# Questions tagged [exterior-algebra]

The exterior-algebra tag has no usage guidance.

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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 ...

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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 ...

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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 (...

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236 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 ...

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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:...

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### Reference request: 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) = ...

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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 $...

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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 \...

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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(...

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168 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) > ...

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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 \...

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357 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) \, \, \, \, , \, \, ...

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### Is a Sobolev map with smooth minors smooth on the whole domain?

This question is related (but not identical) to this question.
Let $d>2$ be an integer and let $2 \le k \le d-1$ be a fixed integer. Suppose that at least one of $k,d$ is not even. Let $\Omega$ be ...

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337 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 ...

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199 views

### Is a Sobolev map with smooth minors smooth?

$\newcommand{\Cof}{\text{cof}}$
Let $d>2$ be an integer and let $2 \le k \le d-1$ be a fixed integer. Suppose that $k,d$ are both even. Let $\Omega$ be an open subset of $\mathbb{R}^d$, and let $f \...

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143 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 ...

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573 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 $r$-...

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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)...

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178 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 ...

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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 ...

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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?

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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 ...

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540 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 \...

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### 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 ...

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178 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 $\...

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203 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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: ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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122 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)\...

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### 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}, ...

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### 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 ...

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### 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 $C(V)$ ...

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### Equivalence of exterior forms

Let us start with the following definition.
Let $1\leqslant k\leqslant n$ and let $\omega_1,\omega_2\in\Lambda^k(\mathbb{R}^n)$. We say that $\omega_1$, $\omega_2$ are equivalent, if there exists $T\...

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### Definition of modules over $C_\infty$-algebras (“commutative $A_\infty$-algebras”)

Let $\Lambda$ be a finite-dimensional associative algebra. We can think of this as an $A_\infty$-algebra with vanishing $m_i$ for $i \ne 2$ and consider $A_\infty$-modules $M$ over it. A list of ...

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### Are SL(n) Invariants of this wedge product isomorphic to a symmetric product?

In the course of investigating a conjecture about a "strange duality" for sections of line bundles on various models of moduli of sheaves on $\mathbb P^2$, another student and I reduced one special ...

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3k views

### Distributing the Hodge map over the wedge product

Let $(V,\langle,\rangle)$ be a finite dimensional inner product space, $V^{\wedge}$ it exterior algebra, and $\ast$ the Hodge star arising from $\langle,\rangle$. Does there exist any formula to "...

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### Exterior powers and singular values on Hilbert spaces

I am currently writing an article relating to multiplicative ergodic theorems for cocycles of bounded operators acting on a Hilbert space, and in parts of the argument it is necessary for me to refer ...

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### Is the ideal membership problem solvable for differential ideals? Is there a good notion of a Gröbner basis?

Let $K$ be a field of characteristic zero. Let $\Omega = K[x_1, \dots, x_n, dx_1, \dots, dx_n]$ be the differential ring of algebraic differential forms over $K[X_1, \dots, X_n]$.
Is there an ...

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### Analogy between the exterior power and the power set

The symmetric algebra of an object exists in every cocomplete $\otimes$-category. For the category of sets $\mathrm{Sym}(X)$ is the set of multi-subsets of $X$.
The usual definition of the exterior ...

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338 views

### Domain of the wedge product in Little Spivak

Hello! in Little Spivak, p. 79, we find this:
We will therefore define a new
product, the wedge product
$\omega\wedge\eta\in
> \Lambda^{k+\ell}(V)$ by $$
> \omega\wedge\eta =
> \frac{(...

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180 views

### Non-negative Quadratic forms with Exterior Forms

Hello All,
I apologize if the following question is too elementary. Any suggestion is greatly appreciated. Thank you.
Let $n\geqslant 4$, $X$ be an $n$-dimensional inner product space over $\mathbb{...

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993 views

### Representation theory for the exterior algebra

Hello everyone.
Let $V$ and $W$ be finite dimensional vector spaces over some field $K$. Consider $\rho:Sym(V)\to End(W)$ a homomorphism of algebras with unit (i.e., a representation of $Sym(V)$ on $...

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**1**answer

380 views

### Inequalities Involving Wedge Product (Reference Request)

Hello,
I'm looking for references on various inequalities involving the wedge product and exterior forms. The only references I could hunt down are references on Hadamard-Schwarz Inequality. The ...

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208 views

### Decomposability of exterior two-forms

Hello,
The following question appears as a step in my proof. It seems easy but somehow I have not been able to prove this. I could solve few special cases though. Any help in this context is welcome....