# 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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### Relationship between exterior power of representation and variance?

I was reading the question: Symmetric and exterior power of representation regarding how to determine the character of an exterior power of a representation from the original representation. One of ...
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### How to compute Ext over an exterior algebra

I found this question in several places (even on mathoverflow and mathstackexchange), but I never found a satisfying answer. Let $k$ be a field and $V$ a finite dimensional $k$-vectorspace. I ...
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### Hodge-Star-Operator on arbitrary oriented basis

Assume that $V$ is oriented finite dimensional vectorspace with dimension $n$, $g \in T^0_2(V)$ a given symmetric and nondegenerate tensor. Let $\mu$ be the corresponding volume element of $V$. ...
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### How many matrix minors determine all the minors?

Let $n$ be a positive integer, and let $1<k<n$. Suppose we have an "unknown" real $n \times n$ matrix $A$. (we do not know the entries of $A$). Can we recover all the $k$-minors of $A$ from ...
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### A characterization of the subgroup of $\text{GL}(\bigwedge^k V)$ which preserves pure tensors?

Let $V$ be a $d$-dimensional real vector space, and let $1<k<d$. Set $$H=\{B\in\text{GL}(\bigwedge^k V) \, | \, B \,\text{ preserves pure tensors }\}$$ (i.e. $B \in H$ if it maps ...
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### Does this geometric rigidity condition forces the map to be the identity?

$\newcommand{\Cof}{\operatorname{cof}}$ $\newcommand{\id}{\operatorname{Id}}$ $\newcommand{\End}{\operatorname{End}}$ $\newcommand{\GL}{\operatorname{GL}}$ Let $V$ be a real $d$-dimensional vector ...
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### Classifying the orbits of the natural $\text{GL}(V)$-action on the exterior power $\bigwedge^k V$

Let $V$ be a real $d$-dimensional vector space, and let $1 < k < d$. Consider the following action of $\text{GL}(V)$ on $\bigwedge^k V$: $(T,\omega) \to (\bigwedge^k T) \omega$. Can we ...
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### Vector Laplace Beltrami operator on surface tangent and surface normal vector field

Consider a closed, compact, embedded surface $f:M \rightarrow \mathbb{R}^3$ and a vectorfield $X$ on the surface that can be decomposed in the surface frame basis $\{e_1,e_2,e_3\}$, where $\{e_1,e_2\}$...
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### Wedge product for composite of linear transformations

I am studying the wedge product for multivariable analysis, but I feel that the operations are not so intuitive in general. I am looking at the following question which deals with linear ...
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### Clarifying a step in an answer about exterior products of coherent sheaves

The question is about the accepted answer here. I decided not to ask in a comment there since the original asker is no longer active. In "Step 2" of the answer given there, Roland claims, "This is ...
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### Finding the Wedge Product of Two Multivariable Vectors and Making Sense of It

I was attempting to solve: $$-2dx_{1} \land dx_{4}\left( \begin{bmatrix} 2 \\ 3 \\ -5 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 4 \\ -5 \end{bmatrix}\right)$$ I solved this by pulling out the ...