# Questions tagged [exterior-algebra]

The exterior-algebra tag has no usage guidance.

99
questions

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### When is a symmetric tensor equal to gradient times gradient?

On a ball in $\mathbb R^n$, a vector field $v_j$ is a gradient of a function when its exterior derivative vanishes. In other words, if $$\partial_i v_j-\partial_j v_i=0$$
then there exists a function $...

0
votes

0
answers

115
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### Koszul exterior connections

Let $(E,M)$ be a vector bundle over a riemannian manifold $M$ which is a module for the exterior forms of $M$. I define a Koszul exterior connection as an operator $\nabla$ such that:
$$
\nabla : E \...

1
vote

1
answer

73
views

### nth-power of the dual Lefshetz operator

Let $(X,\omega)$ be a Kahler manifold, denote by $\Lambda$ the dual of the Lefshetz operator $\omega\wedge$ (see e.g. Dual Lefschetz Operator and Contraction with the Fundamental Form). Let $\zeta\in\...

1
vote

0
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74
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### Exterior algebra of free modules over Hopf algebras

Let $H$ be a commutative, cocommutative Hopf algebra over a field $\mathbb{K}$, and $M$ a free Hopf module over $H$. Is the exterior algebra $\Lambda^k_\mathbb{K} M$ with the diagonal $H$-action
$$h \...

2
votes

0
answers

91
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### Is there a non-degenerate solution for this PDE on $\mathbb{R}^3$?

$\newcommand{\tr}{\operatorname{tr}}$
$\newcommand{\R}{\mathbb{R}}$
Does there exist a smooth map $f:\mathbb{R}^3 \to \mathbb{R}^3$, which satisfies
$$\tr \big( df \otimes \delta(df \wedge df) \big)=0,...

2
votes

0
answers

201
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### What comes next in the sequence "symmetric algebras, exterior algebras, divided power algebras, ..."?

This question was posed by A Rock and a Hard Place in this discussion, where they mentioned the isomorphisms
\begin{align*}
\mathrm{L}\,\mathrm{Sym}^n_R(M[1]) &\cong (\mathrm{L}\,{\...

4
votes

0
answers

129
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### Structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ with $A$ a local integral domain

I am trying to see the structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ where $A$ is a local integral domain with small residue field.
Let $A$ be a local integral domain with maximal ideal $M$, residue ...

2
votes

1
answer

122
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### Intersecting a ideal generated in degree $\leq a$ with one generated in degree $\leq b$ in a polynomial ring

Question 1. Let $R$ be a polynomial ring over a field $k$. Assume that $R$ is graded in the usual way (i.e., each variable has degree $1$). Let $I$ and $J$ be two ideals of $R$ such that $I$ is ...

0
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0
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49
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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 ...

1
vote

1
answer

389
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### Simple properties of the codifferential

The exterior derivative $d$ has many very nice algebraic relations. For example
$d(\alpha\wedge\beta) = (d\alpha)\wedge \beta +
(-1)^k\alpha\wedge(d\beta)$
$f^*(d \alpha)=d f^*(\alpha)$.
$d\circ ...

2
votes

0
answers

105
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### What's known about the matroid induced by the Plücker coordinates of the representation of a matroid?

Let $M$ be a linear matroid with ground set $E$ and independent subsets $\mathcal I$, represented by $\rho: E \rightarrow V$.
This induces a map
$$
\hat\rho: \mathcal I \rightarrow \mathbf P(\Lambda V)...

5
votes

0
answers

123
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### Rational cohomology of p-adic general linear groups

I wanted to compute the cohomology ring $H^*(GL_n(\mathbb{Z}_p); \mathbb{Q}_p)$ (with $p$ fixed prime as usual). I found some incomplete notes stating that the computation should go as follows.
First ...

5
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0
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161
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### Federer's questions on the mass and comass norms

In Federer's book "Geometric measure theory", he says in section 1.8.3 (where $\|\cdot\|$ is the comass norm):
Very little appears to be known about the structure of the convex sets $\wedge^...

4
votes

1
answer

221
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### Exterior algebra of normed spaces

This question is related to my prior question, but this one is aimed, even though it's more general. If $V$ is a vector space, we define the exterior algebra of $V$ do be:
$$\bigwedge V := \bigoplus_{...

6
votes

1
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685
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### What is the role of topology on infinite dimensional exterior algebras?

Wedge products and exterior powers are discussed in W. Greub's book Multilinear algebra as follows.
Definition: Let $E$ be an arbitrary vector space and $p \ge 2$. Then a vector space $\bigwedge^{p}E$ ...

3
votes

1
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161
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### 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) ...

16
votes

0
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520
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### Are $0, 1, 4, 7, 8$ the only dimensions in which a bivector-valued 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

2
answers

316
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### 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

1
answer

444
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### 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

1
answer

520
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### 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

1
answer

595
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### 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
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0
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60
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### 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 ...

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0
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170
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### 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 \...

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0
answers

73
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### 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 ...

7
votes

1
answer

268
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### 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 $\...

6
votes

1
answer

331
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### 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{...

4
votes

2
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767
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### 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

1
answer

286
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### 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 ...

11
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1
answer

340
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### 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

0
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131
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### 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

3
answers

413
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
votes

0
answers

53
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### 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

1
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263
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### 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
votes

1
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569
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### 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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0
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### 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
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0
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266
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### 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

1
answer

200
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### 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

1
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### 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
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1
answer

392
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### 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

1
answer

198
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### 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

2
answers

403
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### 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 ...

6
votes

0
answers

250
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### 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

1
answer

215
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### 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

2
answers

1k
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### 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

0
answers

51
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### 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

1
answer

223
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### 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

0
answers

121
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### 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 ...

20
votes

1
answer

2k
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### 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

1
answer

256
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### 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
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3
answers

641
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### 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 \...