# 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.

214 questions
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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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### 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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Let $V$ be a graded vector space, thought of as a collection $\{ V^n \}_{n \ge 0}$ of vector spaces. Let $V_{odd} = \bigoplus_{n \text{ odd}} V^n$ and $V_{even} = \bigoplus_{n \text{ even}} V^n$. I am ...
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### Is there a linear injection $\Lambda^k V^* \otimes \Lambda^k V^* \to \Lambda^k (V^* \otimes V^*)$ which preserves decomposability?

Let $V$ be an $n$-dimensional real vector space, and let $2 \le k \le n-2$. Definitions We say an element $\omega \in \Lambda^k V$ is decomposable if $\omega=\alpha_1 \wedge \dots \wedge \alpha_k$, ...
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### What is the relation between the determinant of the adjoint representation, and volume or eigenvalues?

I'm relatively new to Lie Groups, and finding in a text the following statement, as it is supposed to be obvious. I would love some reference or context to either of these two (probably related) ...
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### Why is the symbol for the exterior product a meet rather than a join?

I've moved this over to HSM. It seems odd that something that looks so much like a join [see below] would get given "the wrong symbol". It's even worse when you dualise it and get something called (...
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### What is the difference between tensor calculus and exterior derivative type concepts?

I am trying to clarify terms in order to help me figure out what I'd like to study. I understand that $p$-forms and $p$-vectors are used with things like wedge products, exterior algebras, and a ...
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### Is there a non-degenerate solution for this PDE on $\mathbb{R}^3$?

$\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\R}{\mathbb{R}}$ I am looking for a smooth map $f:\mathbb{R}^3 \to \mathbb{R}^3$, which satisfy \delta \big( df \wedge df \big) \neq 0, \, \...
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### Uniqueness of connection $1$-forms and curvature $2$-forms

If $(M,\langle\cdot,\cdot\rangle)$ is a pseudo-Riemannian manifold and $\nabla$ denotes the Levi-Civita connection of the metric, we can define the connection $1$-forms and curvature $2$-forms ...
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### Wedge product of vector and tensor

As near as I can figure it: (a⊗b)∧c = a⊗b⊗c - a⊗c⊗b + c⊗a⊗b But I'm not sure if thats right. a, b, and c are vectors. I know that a∧b = a⊗b - b⊗a which is a bivector. The equation I am most ...
What is the form of the Lie derivative in Clifford algebra? Context: Consider the Clifford algebra $\mathcal{C}l (p,q)$ with basis $\{e_i \}$. The geometric derivative following Hestenes is defined ...
### Wedge product of regular $r$-forms
I need to solve this exercise: A continuous form $w$ of degree $r$ in the open $U\subset\mathbb{R}^m$ is regular $\iff$ for all open $V$ with compact closure $\overline{V} \subset U$, there is a ...