldots\geq \lambda_n)$. Is there any physical ...
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303 views

Converse of Schur's First Lemma

Suppose $G$ is a closed subgroup of $SU(d)$, $d>1$, and let $\rho$ be a $d$-dimensional special unitary representation of $G$. Suppose that if a matrix $A$ commutes with all of $\rho(G)$ for all ...
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76 views

Schur's first lemma for finitely generated continuous groups of $SU(d)$

Suppose that a finite set $S$ of $d\times d$ special unitary matrices densely generates a representation $\rho$ of a continuous subgroup of $G$ of $SU(d)$. That is, for every $\epsilon>0$ and ...
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329 views

Schur-Weyl Duality ( Classical ) and the Double Commutant reference request

I would like to ask for any reference suggestions on the topic of Schur-Weyl Duality for GLn ( directly GLn, not through the lie algebra ) and the double commutant theorem. The section on this ...
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82 views

Monodromy Representations

Let, be $V$ a connected smooth manifold and $q_1,q_2\in V$ and $F:U\to V$ a connected covering of degree $d$. This covering induces two monodromy representations $\rho_1:\pi_1(V,q_1)\to S_d $ ...
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80 views

Representations of direct sums of matrix algebras

i'm reading Introduction to representation theory by Pavel Etingof etc. (look at http://arxiv.org/pdf/0901.0827v5.pdf) and i want to do most of the exercises. But i must stop by the exercise on page ...
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169 views

Some irreducible characters of the Symmetric group $S_n$

I want to have characters of some irreducible $S_n$-modules corresponding to certain partitions $\lambda$ of $n$, the computations using Frobenius formula get complicated and I am unable to find in ...
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68 views

4D TQFT construction from a modular tensor category

I know the construction of 3D topological quantum field theory (TQFT) from a modular tensor category. I heard that we can even (mathematically) construct 4D TQFT from a modular tensor category. I ...
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2answers
181 views

Is there a general expression for the adjoint representation of $U(N)$ or $u(N)$?

At least for low values of $N$ like $2$ or $3$ and such I would like to know if there are explicit matrices known giving the representation of $u(N)$ or $U(N)$ in the adjoint? (..a related query: ...
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43 views

Product of two squares in finite groups

Let $G$ be a finite group and $g$ an element of order $n$ in $G$. Assume that $g$ is a product of two squares. Moreover, assume that $n$ and $k$ are coprime. Prove that $g^k$ is also a product of two ...
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1answer
69 views

Decomposition of representation of Problem In Artin

I am trying to solve the problem: Decompose the standard representation of the cyclic group $C_{n}$ in $\mathbb{R}^{2}$ by rotations into a direct sum of irreducible representations. What I have ...
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171 views

Problem 4.3, I. Martin Isaacs' Character Theory

Let $G=H\times K$ be the direct product of finite groups. Let $\varphi\in Irr(H)$ and $\eta\in Irr(K)$ be faithful. Show that $\varphi\times\eta$ is faithful if and only if $(|Z(H)|,|Z(K)|)=1$. Here, ...
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190 views

Nice applications of the Haar measure

The existence of the Haar measure is a beautiful result that has a lot of applications. For example, one can prove using the Haar measure that the category of representations of a compact Lie group is ...
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1answer
217 views

Multiplicity of a completely reducible representation in another irreducible representation.

I have got the next question that I am pondering the answer to. Let $\tau$ be a completely reducible representation of finite dimension of a group $G$, and let $\pi$ be another irreducible ...
5
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1answer
87 views

$S_n$ has only four (irred.) representations with degree $<n$ (for $n>6$)

I'm working on the following exercise: For $n\ge 7$, $S_n$ has no irreducible representations of dimension $m$ with $2\le m\le n-2$. There is a solution here but I'd like to follow the ...
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1answer
61 views

Indecomposable L-module

I have the following exercice which I have be trying to solve: Let L be a Lie algebra and $r:L\rightarrow gl_3(F)$ a representation of L such that $im(r)=t_3(F)$ (the upper triangular matrices). Show ...
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246 views

Irreducible Representations

Let $G$ be a group, $F$ a field, and $V$ be an $F[G]$ module (equivalently $F$-representation of $G$). The following definition is well-known. Definition 1. We say that $V$ is irreducible (or simple ...
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1answer
119 views

Non-Isomorphic induced representations (from the same representation of a subgroup)

I believe that it is true that if we have a group $G$, and two copies $H_1$, $H_2$ of some group $H$ as subgroups of $G$, we can fix a representation $V$ of $H$ and have the situation: ...
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1answer
52 views

isomorphism of an irreducible $\mathbb{R}$ represetation

Theorem: Every irreducible $\mathbb{R}$-representation of the real algebra $\mathbb{R}(n)$ is isomorphic to $\mathbb{R}^n$, where the matrix A ∈ $\mathbb{R}(n)$ acts via left matrix multiplication. I ...
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1answer
285 views

Difference between the SU(2) and SO(3) lie groups and their lie algebras

In many places I have seen the SU(2) and SO(3) lie algebras used interchangeably. How are they exactly identical? Moreover, what about their lie groups? Are they identical as well. It would be great ...
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272 views

The Noether-Deuring Theorem

I have to solve the following exercise taken from the book "Introduction to Representation Theory" by P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina and S. ...
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2answers
258 views

Describing all $\rho$-invariant inner products

Let $z$ satisfying the equation $z^3=1$ be a generator of the cyclic group $\mathbb{Z}_3= \{ 1 , z,z^2 \}$. You are given that $\rho : \mathbb{Z}_3 \to GL(\mathbb{C}^2)$ defined by $$\rho(z) = ...
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2answers
56 views

Checking that $\rho$ is a representation

Let $z$ be a generator of the cyclic group $\mathbb{Z}_3 = \{ 1,z,z^2 \}$. Prove that a representation $\rho$ of $\mathbb{Z}_3$ in the $2$-dimensional complex vector space $\mathbb{C}^2$ can be ...
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1answer
77 views

Error in Weyl character formula computation.

I need someone with a keen eye for errors. I am trying to use the Weyl character formula for the symplectic group Sp$(4,\mathbb{C})$ on certain matrices coming from 2x2 quaternion matrices. Summing ...
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2answers
511 views

Finding invariant subspaces

Let $x$ be a variable. Denote by $V$ the vector space consisting of all polynomials $P(x)=ax^2+bx+c$ of degree not more than 2, with complex coefficients. For any real number $t$ determine an operator ...
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73 views

Cartan decomposition of unitary group

For number field $F$, we consider two case 1) $E$=quadratic extension of $F$ , 2) $E = F \times F$. Let V be a 2-dim hermition space over E. In 1) case, by Cartan decompostion $U(2)$ can be ...
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659 views

Proofs that the degree of an irrep divides the order of a group

It is a theorem in basic representation theory that the degree of an irreducible representation on $G$ over $\mathbb{C}$ divides the order of $G$. The usual proof of this fact involves algebraic ...
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1answer
100 views

Why is the tensor product of two pseudoreal representations real?

Let G be a group, and $\rho : G \to GL(n, \mathbb{C})$ be a representation of $G$. Then we also get the conjugate representation $\rho^* : G \to GL(n, \mathbb{C})$, where $\rho^*(g) = ...
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1answer
74 views

Dimension of the GL-orbit of d-forms in one less variable

Let $V:=k[x_0,\ldots,x_n]_d$ be the $k$-vector space of homogeneous polynomials of degree $d$. Let $G:=\mathrm{Gl}(n+1,k)$ act on $V$ induced by the canonical action on the linear forms: For ...
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1answer
103 views

Cartan or Coxeter matrix of an algebra of infinite global dimension

Let $(Q, I)$ be a bound quiver such that $A=KQ/I$ has infinite global dimension. I want to ask the following questionss: (1) Is the Cartan matrix $C_A$ of $A$ invertible in the matrix ring ...
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76 views

harish chandra for sl(2,C)

Is it true that each irreducible sl(2,$\mathbb{C}$)-module, $P(\lambda,\mu)$ with $\lambda \in \mathbb{Z}$ appears as the harish chandra module of some $(\pi_{\chi},V_{\chi})$ And given ...
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Exactness Properties of Schur Functors

The title says it all: What are the exactness properties of Schur Functors? Thanks!