Questions tagged [exterior-algebra]
The exterior-algebra tag has no usage guidance.
103
questions
19
votes
3
answers
899
views
How big can a wedge of $n$ 2-forms in $\mathbb{R}^{2n}$ be?
$\def\RR{\mathbb{R}}$Let $\omega$ be a $2$-form on $\RR^{2n}$, where $\RR^{2n}$ has the usual Euclidean norm. The comass of $\omega$ is defined to be $\max_{|u|, |v| \leq 1} \omega(u,v)$. Here is the ...
- 14.1k
7
votes
1
answer
391
views
How big can a wedge of 2-forms be?
The comass of a 2-form $\alpha$ is the maximal value of $\alpha(u,v)$
for a pair of unit vectors $u,v$. The symplectic form $\alpha$ on
$\mathbb R^{2n}$ has the property that $|\alpha^{\wedge n}| = n!...
- 14.1k
0
votes
0
answers
72
views
Intersection of stabilizer group orbits and algebraic variety of decomposable forms
I have been trying to prove/come up with counter examples to the following situation, any help would be very much appreciated.
Let $\{E_I\}$ be a basis of $\mathbb R^6$, so that any vector $V\in\...
3
votes
1
answer
365
views
What is the name for algebras generated by elements, all of whose cubes vanish?
Given a ring $R$ with identity $1$, we can define the exterior algebra of order $k$ over $R$ to be the algebra over $R$, generated by elements $x_1, \dots, x_k$ satisfying $x_i^2 = 0$ for each index $...
- 397
4
votes
0
answers
112
views
The set of strongly positive forms is a closed cone
This question comes from the Complex Analytic and Differential Geometry by Demailly. Let $V$ be a $n$ dimensional complex space. Consider the exterior algebra $\Lambda V^* = \oplus \Lambda^{(p,q)}V^*$....
- 98
5
votes
1
answer
130
views
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 $...
- 51
0
votes
0
answers
147
views
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:
$$
\nabla : E \...
- 377
1
vote
1
answer
92
views
nth-power of the dual Lefshetz operator
Let $(X,\omega)$ be a Kahler manifold, denote by $\Lambda$ the dual of the Lefshetz operator $\omega\wedge$ (see e.g. Dual Lefschetz Operator and Contraction with the Fundamental Form). Let $\zeta\in\...
- 747
1
vote
0
answers
95
views
Exterior algebra of free modules over Hopf algebras
Let $H$ be a commutative, cocommutative Hopf algebra over a field $\mathbb{K}$, and $M$ a free Hopf module over $H$. Is the exterior algebra $\Lambda^k_\mathbb{K} M$ with the diagonal $H$-action
$$h \...
2
votes
0
answers
99
views
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
$$\tr \big( df \otimes \delta(df \wedge df) \big)=0,...
- 6,571
2
votes
0
answers
231
views
What comes next in the sequence "symmetric algebras, exterior algebras, divided power algebras, ..."?
This question was posed by A Rock and a Hard Place in this discussion, where they mentioned the isomorphisms
\begin{align*}
\mathrm{L}\,\mathrm{Sym}^n_R(M[1]) &\cong (\mathrm{L}\,{\...
- 9,565
4
votes
0
answers
133
views
Structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ with $A$ a local integral domain
I am trying to see the structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ where $A$ is a local integral domain with small residue field.
Let $A$ be a local integral domain with maximal ideal $M$, residue ...
- 259
2
votes
1
answer
189
views
Intersecting a ideal generated in degree $\leq a$ with one generated in degree $\leq b$ in a polynomial ring
Question 1. Let $R$ be a polynomial ring over a field $k$. Assume that $R$ is graded in the usual way (i.e., each variable has degree $1$). Let $I$ and $J$ be two ideals of $R$ such that $I$ is ...
- 32.2k
1
vote
1
answer
683
views
Simple properties of the codifferential
The exterior derivative $d$ has many very nice algebraic relations. For example
$d(\alpha\wedge\beta) = (d\alpha)\wedge \beta +
(-1)^k\alpha\wedge(d\beta)$
$f^*(d \alpha)=d f^*(\alpha)$.
$d\circ ...
- 4,286
2
votes
0
answers
124
views
What's known about the matroid induced by the Plücker coordinates of the representation of a matroid?
Let $M$ be a linear matroid with ground set $E$ and independent subsets $\mathcal I$, represented by $\rho: E \rightarrow V$.
This induces a map
$$
\hat\rho: \mathcal I \rightarrow \mathbf P(\Lambda V)...
- 193
5
votes
0
answers
139
views
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 ...
- 737
5
votes
0
answers
238
views
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^...
- 2,159
4
votes
1
answer
359
views
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:
$$\bigwedge V := \bigoplus_{...
- 1,087
6
votes
1
answer
1k
views
What is the role of topology on infinite dimensional exterior algebras?
Wedge products and exterior powers are discussed in W. Greub's book Multilinear algebra as follows.
Definition: Let $E$ be an arbitrary vector space and $p \ge 2$. Then a vector space $\bigwedge^{p}E$ ...
- 1,087
3
votes
1
answer
191
views
action of symmetric group on the second exterior power
Let $e_i \wedge e_j \ (i < j)$ be a basis for the $\mathbb Z$-module $\wedge^2 \Gamma$, where $\Gamma = \mathbb Z^n$.
Clearly $S_n$ acts on the module $\wedge^2 \Gamma$ via
$$\pi(e_i \wedge e_j) ...
- 311
16
votes
0
answers
546
views
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 ...
- 976
9
votes
2
answers
363
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 ...
- 6,571
5
votes
1
answer
687
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 ...
- 337
3
votes
1
answer
656
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
$$
e_{i_1} \...
- 15.4k
2
votes
1
answer
774
views
Exterior derivative independence from coordinate systems
In the book Mathematical Methods of Classical Mechanics by V.I. Arnold, the author introduces (p.189) the concept of exterior derivative as "the principal linear part of the increment" of the function ...
- 131
1
vote
0
answers
63
views
Function for unique volume element
This is an issue that I'm am trying to solve for a fine-tuning measure in particle physics, but it is purely mathematical. Consider three vectors $\{v_1, v_2, v_3\}$ in $\mathbb{R}^3$. I would like a ...
- 11
0
votes
0
answers
178
views
A vector calculus formula
Let me answer my own question, hoping to be forgiven for that.
I asked unsuccessfully that question on Mathematics. Let $A, B$ be vector fields in $\mathbb R^3$.
We have
$$
\text{curl}\bigl((A\cdot \...
- 14.6k
1
vote
0
answers
91
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 ...
- 11
7
votes
1
answer
286
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 $\...
- 6,571
6
votes
1
answer
363
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{...
- 6,571
4
votes
2
answers
1k
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 ...
- 151
4
votes
1
answer
372
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 ...
- 1,152
12
votes
1
answer
443
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 ...
- 32.4k
7
votes
0
answers
136
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 (...
- 6,571
3
votes
3
answers
461
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 ...
- 6,571
2
votes
0
answers
60
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:...
- 17.1k
7
votes
1
answer
279
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
$$
p(y) = ...
- 193
17
votes
1
answer
612
views
An explicit reconstruction of a matrix from its minors
$\newcommand{\End}{\operatorname{End}}$
$\newcommand{\GL}{\operatorname{GL}}$
$\newcommand{\Cof}{\operatorname{cof}}$
Let $V$ be a $d$-dimensional real vector space. ($d \ge 4$). Fix an odd integer $...
- 6,571
2
votes
0
answers
31
views
Expression for the lattice operations on subspaces in Plucker embedding
Suppose that $V$ is a finite dimensional $\mathbb Q$-vector space. To each subspace $S$ of dimension $k$, we can associate the line from the origin of $\Lambda^k(V)$ through the point $s_1\wedge \...
- 570
3
votes
0
answers
293
views
Is the image of the map $A \to \bigwedge^{k}A $ a weakly embedded submanifold?
$\newcommand{\End}{\operatorname{End}}$
$\newcommand{\GL}{\operatorname{GL}}$
Let $V$ be a $d$-dimensional real vector space. ($d \ge 4$). Fix an odd $2 \le k \le d-2$. Define
$H_{>k}=\{ A \in \End(...
- 6,571
4
votes
1
answer
208
views
Is the map $A \to \bigwedge^{k}A $ from matrices above rank $k$ proper?
$\newcommand{\End}{\operatorname{End}}$
Let $V$ be a $d$-dimensional real vector space. ($d \ge 3$). Fix an odd $2 \le k \le d-1$. Define
$H_{>k}=\{ A \in \End(V) \mid \operatorname{rank}(A) > ...
- 6,571
5
votes
1
answer
140
views
If $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$ equals $\bigwedge^k B$ for some complex matrix $B$, does it have a real source?
Let $1<k<d$ be an integer. Let $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$, and suppose that $A=\bigwedge^k B$ for some complex $B \in \text{End}(\mathbb{C}^d)$.
Does there exist $M \in \...
- 6,571
12
votes
1
answer
401
views
Is the image of the map $A \to \bigwedge^k A$ closed over $\mathbb{R}$?
Let $V$ a real vector space of dimension $d$. Let $1<k < d-1$. Consider the map induced by the exterior algebra functor:
$$ \psi:\text{End}(V) \to \text{End}(\bigwedge^kV) \, \, \, \, , \, \, ...
- 6,571
7
votes
1
answer
203
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 $p \ge 1$.
...
- 6,571
8
votes
2
answers
428
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 ...
- 6,571
6
votes
0
answers
253
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 ...
- 6,571
6
votes
1
answer
248
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 ...
- 6,571
7
votes
2
answers
2k
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 $r$-...
- 54.3k
1
vote
0
answers
53
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)...
- 147
3
votes
1
answer
266
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 ...
- 139