Questions tagged [exterior-algebra]

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3
votes
1answer
115 views

action of symmetric group on the second exterior power

Let $e_i \wedge e_j \ (i < j)$ be a basis for the $\mathbb Z$-module $\wedge^2 \Gamma$, where $\Gamma = \mathbb Z^n$. Clearly $S_n$ acts on the module $\wedge^2 \Gamma$ via $$\pi(e_i \wedge e_j) ...
3
votes
0answers
122 views

Are $0, 1, 4, 7, 8$ the only dimensions in which a bivector cross product exists?

It is a well-known mathematical curiosity that ordinary (vector-valued) cross products over $\mathbb{R}$ exist only in dimensions $0, 1, 3$ and $7$ (this fact is related to Hurwitz's theorem that real ...
9
votes
2answers
266 views

Can we recover all $k$-minors of a square matrix from some of them?

This is a cross-post. Let $k,n$ be natural numbers, $1<k<n$. Suppose we have an "unknown" invertible $n \times n$ matrix $A$ over a field of characteristic zero. (we do not know the entries of ...
5
votes
1answer
218 views

Extension of a bilinear form to the exterior algebra

In Serre's Local Fields, at the beginning of the chapter III section 2, he has wrote "it is known that $T$ extends to a non-degenerate bilinear form on the exterior algebra of $V$", where $T$ is a ...
3
votes
1answer
203 views

Trace and exterior product

Let $V$ be a $2n$-dimensional complex vector space with base $\{e_1,\dotsc,e_n,f_1,\dotsc,f_n]\}$ Let $W \subset \wedge^n V$ be the subspace in the exterior product, with basis vectors $$ e_{i_1} \...
2
votes
1answer
237 views

Exterior derivative independence from coordinate systems

In the book Mathematical Methods of Classical Mechanics by V.I. Arnold, the author introduces (p.189) the concept of exterior derivative as "the principal linear part of the increment" of the function ...
1
vote
0answers
53 views

Function for unique volume element

This is an issue that I'm am trying to solve for a fine-tuning measure in particle physics, but it is purely mathematical. Consider three vectors $\{v_1, v_2, v_3\}$ in $\mathbb{R}^3$. I would like a ...
0
votes
0answers
155 views

A vector calculus formula

Let me answer my own question, hoping to be forgiven for that. I asked unsuccessfully that question on Mathematics. Let $A, B$ be vector fields in $\mathbb R^3$. We have $$ \text{curl}\bigl((A\cdot \...
1
vote
0answers
50 views

Poincaré lemma for gradient times its transpose

Poincaré lemma states that a vector $v_i(x)$ defined on a ball in $R^n$ is the gradient of a function if and only if \begin{equation} \partial_i v_j = \partial_j v_i \end{equation} or equivalently ...
6
votes
1answer
241 views

Is every basis for $\bigwedge^kV$ satisfying a “complementary” property a rescaling of a “standard” basis?

This is a cross-post. Let $V$ be a $4$-dimensional real vector space. Let $\omega_{i_1,i_2}$ be a basis for $\bigwedge^2V$, where each $\omega_{i_1,i_2}$ is decomposable. Suppose that for every $\...
5
votes
1answer
297 views

Do non-continuous Sobolev maps pull back closed forms to weakly closed forms?

$\newcommand{\R}{\mathbb R}$ $\newcommand{\N}{\mathbb N}$ $\newcommand{\de}{\delta}$ $\newcommand{\sig}{\sigma}$ $\newcommand{\Average}[1]{\left\langle#1\right\rangle} $ $\newcommand{\IP}[2]{\Average{...
3
votes
2answers
438 views

What are the necessary and sufficient conditions for a two-form to be an exterior product of two one-forms?

Consider a two-form $\gamma \in \Lambda^2(V)$ where $V$ is a real vector space. Now I would like to know the necessary and sufficient conditions for $\gamma$ to be expressible as an exterior product ...
4
votes
1answer
190 views

Taylor spectrum of commuting operators

Taylor spectrum of commuting operators Fom the following paper (M. Ch—o, H. Motoyoshi, B. Na¡cevska Nastovska: On the joint spectra of commuting tuples of operators and a conjugation) we have Let ...
10
votes
1answer
224 views

Representations of degenerate Clifford algebras

Given a real finite-dimensional vector space $V$ with a symmetric bilinear form $b$, we define the Clifford algebra $Cl(V,b)$ as the quotient of the tensor algebra $\bigotimes V$ by the two sided ...
7
votes
0answers
111 views

A generalization of matrix minors to non-integer values

I am interested to know if there exist a notion of $k$-minors of a real square matrix, for non-integer positive values of $k$ One approach I thought of was to use the fact that the $k$-minors are (...
3
votes
3answers
354 views

Can we specify the value of harmonic forms at a point?

Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $1 < k < d$ be fixed. Let $p \in M$, and let $\alpha_p \in \bigwedge^k(T_pM)^*$. Does there exist an open ...
2
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0answers
42 views

Ranks of cycles with base field coefficients as a generalization of ranks of multivectors?

This must be probably only reference request since I am inclined to believe that I am asking about something well known but just cannot pin down appropriate keywords for searching. The starting point:...
7
votes
1answer
234 views

A commutative variant of the exterior algebra

Consider $A = \mathbb{R}[t_1,\ldots,t_{k}]$, the ring of real $k$-variate polynomials in the indeterminates $t_1,\ldots,t_k$. For $y\in\mathbb{R}$, define the element $p(y) \in A$ through $$ p(y) = ...
17
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1answer
506 views

An explicit reconstruction of a matrix from its minors

$\newcommand{\End}{\operatorname{End}}$ $\newcommand{\GL}{\operatorname{GL}}$ $\newcommand{\Cof}{\operatorname{cof}}$ Let $V$ be a $d$-dimensional real vector space. ($d \ge 4$). Fix an odd integer $...
2
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0answers
30 views

Expression for the lattice operations on subspaces in Plucker embedding

Suppose that $V$ is a finite dimensional $\mathbb Q$-vector space. To each subspace $S$ of dimension $k$, we can associate the line from the origin of $\Lambda^k(V)$ through the point $s_1\wedge \...
3
votes
0answers
215 views

Is the image of the map $A \to \bigwedge^{k}A $ a weakly embedded submanifold?

$\newcommand{\End}{\operatorname{End}}$ $\newcommand{\GL}{\operatorname{GL}}$ Let $V$ be a $d$-dimensional real vector space. ($d \ge 4$). Fix an odd $2 \le k \le d-2$. Define $H_{>k}=\{ A \in \End(...
4
votes
1answer
187 views

Is the map $A \to \bigwedge^{k}A $ from matrices above rank $k$ proper?

$\newcommand{\End}{\operatorname{End}}$ Let $V$ be a $d$-dimensional real vector space. ($d \ge 3$). Fix an odd $2 \le k \le d-1$. Define $H_{>k}=\{ A \in \End(V) \mid \operatorname{rank}(A) > ...
5
votes
1answer
134 views

If $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$ equals $\bigwedge^k B$ for some complex matrix $B$, does it have a real source?

Let $1<k<d$ be an integer. Let $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$, and suppose that $A=\bigwedge^k B$ for some complex $B \in \text{End}(\mathbb{C}^d)$. Does there exist $M \in \...
12
votes
1answer
385 views

Is the image of the map $A \to \bigwedge^k A$ closed over $\mathbb{R}$?

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) \, \, \, \, , \, \, ...
7
votes
1answer
188 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 $p \ge 1$. ...
8
votes
2answers
384 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 ...
6
votes
0answers
238 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 ...
6
votes
1answer
165 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 ...
7
votes
2answers
980 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 $r$-...
1
vote
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)...
3
votes
1answer
199 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 ...
1
vote
0answers
101 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 ...
19
votes
1answer
1k 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?
6
votes
1answer
252 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 ...
13
votes
3answers
587 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 \...
6
votes
2answers
634 views

When is the exterior algebra a Hopf algebra?

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 ...
0
votes
2answers
185 views

Can one reconstruct a diagonalisable endomorphism from its action on the exterior algebra?

Let $K$ be a field. Let $V$ be a finite-dimensional $K$-vector space; and let $f$ be an automorphism of $V$. Assume that $f$ is diagonalisable over some extension of $K$. Form the exterior algebra $\...
2
votes
1answer
213 views

Invariant characterization of isometric embeddings

$\newcommand{\Cof}{\operatorname{Cof}}$ $\newcommand{\Det}{\operatorname{Det}}$ $\newcommand{\Lam}{\operatorname{\Lambda}}$ Motivation (and the "classic" case): I am trying to find a coordinate-free ...
2
votes
0answers
196 views

Exterior power of a torsion-free sheaf on a DVR

Let $R$ be a discrete valuation ring and $X$ be a regular, integral. projective $R$-scheme, flat over $R$. Let $F$ be a torsion-free coherent sheaf on $X$ of rank $n$, flat over $\mathrm{Spec}(R)$. Is ...
4
votes
1answer
230 views

Ideals in exterior algebras over the field with two elements

Suppose we have an exterior algebra over $\mathbb{F_2}$, say $R = \Lambda_{\mathbb{F_2}}V$, where $V$ is an $n$-dimensional $\mathbb{F}_2$ vectorspace. Let $x_1,\ldots,x_n$ be a basis of that ...
4
votes
0answers
324 views

Hodge duality and the determinant of the product of two matrices

I stumbled onto the following identity, and I would like to know: Is it known by some name and are there some references I might cite (or is it actually too trivial to be mentioned anywhere)? Are ...
6
votes
1answer
226 views

Is the (super-)symmetric power of the exterior algebra free?

Let $V$ be a vector space over $k$ of dimension $m$. (I'm only interested in the case $k=\mathbb{Q}$.) Let $R:=\Lambda^*V$ be the exterior algebra. It carries the structure of a supercommutative ring: ...
4
votes
1answer
230 views

Exterior derivative as only (up to multiple) natural operator $\Lambda ^kT^\ast \rightsquigarrow \Lambda ^{k+1}T^\ast$

In Kolar, Michor, & Slovak's book Natural Operations in Differential Geometry, it is proved the exterior derivative is universal in the following sense. Proposition 25.4. For $k>0$ all natural ...
3
votes
1answer
203 views

Supercommutator of exterior multiplication operators and their adjoints

Let $\mathfrak{h}$ be a complex Hilbert space and consider Grassmann algebra $\mathcal{F}=\bigwedge\mathfrak{h}$ with its induced inner product. For $\omega\in\mathcal{F}$ we also consider the ...
4
votes
0answers
128 views

Cannot multivectors be classified more easily than general tensors?

This is sort of a spinoff of Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces? - seems to be almost hopeless, but maybe some partial ...
5
votes
0answers
142 views

Counting square zero forms over finite fields

Let $p$ be an odd prime and let $R=\Lambda_{\mathbb{F}_p}[x_1,\dots,x_n]$ be the exterior algebra on $n$ generators over the finite field with $p$ elements. This is a graded-commutative ring. Is ...
1
vote
1answer
137 views

Are sections of $\tau M$ differential operators on the exterior algebra?

Let $M$ be a smooth manifold and $\tau M$ its second-order tangent bundle. A second-order vector field $A\in \Gamma(\tau M)$ can locally be expressed as a finite sum of operators $C^\infty(M)\...
8
votes
1answer
551 views

Deligne's exterior power

In "Catégories Tannakiennes", Deligne defines the $n$th exterior power of an object $A$ of an abelian tensor category $\mathcal{C}$ as the image of the morphism $$p : A^{\otimes n} \to A^{\otimes n}, ...
4
votes
0answers
277 views

Differential ideals of Pfaffian forms on jet bundles (Integrability)

(I asked this question on math.stackexchange, but got no reaction in several weeks. So, my conclusion is, that it is harder to answer than I thought, and maybe admissible for the attribute 'research ...
4
votes
0answers
190 views

Explicit description of graded (counital) cofree cocommutative coalgebras

Let $k$ be a field of characteristic $p \neq 2$, and $V = \oplus V_{n}$ be a graded vector space over $k$. Then, can one compute the graded (counital) cofree cocommutative coalgebra $C(V)$ ...