Tagged Questions

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

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Character of dual Representation?

Let $G$ be a finite group and consider the group ring $\mathbb C[G]$. If $M$ is a $\mathbb C[G]$-module consider the dual representation in $M^*=\textrm{End}(M, \mathbb C)$ given by $$(g\cdot ...
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1answer
29 views

A converse of schur's lemma

Suppose $\rho: G \rightarrow GL(V)$ is a representation. and if $T: V \rightarrow V$ is a linear operator such that $T\circ \rho_g= \rho_g\circ T$ for all $g\in G$ implies $T=k\cdot Id$ for some ...
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19 views

If $G$ is a finite non-trivial group of odd order, it has an irreducible representation not realisable over the reals.

$\textbf{Lemma }$If $V$ is a representation of a finite group $G$, then $V$ is of real type if and only if $V$ is the complexification of a representation $V_{\mathbb{R}}$ over the field of real ...
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34 views

Is quotient under $S_4$ action on “cube” representation a flat morphism?

Consider a three-dimensional irreducible representation $V$ of $S_4$, corresponding to symmetries of cube. Let $p$ be canonical projection $p: V \rightarrow V/S_4$. My question: is $p$ flat? I want ...
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1answer
27 views

Relationship between simple group and character table

Prove that a group $G$ is not simple iff $\chi (g)=\chi (1)$ for some nontrivial character $\chi$ and some $g\not= 1$. I have no idea how to do this, please help, thanks.
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1answer
13 views

The Weyl group and eigenspaces

Let $V$ be a representation of the Weyl group. For any reflection $\sigma_{\alpha}$ (where $\alpha$ is a root), we know that $V$ has two eigenspaces with eigenvalues $1$ and $-1$. The ...
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Is the map from representation ring to class functions a isomorphism?

I have a questions from representation theory. ($G$ is finite group) Fulton and Harris in "Representation Theory. A first Course" write that: the character defines a map $$\chi : R(G) \to ...
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1answer
31 views

What is the center of the Valentiner group $\mathcal{V}=\langle I, Q \rangle$?

(Please refer to this question first: Is $\langle(26543),(34)(56),(12)(3654)\rangle $ isomorphic to $A_6$? ) I want to understand the center of the Valentiner group: $$\mathcal{V}=\langle I, Q ...
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1answer
19 views

character group of finite abelian group and induced homorphism

This is ex 5.7 of chapter 10 of artin's algebra (2nd edition) Suppose $\varphi:G \rightarrow G'$ is a homomorphism of abelian groups. Define an induced homomorphism $\hat{\varphi}" \hat{G'} ...
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1answer
68 views

Is $\langle(26543),(34)(56),(12)(3654)\rangle $ isomorphic to $A_6$?

My question is the one in the title. If you want to understand the context of the problem, please read further. I reduced a problem to proving the question. Background is: Valentiner group ...
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1answer
24 views

Is the center of a compact Lie algebra precisely the set of vectors on which the Killing form is zero?

Suppose a Lie algebra $\frak{g}$ has a killing form, $B$, which is negative semidefinite. Suppose $B(X,X)=0$ for some $X\in \frak{g}$. Is $X$ necessarily in the center of $\frak{g}$?
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26 views

Partial generalisation to Whitehead's second Lemma

Let $k$ be an algebraically closed field of characteristic $0$ and let $\mathfrak{g}$ be a finite dimensional semisimple $k$-Lie algebra. By Whitehead's second Lemma, we know that $H^{2}(\mathfrak{g}, ...
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24 views

action of $SU(2)$

Let $L=\mathbb C[u,v]$ (the $\mathbb C$-algebra of polynomials over two commuting variables $u,v$. For each non negative integer $n$ let $L_n$ be the linear subspace of homogeneous polynomials of ...
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4answers
114 views

Alternate proof of Schur orthogonality relations

I am trying to find an alternate proof for Schur orthogonality relations along the following lines. Let $G$ be a finite group, with irreducible representations $V_1$, $V_2$, $\cdots$, $V_d$. Let $V$ ...
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0answers
27 views

Irreducible representations of $S_3$

I am trying to do a problem from artin's algebra 2nd ed (Chapter 10, Exercise 2.3) but having trouble: Let $(\rho , V)$ be a representation of the symmetric group $S_3$. Let $x=(123), y=(12)$ be the ...
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1answer
23 views

How does one define weights for a semisimple Lie group?

For compact Lie groups one considers a maximal torus to define the weight space decomposition of a representation. For a complex semisimple Lie algebra one considers a Cartan subalgebra. How does ...
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1answer
51 views

An irreducible character of degree prime affords a faithful representation.

This is a long question, but hopefully someone can give me a suggestion, as I've been hitting my head against the wall... Take a non-abelian group $P$, of order $p^3$, $p$ prime. We've already ...
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2answers
13 views

Do elementary row operations give a similar matrix transformation?

So we define two matrices $A,B$ to be similar if there exists an invertible square matrix $P$ such that $AP=PB$. I was wondering if $A,B$ are related via elementary row operations (say, they are ...
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19 views

Reference request: classification of simple Lie groups and simple real Lie algebras

I am trying to understand the classification of simple Lie groups and the theory of highest weights for semisimple Lie groups by first understanding the case for complex Lie algebras, then relating to ...
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1answer
22 views

Character of the algebra $\mathbb{C}[G]$ as $G \times G $-module

Let $G$ be a finite group. We can define an action of $G\times G$ on the group algebra $\mathbb{C}[G]$ in the following way: If $x \in \mathbb{C}[G]$ then $(g,h)\cdot x=gxh^{-1}$. Now, what about ...
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37 views

Convolution of matrix coefficients is also a matrix coefficients

I have a question about the convolution of matrix coefficients as follows: Let $G$ be a compact Lie group. A Map $f:G\rightarrow \mathbb{C}$ is called a matrix coefficient if there is a finite ...
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1answer
20 views

Why if we have representation $\rho$ of finite group then $\rho(g)$ is diagonalisable matrix?

Why if we have representation $\rho:G \to GL(V)$ of finite group $G$ then $\rho(g)$ is diagonalisable matrix? I read that it's because $x^{o(g)} -1$ splits, but I don't understand how this fact is ...
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1answer
29 views

Decompose the following representation of $A_5$ in irreducible representations

Denote the space of functions on the set of faces of the icosahedron by $V_f$, this is a 20-dimensional representation of $A_5$ which acts here on as the group of rotational symmetries. Here follows ...
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1answer
23 views

Dimension of subspace stabilized by group and principle character

Let $\phi: G \rightarrow GL(V)$ be a representation with character $\chi$. Let $W$ be the subspace $\{v \in V : \phi(g).v = v$ for all $g \in G \}$ of $V$. Prove that $\dim W = \langle \chi, ...
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1answer
57 views

Behaviour of Ext with Hom

Let $K=SO(2,\mathbb{R}), \mathfrak{g}=gl(2,\mathbb{R})$. Let $V$ be a finite dimensional irreducible $(\mathfrak{g},K)$-module, W be a $(\mathfrak{g},K)$ module. I want to prove that ...
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0answers
22 views

How to construct explicit matrix representations of $\mathfrak{su}(3)$

I'd like to implement an algorithm which produces matrix representations of the (complexified) Lie Algebra $\mathfrak{su}(3)$ on carrier spaces with arbitrary highest weight vector; i.e. 8 $n\times n$ ...
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1answer
29 views

Equivalence of induced representation

Let $H$ be a subgroup of $G$. In Wiki, it gives an algebraic construction of induced representation. And it is equivalent to the vector space $Hom_{H}(\mathbb{C}[G],V)$ i.e. $ \{ f:G \rightarrow V ...
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if $\rho: H \to \text{GL}_m(\mathbb{C})$ faithful then $\text{Ind}_H^G \rho$ faithful [closed]

I need help with this problem. Prove that if $\rho: H \to \text{GL}_m(\mathbb{C})$ is faithful then $\text{Ind}_H^G \rho$ is faithful.
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38 views

Borel subalgebras inside the grassmannian

This is probably something standard and I just don't know where to look (so a reference would be just as appreciated as an answer), but... Let $\mathfrak{g}$ be a finite dimensional semisimple Lie ...
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Irreducible representations of inhomogeneous linear transformations over $\mathbb{F}_q$

Let $\mathbb{F}_q$ be a finite field with $q$ elements and let $G$ be the group of inhomogeneous linear transformations over $\mathbb{F}_q$, that is $x\mapsto ax+b$ for some $a\in \mathbb{F}_q^*$ and ...
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1answer
40 views

Representation of the symmetry group (rotations) of the icosahedron

Suppose $I$ is the set of vertices of the regular icosahedron, here is a link of the icosahedron: http://www.werheit.mynetcologne.de/icosaeder.gif Let $F(I)$ be the space of complex functions on $I$, ...
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1answer
29 views

Algebraic invariant theory

Deal all, I am looking for a gentle introduction to algebraic invariant theory (for a Bachelor project) with some simple (but interesting) applications in representation theory (of finite groups, of ...
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2answers
57 views

Action on sheaf cohomology in Bott-Borel-Weil theorem

Let $G$ be simply connected complex semisimple Lie grou and $P \subseteq G$ parabolic subgroup. Suppose $V$ is finite dimensional irreducible representation of $P$ with highest weight $\lambda$, and ...
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How to tackle a research journal - level course in Lie Theory and Representation Theory?

I am taking a course in Lie Theory and Theory of Representations this year, where starting from the second lecture, Lie Theory is heavily bundled with Theory of Representations. It is pretty much a ...
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1answer
54 views

Base for symmetric group

Given symmetric group $S_n$, is it possible to find $k=\lceil\log_2S_n\rceil=\lceil\log_2n!\rceil$ members $\{\alpha_i\}_{i=1}^{k}$ in $S_n$ such that every member of $S_n$ can be written as ...
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33 views

Character of a Representation

Suppose that we have a representation $V$ of a group $G= SU(2)$ . Is it true that if $ \chi_V \cdot c \neq 0$ for some non-zero constant c, then the trivial representation must be one of the ...
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125 views

Is the left regular representation of an algebra, always faithful?

Let $\mathcal{A}$ be a unital associative algebra with a countable basis $\mathcal{b}$ over $\mathbb{C}$. Let $H=l^2(b)$ be the Hilbert space generated by $\mathcal{b}$. Let $H_0 = \{v \in H \ \vert \ ...
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28 views

Root Systems and Dynkin diagrams.

On page 142, the textbook An Introduction to Lie Groups and Lie Algebras (by Kirillov) determines the fundamental group of the root system $A_2$. Basically, the author says we have two simple roots ...
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A compact group with a finite dimensional faithful representation [duplicate]

Theorem: If $G$ a compact group has a finite dimensional faithful representation $W$, then any irreducible representation $V$ is contained in $W(k,l) = W^{\otimes k} \otimes (W^*)^{\otimes l}$ for ...
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1answer
80 views

Construction of a triple cover of $A_6$ in “Finite Simple Groups” by Wilson

I am reading The Finite Simple Groups by Robert Wilson: see page 29. I want to understand a construction of triple cover of $A_6$. On section 2.7.3., I don't understand the second paragraph, which is ...
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33 views

Schur-Weyl Duality - references

I'm trying to understand the Schur-Weyl duality. Unfortunately the lecture notes I have don't describe it very detailed. Any good references?
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1answer
48 views

Multiplicity of a dual simple module in the dual module?

Let $A$ be a finite dimensional $k$ algebra. Let $S$ be a simple left $A$-module and $M$ be any left $A$ module. Then my first question is that is it true $$[M:S]=[M^*:S^*]$$ where ...
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1answer
22 views

Is a virtual vector bundle the same as a vectorial bundle?

What is a virtual vector bundle? Is a virtual vector bundle the same as a vectorial bundle? The current entry in nLab states the following: "In one class of models for K-theory – generalized ...
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1answer
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Are the weights of an irreducible representation of a simple Lie algebra in a single Weyl orbit?

When we consider the weights of an irrep of a simple Lie algebra, are they always in a single orbit under the Weyl group of the Lie algebra, or do they form a set of disjoint orbits? If they form ...
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How to construct $\operatorname{End}(V_{\pi})$ using a representation $\pi$

Let $(\pi, V)$ be a representation of the group $G$. To make the setting as general as possible, I will not put any restrictions on $\pi, V$, and $G$ from the beginning. By the very definition, for ...
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unique part of induced representation.

In the book about representation by Harris and Fulton . It proofs a proposition(3.17), where H is a subgroup of G: It only proofs existence at glance, why is it unique?
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2answers
60 views

A question in representations theory

My question is about irreducible representations of groups over the field $\mathbb{Q}$. Let $G$ be a cyclic or an abelian group. I want to check that under what conditions we have a ...
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1answer
32 views

Eigenspace of finite abelian group

Let $\rho: G\to {\rm GL}_n(\mathbb{C})$ be faithfull representation of finite abelian group $G$ and $V$ is the eigenspace of some $g\in G$. Is it true that $V$ is also eigenspace for all $G$ (that ...
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4answers
294 views

Polynomials as module over symmetric polynomials

Consider ring of polynomials $\mathbb{k} [x_1, x_2, \ldots , x_n]$ as a ring over symmetric polynomials $\Lambda_{\mathbb{k}}$ Is $\mathbb{k} [x_1, x_2, \ldots , x_n]$ free $\Lambda_{\mathbb{k}}$ ...
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1answer
41 views

A problem similar to Maschke's theorem

My question is about a problem that its assumptions are like Maschke's theorem in some ways. Let $\mathbb{F}$ be a field that $char \mathbb{F}$ doesn't divide $|G:H|<\infty$ and $M$ be an ...