0
votes
0answers
28 views

Orbits of $Sp(n,R)$ under action of $Gl(2n,R)$ by conjugation

These questions arose from a question related to K-theory, I am hoping for (big) results from the theory of linear algebraic groups to be helpful. Maybe somebody with a better background there can ...
1
vote
0answers
23 views

matrix of the dual representation: inverse of the transpose

I have a doubt concerning the dual representation. Can someone check that what I wrote is correct please? Let $A: V \longrightarrow V$ be linear, the dual map $A^T : V^* \longrightarrow V^*$ is ...
1
vote
1answer
51 views

Question concerning a list sorting problem

I have the following question: Let $a,b,c,d$ be four natural numbers with $a \leq b$ and $c\leq d$. I have written a program that produces a list, which has as entries all 2-tuples $(x,y)$ with ...
0
votes
0answers
31 views

Isomorphic Dual and Conjugate Representations of a Lie Algebra

Let $\frak{g}$ be a complex Lie algebra $\frak{g}$, and $R:\frak{g} \to $End$(V)$, a representation for some finite dimensional complex vector space $V$. As is well-known, we can construct from $R$ ...
3
votes
2answers
78 views

Is the tensor product of two representations a representation?

I am a little bit uncertain about an argumentation showing that a given map of a topological group is somehow obviously continuous. In the following I will rely on the book of Anthony W. Knapp „Lie ...
0
votes
1answer
28 views

Unitary matrix for matrix representation

In the book The Symmetric Group the author says: Let $\chi$ and $\psi$ be characters of the $G$-module $V$. By picking an orthonormal basis for $V$, we obtain a matrix representation $Y$ for ...
2
votes
1answer
26 views

Irreducible representation - Eigenvalues of Matrix

I am currently working at Bruce Sagan's "The Symmetric Group". The following example is an illustration to show that Maschke's Theorem is not true for infinite groups. The following paragraphs are ...
3
votes
2answers
93 views

Inner product in Maschke's Theorem

I am working through Maschke's Theorem on page 16 in Bruce Sagan's The Symmetric Group: In order to prove the theorem the author constructs an inner product $\langle v, w \rangle' = \sum_{g \in G} ...
1
vote
0answers
32 views

Derived series of a Lie algebra

I've been studying semisimple Lie algebras and solvability and was wondering if someone could explain to me the meaning of the derived series of a Lie algebra L and this part: $$L^{(1)}=[LL]$$ I don't ...
1
vote
1answer
45 views

proof of basic fact that torus actions are diagonalizable

Suppose a torus $T=(\mathbb{C}^\ast)^n$ acts on a finite dimensional vector space $W$, and define for $m \in M$ ($M$ is the character lattice of $T$) the eigenspace $W_m$ by $$W_m = \{w \in W \mid ...
2
votes
1answer
51 views

Indecomposable quiver representations

Is there are any way to found indecomposable representation of a given quiver explicitely if it's dimention vector is given?
1
vote
0answers
30 views

Finding the adjacency matrix for any given quiver and some collection of words.

For a directed graph (quiver) $Q$ with $n$ vertices and without multiple arrows, we have the adjacency matrix $A$, in which $A(i,j)=1$, if there is an arrow from $i$ to $j$, and $0$ elsewhere. This ...
1
vote
4answers
69 views

Representation of $S_n$ by $V^{\otimes n}$,

Let $V$ be a real and finite dimensional vectorspace. Then $$ \sigma.(v_1 \otimes \cdots \otimes v_n) := (-1)^{\sigma} v_{\sigma(1)} \otimes \cdots \otimes v_{\sigma(n)}. $$ My question: Why is this ...
1
vote
2answers
44 views

Non-unitary representation

How to prove $\pi :\mathbb R\to \mathbb C^2$, defined by $t\mapsto \begin{pmatrix} 1 & t\\ 0 & 1\end{pmatrix}$ is a non-unitary representation? Is the following correct? $\pi$ is a ...
0
votes
0answers
34 views

Why is character sum of eigenvalues?

Working my way through a first course in Representation theory, I run into some difficulties (due to bad knowledge of linear algebra) with that said I am wondering about the following. Let $\Theta : ...
0
votes
1answer
12 views

Consider the action of $S_3$ on $C^3 = \{ (x,y,z) | x + y + z = 0\}$. Show that $\rho$ is irreducible.

The action is defined as $\rho_g (x_1, x_2, x_3) = (x_{g(1)}, x_{g(2)}, x_{g(3)})$. For example: if $g=(12)$, then $g(2,3,-5) = (3,2,-5)$. I understand that the action just permutes the elements, ...
0
votes
1answer
39 views

Finding the dimension of $Alt^2(V)$ and $Sym^2 (V)$, given that $V = \mathbb{C}^2$.

The question is quite clear, I think. I know that if I can count the basis elements, then I am done. Here is the information I was given about these two spaces: $Sym^2(V) = < a \otimes b + b ...
0
votes
1answer
34 views

Prove that $\chi_{V_1 \otimes V_2} (g) = \chi_{V_1} (g) \cdot \chi_{V_2} (g).$

Here, $\chi$ is the character of the sub-representation, i.e., Given $\rho : G \to GL(V)$ is a representation, then the function $\chi_{\rho}: G \to \mathbb{C}: \chi_{\rho}(g) \to Tr(\rho_g)$. I ...
0
votes
0answers
28 views

Powers of traces, integrals over spheres and class functions

Let $V$ be a complex vector space of dimension $\operatorname{dim}_{\mathbb C} V = n$, equipped with a Hermitian inner product $\langle \,\cdot\,,\,\cdot\, \rangle$. Let also $A$ be an endomorphism of ...
0
votes
1answer
20 views

Linear Representations: Show that no $W^0$ exists.

Given the following linear representation and subrepresentation $W$, show that there exists no $W^0$ such that $\mathbb{R}^2 = W \oplus W^0$. Let $\rho: (\mathbb{Z}, +) \to GL(\mathbb{R}^2)$ be ...
0
votes
0answers
46 views

Matrix coefficients of representations of finite groups

In finite-dimensional complex representations of finite groups, I would like to understand what I can learn by looking at a single matrix coefficient. In particular, I would like to look at "diagonal" ...
5
votes
1answer
71 views

Construct a rational matrix $A$ s.t. $A^m = I$

Let $K$ be a field of either $\mathbb{C}$, $\mathbb{R}$ or $\mathbb{Q}$, Let $V$ be a $n$ dimensional vector space over $K$. I want to construct a matrix $A \in GL(V)$ s.t. $A^m = I$ for some $m$ and ...
0
votes
1answer
75 views

Dimension of $V\cap V^{\perp}$ over field extension

I'm wondering if this is true: Let $F \subset K$ be fields $V$ an $K$-vector space. If $U\subset V$ then $$\dim_{F}(U\cap U^{\perp}) \leq \dim_{K}(U\cap U^{\perp})$$ where the $U^{\perp}$ ...
2
votes
1answer
25 views

Questions about eigenvalues of matrices in $GL_2(\mathbb{F}_q)$.

I have some questions about eigenvalues of matrices in $GL_2(\mathbb{F}_q)$. Since $\mathbb{F}_q$ is not algebraically closed, it is possible that some $g \in GL_2(\mathbb{F}_q)$ has eigenvalues which ...
2
votes
1answer
81 views

Finding a matrix representation for two Grassmann numbers.

This question is more general in the sense that I want to know how one finds a particular (say matrix) representation for any object. For the case of Grassmann numbers we have from Wikipedia the ...
5
votes
1answer
153 views

Commuting matrices and simultaneous diagonalizability

It is a known fact from linear algebra that if a set of matrices is pairwise commutable then they are simultaneously diagonalizable. A problem in the book I am currently studying asks to prove this ...
3
votes
0answers
64 views

Computing the decomposition of a representation of $S_n$

I have an explicitly defined representation of the symmetric group that I would like to decompose into irreducibles. How to do this most easily? The best approach I have so far is as follows: Find a ...
0
votes
1answer
47 views

find invariant subspace of polynomials

$(L(t)f)(x)=f(x-t)$ I know that $L$ is representation of the group $\mathbb{R}$ in space continuous functions defined on the real line. Find all the invariant subspaces of polynomials of $L$. ...
0
votes
0answers
98 views

Why Bruhat decomposition in $GL_n$ case is the Gauss decomposition?

Gauss decomposition of a matrix is also called LU decomposition. Let $A$ be a matrix. Then $A=LU$ for some lower triangular matrix $L$ and upper triangular matrix $U$. This can be obtained using Gauss ...
0
votes
1answer
41 views

decompose a real representation of a group $C_2\times C_2

Let $\Phi$ be a real representation of the group $C_2\times C_2=\{e,a\} \times \{e,b\}$ such as $ \Phi(a)=\begin{bmatrix}5 & -4 & 0\\6 & -5 & 0 \\0 & 0 & 1\end{bmatrix}$ and $ ...
1
vote
1answer
54 views

complex irreps is in bijective correspondence with sequences

Let $\{a_n\}$ be a sequences of positive integers such that $$0 \leq a_n\leq p^n - 1,$$ $$a_n \equiv a_{n +1} \bmod p^n \quad \text{for all $n$}$$ Prove that the complex irreps of the group $ ...
3
votes
3answers
50 views

irrep of a non unit element in the finite group

Let $G$ be a finite group. Prove following statement. $\forall g \in G$ such that $g \neq e$ $ \exists$ a complex irrep $\rho$ such that $\rho(g) \neq E$ I have no idea how to start it. I can prove ...
2
votes
1answer
46 views

On the structure of a vector bundle

Let $P \rightarrow X$ be a principal $G$-bundle, $\rho: G\rightarrow GL(V)$ and $\sigma: G\rightarrow GL(W)$ be two finite dimensional linear representations of $G$. Let $E=P\times_\rho V$ and ...
0
votes
1answer
56 views

Irreducible representations of nonabelian group generated by $3$ elements

My question is rather commonplace, but nevertheless I'd like to discribe irreducible representations of the so called Heisenberg group (I suppose this one is just a special case of Heisenberg group). ...
3
votes
1answer
132 views

Representation theory and direct sum

I came across the following theorem in one of the online notes regarding representation theory which I thought should have a simple proof. I am trying to prove it using basic linear algebra tools: ...
0
votes
0answers
59 views

A corollary to the Wedderburn-Artin theorem.

Suppose we proved the Wedderburn-Artin theorem, i.e. we have the fact that if S is a semisimple algebra over a field $F$, then $$ A \cong M_{n_1} (D_1) \times ... \times M_{n_k} (D_k), $$ where ...
3
votes
2answers
276 views

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 ...
1
vote
0answers
32 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 $V$. ...
1
vote
1answer
60 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 $$ ...
3
votes
1answer
75 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. ...
4
votes
1answer
144 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 ...
1
vote
1answer
50 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 $V/U$ ...
1
vote
1answer
154 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 ...
1
vote
0answers
80 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 = ...
1
vote
1answer
242 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 ...
0
votes
1answer
50 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. ...
1
vote
1answer
44 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 ...
1
vote
1answer
41 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 ...
1
vote
0answers
255 views

Spinor representation and Clifford modules

Let $V$ be an even-dimensional real inner product space. We denote the Clifford algebra of $V$ by $C(V)$ and the spinor representation by $S$. For a finite-dimensional $\mathbb Z_2$-graded complex ...
1
vote
1answer
38 views

Representations of the Nil-Coxeter algebra

For $i=1,\ldots,n$, let $u_i$ belong to the Nil-Coxeter algebra $\mathcal{N}_n$ which is defined through: \begin{align} u_i^2&=0\\ u_iu_j&=u_ju_i, \ \ \ \ \ \ \ |i-j|\geq 2\\ ...