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Questions tagged [tensor-decomposition]

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Decomposition of the tensor product $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}[i]$ into a product of fields

I am trying to solve the following problem: For each rational prime $p$, describe the decomposition of the tensor product $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}[i]$ into a product of fields, ...
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Does every subspace of the exterior algebra of dimension $>1$ contain a decomposable element?

Let $V$ be a real $n$-dimensional vector space, and let $W \le \bigwedge^k V$ be a subspace . Suppose that $\dim W \ge 2$. Does $W$ contain a non-zero decomposable element? If $\dim W=1$, then ...
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Tensor product and linear dependence of vectors

Let $V_1, \ldots, V_k$ be complex vector spaces. Given $k$ vectors $v_1 \in V_1, \ldots, v_k \in V_k$, we define that the tensor product $v_1 \otimes \ldots \otimes v_k$ has rank 1. For any tensor $T \...
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Why is the exterior power $\bigwedge^kV$ an irreducible representation of $GL(V)$?

$\newcommand{\GL}{\operatorname{GL}}$ Let $V$ be a real $n$-dimensional vector space. For $1<k<n$ we have a natural representation of $\GL(V)$ via the $k$ exterior power: $\rho:\GL(V) \to \GL(\...
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Is every decomposable basis for $\bigwedge^kV$ “standard”?

This is a curiosity: Let $V$ be a $d$-dimensional real vector space, and let $1<k<d$. Set Let $\omega^{i_1,\ldots,i_k}$ be a basis for $\bigwedge^kV$, whose elements are all decomposable. Is $\...
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1answer
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Can we always span a decomposable form via constant coefficients?

Let $M$ be a smooth manifold of dimension $d$. Let $1 < k <d$ be an integer. Let $\omega^i$ be a local frame of the exterior power bundle $\Lambda_{k}(T^*M)$. Does there exists numbers $a_i ...
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2answers
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Comparing the representation $T \otimes T $ in terms of matrices and $T^2$

Interpret the representation $ T \otimes T$ in terms of matrices, and compare it with $T^2$. Could anyone give me a hint on how to solve this please ? EDIT: enter image description here enter ...
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Is “being decomposable” preserved under taking a subspace?

Let $V$ be a vector space over some field, and $W \le V$ a vector subspace. Let $1<k<\dim V$ be an integer. Suppose $\omega \in \bigwedge^k W$ is decomposable as an element in $\bigwedge^k V$....