# Questions tagged [exterior-algebra]

It is a quotient - of the tensor algebra, obtained by taking graded sum over whole numbers $n$ of $n$-fold tensor products - by the ideal generated by elements of the form $a\otimes a$. We write the residue class of $a\otimes b$ in this algebra, as $a\wedge b$ and call it the wedge product.

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### Koszul sign convention and symmetric group action on the graded n-th tensor product

Let $V_\bullet = (V_k)_{k \in \mathbb{Z}}$ and $W_\bullet = (W_k)_{k \in \mathbb{Z}}$ be two graded vector spaces on 0 caracteristic field. We define the tensor product of $V_\bullet$ by $W_\bullet$ ...
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### Eigenvalues of the second exterior power of a linear operator

Let $K$ be a field of characteristic zero and $V=K^n$. Let $T: V \to V$ be a linear map with eigenvalues $\lambda_1,...,\lambda_n \in K$ , not necessarily all distinct. Let $\wedge^2 V$ be the second ...
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### Determinant of transpose intuitive proof

We are using Artin's Algebra book for our Linear Algebra course. In Artin, det(A^T) = det(A) is proved using elementary matrices and invertibility. All of us feel that there should be a 'deeper' or a ...
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### Is a pointwise decomposable differential form smoothly decomposable?

Let $\omega$ be a smooth differential form on a smooth manifold $M$. Suppose $\omega$ is pointwise decomposable, that is for every $p \in M$, $\omega_p=e^1_p \wedge e^2_p \wedge \dots \wedge e^k_p$ ...
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### Exterior product generates the infinitesimal rotations — what is the geometric significance?

Question: What is the geometric significance of the fact that the exterior products of the unit basis vectors in $\mathbb{R}^3$ generate a basis for the Lie algebra of the pure rotation group $SO(3)$? ...
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### The exterior derivate and pullback commute

The above question is from a past exam. I am having trouble with the fine details, ie what $F*dw$ and $dF*w$ actually look like. Can anybody show me how this question is solved? I have solved it using ...
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### What is a more formal name for the wedgey group?

The rule $(v_1,w_1)⋅(v_2,w_2)=(v_1+v_2,w_1+w_2+(v_1∧v_2))$ defines a group structure on the vector space $V⊕(V∧V)$ whenever $V$ is itself a vector space over some field $F$. What is a more common ...
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### Is there a natural way to view $\bigwedge^k_{\mathbb{C}}\mathbb{C}^d$ as a subspace of $\bigwedge^k_{\mathbb{R}}\mathbb{C}^d$

Does there exists an $\mathbb R$-linear embedding $\bigwedge^k_{\mathbb{C}}\mathbb{C}^d \to \bigwedge^k_{\mathbb{R}}\mathbb{C}^d$ which maps decomposable tensors to decomposable tensors? (The ...
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### If $\omega \in \bigwedge^k\mathbb{R}^d$ is decomposable over $\mathbb C$, is it decomposable over $\mathbb R$?
Let $k,d$ be positive integers, and let $\omega \in \bigwedge^k\mathbb{R}^d$ be decomposable in $\bigwedge^k\mathbb{C}^d$. Is $\omega$ decomposable in $\bigwedge^k\mathbb{R}^d$? Edit: Let me be ...