Tagged Questions

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Indecomposable quiver representations

Is there are any way to found indecomposable representation of a given quiver explicitely if it's dimention vector is given?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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" ...
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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 ...
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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 $U^{\perp}$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $L$. ...
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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 ...
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