0
votes
0answers
21 views

Iwasawa decomposition of $GL_n\times GL_m$

One knows that any reductive group, in particular GL$_n$, has an Iwasawa decomposition $G=NAK$. Is the Iwasawa decomposition of $GL_n\times GL_m$ simply the diagonal decomposition, $$GL_n\times ...
3
votes
1answer
31 views

Every representation of a finite group is reducible?

I somehow "proved" that every representation of a finite group is reducible. While I'm fairly sure the error is something silly, I can't seem to place it. Could someone please help me figure out what ...
4
votes
0answers
43 views

Counting the number of elements in a double coset

Let $G$ denote the groups of $n\times n$ invertible matrices and $H$ be the subgroup of invertible upper triangular matrices. For $n=2$, by row reduction, or equivalently LU decomposition, it is ...
1
vote
1answer
39 views

Endomorphism ring of finite-dimensional representation

If $G$ is any group and $V$ is a finite-dimensional representation of $G$, then we can form the endomorphism ring $E = End_G(V)$. Suppose that $V$ is indecomposable, i.e. not a direct sum of ...
0
votes
0answers
22 views

Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course. Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
1
vote
2answers
42 views

Show that the homomorphism $\lambda: k[X] \to End_k(V) : p \mapsto p(A)$ corresponding to the $k[X]$-module strucutre of $V$ has a nontrivial kernel.

$\DeclareMathOperator{\End}{End}$ I'm trying to show that: Show that the homomorphism $\lambda: k[X] \to \End_k(V) : p \mapsto p(A)$ corresponding to the $k[X]$-module strucutre of $V$ as in (see ...
0
votes
1answer
27 views

Quaternionic representation

Let $V$ be $G$-representation over quaternions $\mathbb{H}$. How to show that $$ \mathbb{H} \otimes_\mathbb{C} V $$ is canonically isomorphic to $V \oplus V$ as representation over $\mathbb{H}$? In ...
2
votes
0answers
91 views

Tensor product of algebras which is Frobenius.

Let $A$ and $B$ be two finite dimensional algebras over a field $k$. Let us suppose that the $k$-algebra $A\otimes_{k} B$ is Frobenius (or symmetric). Is it true that $A$ and $B$ are two Frobenius ...
2
votes
1answer
48 views

isomorphism classes of representations of a quiver

Classify all isomorphism classes of representations of dimension vector 1 and 2 of the following quiver The professor briefly did the solution, but I could not understand what was going on. What he ...
0
votes
0answers
34 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
32 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
55 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
42 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
82 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
31 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
28 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
95 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
54 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
61 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
34 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
35 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
40 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
49 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
78 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
95 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
164 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
108 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
48 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
57 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
139 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
278 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
33 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
76 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
153 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 ...