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

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Finding the irreducible subrepresentations.

Let $V_d$ be the vector space of homogeneous polynomials of degree $d$ in three variables $x, y,$ and $z$, and let the symmetric group $S_3$ act on $V_d$ by permuting the variables. Find the ...
2
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1answer
346 views

Is the Hodge star an $SO(n)$-equivariant isomorphism to the dual representation?

Let $V$ be an oriented inner product space of dimension $n$. The Hodge star operator maps $\Lambda^k V\to \Lambda^{n-k}V$. In particular it maps $V\to \Lambda^{n-1}V.$ $V$ carries a representation of ...
5
votes
1answer
298 views

Faithful representations, centres and reducibility

Take all fields to be algebraically closed. Show that if $G$ is a finite group with trivial centre and $H$ is a subgroup of $G$ with non-trivial centre, then any faithful representation of $G$ is ...
3
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1answer
463 views

Showing that a given representation is not completely reducible

Let $\rho : \mathbb Z \to \mathrm{GL}_2(\mathbb C)$ be the representation defined by $\rho(1) = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$. I'd like to show that $\rho$ is not completely ...
4
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1answer
114 views

Why is $\operatorname{Sym}^k(F^n)$ irreducible?

Consider a field $F$ and the standard representation of $SL_n(F)$ on $F^n$. Let $k$ be an integer. Then $SL_n(F)$ acts on $\operatorname{Sym}^k(F^n)$. Why is this latter representation irreducible? (I ...
9
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1answer
369 views

Young diagram for standard representation of $S_d$

I'm working through Fulton-Harris and I'm kind of "stuck" at the following question. I'm looking for representations of $S_d$, the symmetric group on $d$ letters via Young Tableaux. The question is: ...
4
votes
1answer
442 views

Representation theory/Finitely generated abelian groups

This is not homework. Motivation : I have been reading "Representation and characters of groups" by James & Liebeck, and in chapter 9 they introduce Schur's Lemma, which states that for a finite ...
6
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2answers
725 views

character table of nonabelian group of order 21

I've just started studying representation theory of finite groups and I'm having trouble finding the character table of the group $G:=\left\langle x,y | x^7=e=y^3, y^{-1}xy=x^2\right\rangle$. This ...
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1answer
88 views

Representation of a fundamental group.

Consider the fundamental group $\pi_1(\mathbb{CP}^1\backslash \{a_1, \ldots, a_n\})$. It is said that there is a representation: $\pi_1(\mathbb{CP}^1\backslash \{a_1, \ldots, a_n\}) \to GL(n, ...
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0answers
96 views

About the realization of $SO(N)$ given by Daniel Bump in his book Lie Group

Thank for your interest for my question and thank you very much if you can answer me. In his book p.187, Daniel Bump says that a realization (a representation) of $SO(2n)$ is given by the unitary ...
7
votes
1answer
311 views

Dynkin diagram automorphisms and weights

Let $\sigma$ be a nontrivial Dynkin diagram automorphism of a finite-dimensional complex simple Lie algebra $\frak g$ (of type A, D or E) and let $\frak h$ be a Cartan subalgebra of $\frak g$. Let $I$ ...
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2answers
805 views

Irreducible Representations of Matrix Algebras

I am currently reading the book Spin Geometry by Lawson/Michelsohn to understand Dirac Operators and related topics. At some point it uses representation theory to classify Clifford Algebras. In ...
9
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1answer
379 views

Exercise 4.7, I. Martin Isaacs' Character Theory

I have been working on this problem for few hours, and have gotten no where. Here it is: If $G$ is a p group, and $G/\phi(G)$ has order at least $p^{2\alpha-1}$, then the number of elements of order ...
2
votes
0answers
122 views

induced representation of tensors of irreducibles

Let $V_{\lambda}$ and $V_\mu$ be representations of the symmetric groups $\mathfrak{S}_d$ and $\mathfrak{S}_m$ respectively where $\lambda$ is a partition of $d$ and $\mu$ is a partition of $m$. It is ...
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vote
1answer
190 views

question on roots and root vectors of a simple lie algebra

Assuming that for each root α there is only one linearly independent root vector, show that if $\alpha$, $\beta$, and $\alpha+\beta$ are roots, then [$e_\alpha$ , $e_\beta$ ] not equal to 0. Here ...
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vote
1answer
85 views

If $\alpha$ is a root of a simple lie algebra, then prove that the only multiples of $\alpha$ which are roots are $\alpha, -\alpha,0$

If $\alpha$ is a root, then the only multiples of α which are roots are $\alpha, -\alpha, 0$. Here $\alpha$ is a root of a simple lie algebra. How do I prove this?
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1answer
64 views

If $\alpha$ is a root of a simple lie algebra, prove that $\langle \alpha,\alpha \rangle \neq 0$

If $\alpha$ is a root of a simple lie algebra, prove that $\langle \alpha,\alpha \rangle$ not equal to $0$. From this, I want to prove that the $\langle,\rangle$ could be used as a scalar product.
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1answer
73 views

Why does rigidity hold only if rank >1?

In simple words, why does Margulis' superrigidity and arithemiticit only hold for lattices in Lie groups of rank $>1$? E.g. what is the reason for it to fail for $SL(2,R)$?
3
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1answer
207 views

What does “2:1 covering” mean?

I am doing an exercise in a representation theory book that asks the following: "For $g$ and $h$ in $SL_{2}\mathbb{C}$, the mapping $A \mapsto gAh^{-1}$ is in $SO_{4}\mathbb{C}$. Show that this gives ...
3
votes
1answer
140 views

How do I find all the roots of a lie algebra and hence its root system diagram given the cartan matrix for that algebra?

If I am given the cartan matrix, I can find the $2\langle αi , αj\rangle/\langle αj , αj\rangle$ of the simple roots where $α$ are the simple roots. But, from this how do I find the $\langle αi , ...
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55 views

$G_1$-Scalar factors for Clebsch-Gordan coefficients for $ U(n)$

when evaluating the $G_1$ scalar factors for CGC's of $U(n)$ it seems that some of the factors are undefined. The explicit formula for the evaluation of the scalar factors is Eq. (6) in 18.2.8 of N.J. ...
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2answers
497 views

Why is the group action on the vector space of polynomials naturally a left action?

When seeking irreducible representations of a group (for example $\text{SL}(2,\mathbb{C})$ or $\text{SU}(2)$), one meets the following construction. Let $V$ be the space of polynomials in two ...
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Isomorphism of representations of the symmetric group

This might be a silly question, but I don't understand why the solution to the following problem implies the result: Let $A = \mathbb{C}S_d$ and let $c_{\lambda}$ denote the Young symmetrizer (with ...
2
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1answer
131 views

Symmetric power and characters

Let $V$ be a 2 dimensional vector space over $\mathbb{C}$. Then $W := Sym^{n}(Sym^{m}V)$ is a representation of $GL(V)$. For $g \in GL(V)$, I consider $\chi_{W}(g)$. Let $x$ and $y$ denote the ...
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817 views

Proof $\mathbb{A}^n$ is irreducible, without Nullstellensatz

As the title suggests, could anyone either provide me with or direct me to a proof that affine n-space $\mathbb{A}^n$ is irreducible, without using the Nullstellensatz? This is an exercise in a ...
3
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1answer
195 views

Property of Young tableaux

Note: All Young diagrams are to use the English notation scheme. Suppose I have two tableaux $T$ and $T'$ on the same Young diagram (we insert the numbers $1, 2, \ldots, n$ in two different ways ...
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1answer
495 views

Why a different tensor product for left $G$-modules (group representations)?

The category of vector (over $k$) representations of a group $G$ is isomorphic to the category of left $k[G]$-modules, where $k[G]$ is the group ring. This isn't just an equivalence of categories, but ...
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2answers
43 views

Showing $R(G) = R(T)^W$

Let $G$ be a compact connected Lie group and $T$ a maximal torus. Let $R(G)$ be the representation ring of $G$. Then restriction of reps gives a map $R(G) \to R(T)^W$, where $R(T)^W$ are the ...
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143 views

How to compute the character of a matrix group operating on homogeneous polynomials?

I have a little problem in representation and/or invariant theory which I need help with. Let's assume $G \leq \mathbb{C}^{n\times n}$ is a finite complex matrix group which operates linearly via ...
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1answer
249 views

Class equation of subgroup of $SL(4,\mathbb{F}_2)$

Can you point me toward a computation-light derivation of the class equation of the subgroup of $SL(4,\mathbb{F}_2)$ consisting of upper-triangular matrices with 1's on the main diagonal? The ...
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0answers
116 views

Reference request: Indecomposable representations of posets

Let $I$ be a finite poset. Definition: A representation of $I$ is a functor $I\to\mathrm{Vect}_{\mathbb C}\ $. Equivalently, a representation of $I$ is a module over its incidence algebra $\mathbb C ...
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1answer
152 views

$\mathrm{Hom}_G(V,W)$ as a subspace of $\mathrm{Hom}(V,W)$

Suppose $V$ and $W$ are both representations of a group $G$, where $V$ and $W$ are $k$-vector spaces. Define $\mathrm{Hom}(V,W)$ to be the space of $k$-linear maps $V \to W$. My notes say that: ...
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1answer
156 views

Decomposing the permutation representation of $S_{4}$

Let $U$ be the trivial representation of $S_{4}$ and $V$ be the standard representation (of $S_{4}$). Why is it that $\mathbb{C}^{4} = U \oplus V$? I see that this is true by character theory, but is ...
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2answers
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Is $f(\operatorname{rad}A)\subseteq\operatorname{rad}B$ for $f\colon A\to B$ not necessarily surjective?

If I have two $K$-algebras $A$ and $B$ (associative, with identity) and an algebra homomorphism $f\colon A\to B$, is it true that $f(\operatorname{rad}A)\subseteq\operatorname{rad}B$, where ...
2
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1answer
274 views

All $\mathbb C$-representations of a finite group $G$ have a $G$-invariant inner product

Let $G$ be a finite group, and suppose we have a complex representation $V$ of $\mathbb C$. Let $\langle , \rangle$ be an arbitrary (Hermitian) inner product on the $\mathbb C$-vector space $V$, and ...
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1answer
46 views

Where can I find an English version of this paper by Gel'fand and Raikov?

It is titled 'Irreducible unitary representations of locally bicompact groups' and the original version is in Russian. Google scholar shows it has been translated into English and once pubulished in ...
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482 views

Tensor products and irreducible representations

Again something from Fulton and Harris I'm having trouble with: Exercise 2.33 (c). If $U$, $V$, and $W$ Are irreducible representations, show that $U$ appears in $V \otimes W$ if and only if $W$ ...
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1answer
345 views

Exercise 2.4 in Fulton and Harris

I'm trying to begin reading Fulton and Harris' Representation Theory and I'm having trouble with the following: Exercise. Show that if we know the character $\chi_{V}$ of a representation $V$, then ...
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1answer
48 views

Dimension of a representation and the order of an element in a group

Let $V$ be a representation of a finite group $G$ with $V$ being finite dimensional. Fix a $g \in G$. Is it necessarily true that $\dim V \geq \operatorname{ord} g$?
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497 views

Being isomorphic as representations of a group G

Let $G$ be a finite group. What is meant by two finite dimensional vector spaces (over $\mathbb{C}$) $V$ and $W$ being "isomorphic as representations of $G$"? To show that we have such an isomorphism, ...
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Understanding the structure of a finite dimensional vector space based on the properties of linear maps to itself

Let $V$ be a finite dimensional vector space over $\mathbb{R}$. What can we say about the dimension of $V$ if we know that there exists some linear map $\phi: V\to V$ such that $\phi^n=-I$, where $I$ ...
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Does the $p$-torsion of an elliptic curve with good reduction over a local field always determine whether the reduction is ordinary or supersingular?

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $E/K$ an elliptic curve with good reduction. Does the $\mathbb{F}_p[\mathrm{Gal}(\overline{K})]$-module $E[p](\overline{K})$ determine whether the ...
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1answer
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unitary representation and denseness.

I have the next unitary representation, $\pi : G\rightarrow \mathcal{U}(H)$, where G is a closed subgroup of $S_{\infty}$ (the group of bijective functions from $\mathbb{N}\rightarrow \mathbb{N}$), ...
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1answer
81 views

Isomorphism between $sl_{4}$ and the orthogonal group of $6$ variables

Let V be the irreducible $sl_{4}$-module with highest weight $\pi_{2}=\lambda_{1}+\lambda_{2}$ (i.e if $H=\left(\lambda_{1},\dots,\lambda_{4}\right)$ is a diagonal matrix in $sl_{4}$ with values ...
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125 views

Characters and permutation matrices

Suppose I represent the group $S_{\,n}$ using $n \times n$ permutation matrices. This is a valid group representation. Let $\chi$ be its character. Since $\chi(g)$ is complex and since ...
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1answer
52 views

What is the systematic way to convert any arbitrary finite dimensional representation into block diagonal form?

Given any arbitrary representation, how do I convert it into block diagonal form, or find its irreducible representation?
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1answer
84 views

Dimension of Lie algebra

Let $o(2l,F)$, with $l \ge2$ and $n=2l$ be the orthogonal lie algebra $\{L\in gl(n,F)|SL=-L^{t}S\}$ where $S=\begin{pmatrix} 0 &I_{l} \\ I_{l} & 0 \end{pmatrix}$. How can I show that ...
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1answer
75 views

Representation of $S_3$ on $\mathbb{C}^6$.

I am asked to decompose $S_3$ into its irreducible unitary representations, on $\mathbb{C}^6$. I wonder how does this differ from analysis the given in the next paper?: ...
0
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1answer
294 views

Induced representation of symmetric group.

Im stuck with this one and I don't even know how to start, I would appreciate any help: Can you describe the induced representation of the standard representation of $S_{n}$ in $S_{n+1}$?
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An algebraic algorithm for finding inverses in the group algebra

This is an extension to my earlier question. Is there a purely algebraic algorithm to find inverses in the group algebra? For example, in the group algebra $\mathbb{C}S_{4}$, how would one go about ...