# Questions tagged [exterior-algebra]

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### Density of a graded algebra

I'm trying to prove the following proposition: If $v \in V$ and $Y \in \mathfrak{so} (V)$ then $[\dot\mu(Y), B(v)] = B(Yv)$. By definition $[\dot\mu(Y), B(v)] = \dot\mu(Y)B(v) - B(v)\dot\mu(Y).$ I ...
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Let $V$ a real vector space of dimension $d$. Let $1<k < d-1$. Consider the map induced by the exterior algebra functor: $$\psi:\text{End}(V) \to \text{End}(\bigwedge^kV) \, \, \, \, , \, \, ... 1answer 192 views ### Is a Sobolev map with smooth minors smooth on the whole domain? Let d\ge 3 and 2 \le k \le d-1 be integers, where at least one of k,d is odd. Let \Omega \subseteq \mathbb{R}^d be open, and let f \in W^{1,p}(\Omega,\mathbb{R}^d), for some p \ge 1. ... 2answers 398 views ### Obstructions for the wedge of coordinate differentials to be harmonic Let (M,g) be a smooth d-dimensional Riemannian manifold, d even. Are there obstructions (I guess in terms of curvature) for g to have the following property: For every p \in M there exist a ... 0answers 244 views ### Is a Sobolev map with invertible smooth minors smooth? \newcommand{\Cof}{\text{cof}} Let k,d be even integers, such that d\ge3 and 2 \le k \le d-1. Let \Omega \subseteq \mathbb{R}^d be open, and let f \in W^{1,p}(\Omega,\mathbb{R}^d), for ... 1answer 183 views ### Which metrics on exterior power are induced from metrics on the base? \newcommand{\id}{\text{id}} \newcommand{\Hom}{\text{Hom}} This is a cross-post. Let V be a d-dimensional real vector space, and let 2 \le k \le d-1. Every inner product on V induces an ... 2answers 1k views ### Determinant of a sub-matrix of the classical adjoint Let A be a square matrix of order n, say with complex coefficients, and let M be the plain matrix of minors of A of order n-1 (no transpose, no sing changes). Let I and J be r-... 0answers 51 views ### Maximum number of matrices satisfying given rank conditions Assume that we have 2k matrices S_1,\ldots,S_k and \Phi_1,\ldots,\Phi_k over some finite field F such that (i) S_i\in F^{l/2\times l} and \dim S_i=l/2 for any i\in\{1,\ldots,k\}; (ii)... 1answer 207 views ### Determine 1-form from volume forms Given a 1-form \omega\in \Omega^1(\mathbb{R}^p) we can construct various non-trivial p-forms (\omega\wedge\star\omega excluded), but using the exterior derivative for all of all the players to ... 0answers 106 views ### The dual space mapping of a p-form I know that the dual of a tensor product is the tensor product of the duals. For example, the dual space of V\otimes V^* is V^* \otimes V. I also know that the induced dual space mapping of a ... 1answer 2k views ### Etymology of “exterior” in “exterior calculus” What is the origin of the term "exterior" in "exterior calculus"? How does this term relate to "interior products" and "inner products", if it does at all? 1answer 255 views ### Simultaneous triangularisation of an exterior power of a set of matrices I'm working on some research problems relating to random matrix products, and this is taking me into areas of mathematics I've not previously studied: Lie groups, representation theory, and real ... 3answers 616 views ### Ring of invariants of \operatorname{SL}_6 acting on \Lambda^3 \mathbb C^6 Let G=\operatorname{SL}_6 act on V=\Lambda^3 \mathbb C^6. I would like to find the ring of invariants \mathbb C[V]^G. There is an obvious invariant$$Sq: V \to \mathbb C, \quad \omega \mapsto \...
I have several questions on the exterior algebra of a vector space: Q1:When has the exterior algebra A (viewed just as an algebra, not considered as a graded algebra) of an $n$-dimensional vector ...