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

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Existence of a G-invariant matrix

Let $\phi: G \to GL(\mathbb{R}^n)$ be a homomorphism, $G$ finite. Prove that there is a positive-definite matrix $M$ such that $\phi(g)^tM \phi(g) =M$ $\forall g \in G $. This looks really ...
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0answers
38 views

If an $H\leq G$ has an irreducible representation of dimension $d$, then show $G$ has an irreducible representation of atleast dimension $d$. [duplicate]

Let $H$ be a subgroup of a group $G$, and let $\rho:H\rightarrow GL(V )$ be an irreducible representation of dimension $d$. Prove that there is an irreducible representation of $G$ whose dimension is ...
3
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1answer
187 views

With the branching rules of subalgebra, how can I write down explicit matrix elements for a representation?

I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not ...
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1answer
105 views

The characters of the irreducible representations of a group

Let $G$ be finite group of order $n$ with $s$ conjugacy classes and let $X_1, . . . ,X_s$ be the characters of the irreducible representations of $G$ over $C$. Prove that the sum $ \sum_{g\in G} X_i(g) ...
3
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177 views

Unitary representations of noncompact groups

Does there exist a noncompact connected Lie group with a finite-dimensional, unitary, faithful, irreducible representation over $\mathbb{C}$? If you remove any of these hypotheses except that of ...
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1answer
53 views

Is there a name for this Lie algebra?

Consider the three dimensional, complex Lie algebra with basis $\{a,a^\dagger, I\}$ and the following structure relations: \begin{align} [a,a^\dagger] = I, \qquad [a,I] = 0, \qquad [a^\dagger, I] = ...
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1answer
77 views

What is $\mathfrak{gl}(\infty)$

As title says, I know what is $\mathfrak{gl}(n,\mathbb{C})$, but what is $\mathfrak{gl}(\infty)$? Where can I find good reference for this?
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87 views

Etymology of the term “weight vector”

I am writing a work on the representation theory of $SU(3)$ in basque and I would like to know the etymology of the term $\textbf{weight vector}$ in order to properly translate it.
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99 views

Verma modules and delta function

What is the relationship between Verma modules and delta functions? Here's the quote from Woit's notes on Lie theory (http://www.math.columbia.edu/~woit/LieGroups-2012/vermamodules.pdf): The ...
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77 views

Dual representation matrix “recycling”

Imagine we have $V$, a finite dimensional vector space endowed with an inner product and its dual space $V^*$. We have also a matrix Lie algebra and a representation of it, $\pi$, that acts on $V$. ...
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1answer
68 views

Prove that $FS_4$-module is simple

I am solving the following problem: Consider a field $F$ with $\operatorname{char} (K)=0$, let $\sigma = (1,2)$ and $\pi = (1,2,3,4)$. An $FS_4$-representation $\rho$ is given by $$ \begin{...
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138 views

Proof of Clifford's theorem for modules

http://en.wikipedia.org/wiki/Clifford_theory#Proof_of_Clifford.27s_theorem I've a very easy question that I just can't seem to find the answer to. I'm self-studying so I can't ask anyone else. ...
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1answer
117 views

Character group of $\mathbb{C}^*$

Let $\mathbb{C}^* = \mathbb{C}-\{0\}$ be the group of multiplication. Then why the character group $G := Hom(\mathbb{C}^*, \mathbb{C}^*)$ is $\mathbb{Z}$?
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1answer
85 views

Real representations of SL(2,C)

Is there a classification of real-linear (rather than complex-linear) finite-dimensional representations of SL(2,C)?
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1answer
763 views

Any irreducible representation of a $p$-group over a field of characteristic $p$ is trivial.

In general, we know that if $G$ is a finite group and $K$ is a field, then $K[G]$ (the group algebra) is semisimple whenever $\operatorname{char}(K)$ does not divide the order of $G$. However, this ...
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0answers
49 views

Intertwiner for $U(n-1) \subset U(n)$

I'm using the notation of Vilenkin and Klimyk, ''Part3: Representations of Lie Groups and Special Functions''', chapter 18. Given an irreducible representation $T_m$ of the complex Lie algebra $U(n)$ ...
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557 views

Young diagram for exterior powers of standard representation of $S_{n}$

I'm trying to solve ex. 4.6 in Fulton and Harris' book "Representation Theory". It asks about the Young diagram associated to the standard representation of $S_{n}$ and of its exterior powers. The ...
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66 views

Induced representation of $G=SL_2$ by $\chi _ w$ is irreducible if $w^2\neq 1$.

This is the question from Serre's book #7.4. $G=SL_2(k)$, where $k$ is a finite field and $H\leq G$ such that $H$ consists of matrices $\begin{pmatrix} a & b \\ 0 & d \end{pmatrix}$. Let $w:k^...
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22 views

The isomorphism of representive functions

$G$ is a compact Lie group with closed subgroup $H$ and $\mathscr{T}(G),\mathscr{T}(H)$ are the sets of their representative functions respectively (with real or complex representation). If the ...
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118 views

How to understand the direct product of group representations (on example)?

The algebra of the Lorentz group $SO(3, 1)$ can be represented as direct product of $SU(2)$ or $SO(3)$ algebras. How to understand this statement?
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1answer
69 views

Prove that the matrix A is positive definite.

A matrix $A$ is defined as: \begin{equation} A := \sum_{g\space \epsilon\space G}{{D}^{\dagger}(g)D(g)} \end{equation} Where the $D(g)$ are representations matrices of the finite group $G$ on a ...
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1answer
143 views

1-Cocycle of an Algebra

Is there a good definition of a 1-Cocycle of an algebra A that is relatively easy to understand? I am rather new to cohomology and representation theory, but it seems like this is a fundamental ...
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0answers
68 views

Isomorphism types of stabilizers of vectors in linear representations of the special linear group

Suppose we have a linear representation of the group $SL_d$ over $\mathbb{C}$. i.e. a finite dimensional vector space $V$ with a linear action of $SL_d$ on it. Let $v\in V$ be some vector and let $H\...
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1answer
115 views

What is dual representation in plain English?

Can someone please explain what is Dual representation in plain English. I read its definition on wikipedia and at many other places but could not develop an intution for it. Please explain in plain ...
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1answer
41 views

If $G$ is solvable, is it true that for any $m,n\in\operatorname{cd}(G)$, there exists a prime $p$ such that $p\mid m,n$?

Let $G$ be a finite group and let $\operatorname{cd}(G)$ be the set of degrees of irreducible characters of $G$. It is known that if for any $m,n \in \operatorname{cd}(G) \setminus \{1\}$, there ...
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327 views

A family of commuting endomorphisms is semisimple if each element is semisimple

If $\phi : V \rightarrow V$ is an endomorphism of a finite-dimensional (say real) vector space, $\phi$ is called "semisimple" if any $\phi$-invariant subspace of $V$ has a complimentary $\phi$-...
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1answer
72 views

Doubt: “A group representation is exactly like a module over the group ring”

It is traditional to say that a representation of a group $G$ over a field $F$ is "exactly like" a module over the group ring $F[G]$. I think it is inaccurate. I think a module over $F[G]$ encodes ...
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1answer
62 views

Any submodule U of V such that the module V/U is completely reducible must contain the radical?

Want to prove: Given an $R$-Module $V$ and $rad(V)$ which I define to be the intersection of all maximal submodules of $V$. I want to show that if, for some submodule $U$ of $V$, we have $V/U$ ...
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41 views

Weight space for a finite-dimensional $\mathfrak{g}-$module $M$.

Let $\mathfrak{g}$ a semisimple Lie algebra, $M$ finite-dimensional $\mathfrak{g}-$module, $\mu\in\mathfrak{h}^*_{\mathbb{Z}}$ and $s_i$ simple reflection such that $\langle\mu+\rho,\alpha_i^{\vee}\...
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1answer
76 views

Irreducibility of Lie algebra representations

Let $\mathfrak{g}$ be a finite dimensional real Lie algebra and $\pi: \mathfrak{g} \to \mathfrak{gl}(V)$ be a homomorphism of real Lie algebras where $V$ is a finite dimensional real vector space. But ...
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104 views

Application of representation theory

I often read that one can use representation theory in the field of quantum physics or for the analysis of symmetries in physics or chemistry. Unfortunately I coundn't find a concrete example for this....
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279 views

Definition of Verma modules

I have a question regarding different (but equivalent!?) definitions of Verma modules of semisimple Lie algebras: Let F be a field and denote the following: $ \mathfrak{g}$ , a semisimple Lie ...
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2answers
164 views

Eigenbundle decomposition

Let $G$ be a finite cyclic group and $X$ a smooth manifold equipped with a trivial $G$-action. It is known that we can decompose every $G$-equivariant vector bundle with respect to the action: ...
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1answer
98 views

Exterior power respects $G$-action

Basic setting: Let $V$ be a $k$-vector space of finite dimension and $V^*$ its dual space. Let $\bigwedge^n V$ denote the $n$-th exterior power of $V$. Now the canonical pairing $$V \times V^{*} \...
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1answer
375 views

Direct sum and tensor product of two representations of a group

Our lecturer gave us a hard exercice to go further in group theory (we stopped at group actions) : Let G be a group, V and W complex vector spaces and $\rho_1 : G \mapsto GL(V) $ be a group ...
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52 views

Finding an orthonormal basis for a gl(3) module

I'm trying to find an orthonormal basis for gl(3)-module V(ε1-ε3), where ε1-ε3 is the weight (1,0,-1) of the highest-weight vector. Using Gelfand-Tsetlin (/Zetlin/Zeitlin) patterns, I'm at the point ...
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1answer
44 views

Regular module endomorphisms into itself

Let $k$ be a field and let $A$ be an algebra over $k$. Denote by $End_A (A)$ the set of all endomorphisms of the regular $A$-module $A$ into itself. Fix $a \in A$, and define the A-module homomorphism ...
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4answers
582 views

Representation Theory book other than Fulton's

Fulton/Harris's book on representation theory seems to be the "definitive" introductory text on the subject. But is there perhaps a lower level introduction to the subject? Most of the very first ...
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137 views

Irreducible modules - semisimple algebras and endomorphism rings

Let $A$ be a finite dimensional, semi-simple $k$-algebra and $V$ and irreducible $A$-module. I am trying to prove the following claim: If $B = \text{End}_A(V^{\oplus r})$ then $$W = \text{Hom}_A(...
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1answer
44 views

Exactness of Hom functor for torus representations?

Given a reductive algebraic group $G$ and a maximal torus $T$. Is it true that the functors $$ Hom_T(-,\lambda) $$ are exact, where $\lambda$ denotes one of the the simple one-dimensional ...
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76 views

Characters of subgroups of finite abelian groups

Let $G$ be a finite abelian group. Let $H$ be a subgroup of $G$. Let $\hat{G}$ be the group of characters of $G$. Is there a character $\chi \in \hat{G}$ such that $\chi(g) = 1$ iff $g \in H$?
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48 views

Volume of a ball for SO(n).

Let us equip the special orthogonal group $SO(n)$ with a normalized Haar measure $\theta_n$ and let $G_r$ be the subset of rotations $\Omega$ which differ from the identity by (sufficiently small) $r$ ...
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Which Field Would You Use to Represent a Group Larger than $\aleph _1$?

I understand that in representation theory we try to understand a group $G$ by studying the homomorphisms $\rho\ \colon G \to $ GL$(V)$ where $V$ is a vector space over some field. I believe complex ...
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35 views

A question about Cayley-Chow forms

I'm reading some papers about $k$-stable theory and I have a question about Cayley-Chow forms. Maybe this question looks silly. Let X be a variety of $\mathbb{P}^N$ with dimension n and degree d. ...
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1answer
452 views

Haar measure of $SO(3)$ obtained from $SU(2)$

I am reading 'Analysis on Lie groups, an introduction' by Faraud and don't understand the following statement … the image by the map Ad of the Haar measure $\mu$ of $SU(2)$ is equal to the ...
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1answer
1k views

Representation Theory of the Dihedral Group $D_{2n}$

So I'm pretty new into Representation Theory having so far covered only a couple of example sheets. I'm thinking about the following question: Suppose we have the group $D_{2n}$ (for clarity this is ...
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1answer
72 views

Algebra Homomorphism

This is a follow-up to a question I asked here yesterday. It's coming from a (non-examinable) exercise sheet and I really can't get my heard around how this question is posed and to be approached. ...
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1answer
69 views

Group algebras, Maschke's lemma and direct sums of matrix algebras

Let $G=\{g_1,g_2,\dots,g_n\}$ be an arbitrary finite group. We consider its representations over $\mathbb{C}$. There is Maschke's theorem which states that each representation of $G$ is a direct sum ...
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57 views

Finite simple nonabelian groups with the same character table

Conjecture: If $G_1$ and $G_2$ are finite simple nonabelian groups, and if $G_1$ and $G_2$ have the same character table, then $G_1\cong G_2$. I am looking for a proof, or at least some intuition. Or ...
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1answer
54 views

Calculations in $K$-Algebras

Suppose we have some field $K$ and non-zero elements $a,b,$ in $K$. Define $H=H(a,b)$ to be the $K$-algebra with basis $\{1,x,y,z \}$ over $K$ satisfying $$x^2=a, \\ y^2=b, \\ z=xy=-yx$$ Question: How ...