Tensors, multivectors, wedge products, multilinear maps, exterior (Grassmann) algebras.

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104 views

### Composition of multilinear forms agreeing on a subset of points

Let $n$ be a perfect square.
Consider multiaffine polynomials $p(x_1,x_2,\dots,x_n),q(x_1,x_2,\dots,x_n),r(x_1,x_2,\dots,x_n),$$\{s_j(x_1,x_2,\dots,x_n)\}_{j=1}^{n}$$\in\mathbb R[x_1,x_2,\dots,x_n]$.
...

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**1**answer

259 views

### 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 ...

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votes

**1**answer

282 views

### What is the total polarization of the determinant?

Let $A\in\mathfrak{gl}(\mathbb{R},n)$ be an endomorphism, and think up to conformal factors (in particular, $\Lambda^n\mathbb{R}^n$ will be the same as $\mathbb{R}$). By the total polarization ...

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vote

**1**answer

370 views

### A multilinear question and its smooth version

Let $E$ and $F$ be two finite dimensional vector spaces. For every $k\in \mathbb{N}$, $E^{k}$ has a natural vector space structure and is isomorphic to $E\otimes \mathbb{R}^{k}$, in a natural way.
...

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**1**answer

119 views

### 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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**3**answers

202 views

### Axiomatizing orientation in the complex plane

Lately I've begun to suspect that a certain ternary relation might play a role in $\bf{C}$ analogous to the role played by the binary relation $>$ in $\bf{R}$, namely, the relation that the ...

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**2**answers

286 views

### Relationship between curvature tensor, algebraic Bianchi identity and sectional curvature

I am currently trying to understand the algebraic Bianchi identity, and I am clearly missing some purely algebraic fact.
Let $M$ be a Riemannian manifold, $R$ its curvature tensor (with index ...

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**3**answers

2k views

### Multiplicative functions $\phi : M_n(F) \longrightarrow F$ with $\phi(I) = 1$

Let $F$ be an infinite field and let $f \in F[x_{11},x_{12},...,x_{nn}]$ be an arbitrary polynomial in $n^2$ variables. Consider the function $\phi : M_n(F)\longrightarrow F$ defined by ...

**1**

vote

**1**answer

132 views

### Eigenvalue problem with quadratic constraints

$\circ$ Consider the following eigenvalue problem : $$Ax=\lambda x \hspace{0.5cm} (1)$$
where matrice $A \in \mathbb{R}_{n \times n}$ is a positive semi-definite with eigenvectors $x = ...

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vote

**1**answer

107 views

### Number of linear independent equations

Is there any general rule to find the number of linearly independent equations such that
$$L_i(T_{\mu\nu},\partial_\eta T_{\mu\nu},\partial_\omega\partial_\eta T_{\mu\nu},...)=0$$
where $L_i$ is a ...

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**0**answers

123 views

### When a hyperplane of symmetric forms is determined by a quadric hypersurface?

Let $L$ be a 2D real vector space, $L^*$ its dual, and $\{V,\omega\}$ the symplectic space with $V=L\oplus L^*$ and $\omega$ unambiguously defined by $\omega(l,\lambda):=\lambda(l)$, for all $l\in L$ ...

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**3**answers

353 views

### A NICE necessary and sufficient condition on positive semi-definiteness of a matrix with a special structure!

Let
$$
A =
\begin{pmatrix}
\sum_{j\ne 1}a_{1j} & -a_{12} & \cdots & -a_{1n}\\
-a_{21} & \sum_{j\ne 2}a_{2j} & \cdots & -a_{2n}\\
\vdots & \vdots & \ddots & ...

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**0**answers

289 views

### Tensors as multilinear maps

I am aware that many books on differential geometry define tensors as multilinear maps. Namely
$$
V\otimes W := L_2(V^* \times W^*,\Bbb F)
$$
I am also aware that this space is isomorphic to the ...

**5**

votes

**0**answers

121 views

### Is there an analogue of spin/oscillator representation for the general linear Lie algebra?

(Work over complex numbers)
Let $V$ be an orthogonal space. Let $Pin(V)$ be the double cover of the orthogonal group $O(V)$. Then $Pin(V)$ has a basic spin representation which we can think of as the ...

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**1**answer

190 views

### para-complex structure [closed]

Definition :Let $V$ be a finite dimensional real vector space. A para-complex structure
on $V$ is an endomorphism $K$ :
$V \to V$ such that:
$K$ is an involution, that is $K^2 = Id_V$ ;
The ...

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votes

**2**answers

254 views

### An expression with an alternating trilinear form, written in terms of the determinant and a symmetric bilinear form

I am trying to understand a line from MacLachlan/Reid's The Arithmetic of Hyperbolic 3-Manifolds (it's in 3.4 if you have the book), that seems it should be elementary but I can't seem to find where ...

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votes

**3**answers

621 views

### A basis of the symmetric power consisting of powers

I have asked this question on math.se, but did not get an answer - I was quite surprised because I thought that lots of people must have though about this before:
Let $V$ be a complex vector space ...

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votes

**1**answer

147 views

### Is there a wedge which operates on multiple vector spaces?

Let's say I have two vector spaces $V,W$ , and we have the graded algebras $\Lambda(V),\Lambda(W)$, each with an operation $\wedge$. I'd like to know if there are "many" $\wedge$ operators, or if ...

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**2**answers

387 views

### basics of classification of trilinear forms (when is it non-discrete)

Consider tri-linear forms, $\{A_{ijk}\}$ where $i=1,..,n_1$, $j=1,..,n_2$, $k=1,..n_3$, over a field of zero characteristic, up to the equivalence $A\to (U_1,U_2,U_3)(A)$, by three matrices.
What is ...

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**0**answers

152 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 ...

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**0**answers

296 views

### Tensor rank of anti-symmetric tensor

Let $V$ be a vector space of dimension $n$. Let us consider $V^{\otimes n}=V\otimes V \ldots \otimes V$. This vector space contains one dimentional vector space $\wedge^n V$. My question is does it ...

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**1**answer

243 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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votes

**1**answer

181 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 ...

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**1**answer

127 views

### Nonlinear matrix equation 2

Solve the following nonlinear equations for $v$ and $w$
$Avv^TAw+Bvv^TBw=\lambda_1v+\lambda_2w$
$Aww^TAv+Bww^TBv=\lambda_1w+\lambda_2v$
$v^Tw=w^Tv=0$
$v^Tv=w^Tw=1$
where $\lambda_1, \lambda_2, ...

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votes

**1**answer

553 views

### Nonlinear matrix equation

Solve the following nonlinear equations for $v$ and $w$
$Avv^TAw=\lambda_1v+\lambda_2w$
$Aww^TAv=\lambda_1w+\lambda_2v$
$v^Tw=w^Tv=0$
$v^Tv=w^Tw=1$
where $\lambda_1, \lambda_2, \lambda_3$ are ...

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votes

**3**answers

451 views

### Example of a form linear in infinitely many variables ?

We all know plenty of examples of multilinear forms in finitely many variables (e.g. determinants). However, I am missing an interesting example of a form in infinitely many variables, linear in each. ...

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**1**answer

120 views

### How to characterize the dual of an isotropic hyperplane?

Hi there! I have a very simple question, which requires an expert in multilinear algebra.
$V$ is an $n$-dimensional vector space, and $\omega\in V^\ast\wedge V^\ast$ is a skew-symmetric form on it. ...

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**0**answers

189 views

### Singular quadratic space

Let $(V,b)$ a symmetric bilinear space. An old theorem of Witt says that if $(V,b)$ is regular, then given a subspace $W$ of $V$ and an isometry $\sigma: W \to V$, there exists an isometry $\Sigma: V ...

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**4**answers

607 views

### Generalization of the Polarisation Formula for Symmetric Bilinear Forms to Symmetric multilinear Forms

Hello,
Given a symmetric bilinear form $f:V\times V \to K$ , where $V$ is a vectorspace and $K$ is an appropriate field, define the quadratic form $q:V \to K$ as $q(v):= f(v,v)$.
The Polarisation ...

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**1**answer

277 views

### Question about decomposition of exterior product

In their paper "New lower bounds for the border rank of matrix multiplication", Landsberg and Ottaviani make use of the fact that
$$\tag{$\dagger$} {\textstyle\bigwedge}^p(V\otimes W) \cong ...

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**3**answers

2k views

### n-dimensional “cross product” reference request

I have written a paper which involves a "cross product" in $\mathbb{R}^n$ and I would like to have a reference to point to.
Let ${\bf e_1}, \dots, {\bf e_n}$ be the standard basis for $\mathbb{R}^n$ ...

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votes

**2**answers

314 views

### Alternating multilinear invariants of GL(n) on End (k^n)

Introduction. Let $k$ be a field of characteristic $0$, and let $n\in\mathbb N$. Let $V=k^n$. The group $\mathrm{GL}_n\left(k\right)=\mathrm{GL} V$ acts on $\mathrm{End} V$ by conjugation, and thus ...

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votes

**2**answers

202 views

### Reference for Tensors on graded spaces needed

Is there a good introduction to
1.) Tensor (co)algebras on graded vector spaces ?
2.) Tensor (co)algebras on graded modules ?
In the research field of $L_\infty$-algebras there is some stuff, but ...

**2**

votes

**1**answer

560 views

### Spectral sequence of symmetric or exterior algebras?

This question is inspired by Hartshorne's exercise II.5.7 (c-d): the problem reads:
Let $0\rightarrow \mathcal{F}'\rightarrow\mathcal{F}\rightarrow\mathcal{F}''\rightarrow0$ be a short exact sequence ...

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votes

**0**answers

423 views

### modified bessel fucntion of the third kind

Hi I'm doing a computation where the modified bessel function of the third kind is the main source of computational strain, we are using a 10,000 dimension's for our distribution, is there any easier ...

**-2**

votes

**1**answer

294 views

### A Matrix equation

Let $A$ and $B$ be two $n \times n$ full-rank matrices.
Let $XAY = B$ be the given equation where $X$ and $Y$ are unknown $n \times n$ matrices. We know that $Vec(B) = (Y^{T} \otimes X)Vec(A)$. Under ...

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votes

**1**answer

896 views

### Multilinear generalization of Cauchy-Schwarz inequality

Let $V$ be a real vector space, and let $(\cdot,\cdot;\cdot,\cdot) : V^4 \to \mathbb{R}$ be a multilinear form with the following properties:
$(x,y;z,w) = (y,x;z,w) = (x,y;w,z)$ (symmetry in the ...

**8**

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**1**answer

705 views

### Exact sequences of bundles on Grassmannians

We're looking for a large set of exact sequences of vector bundles on Grassmannians. Here's the set up:
$V$ and $Q$ are complex vector spaces of dimensions $d$ and $r$ respectively $(d\geq r)$, and ...

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**0**answers

279 views

### Derivative of a homogeneous polynomial map

Let $K$ be a field and $V$ be a linear space over $K$. A map $p\colon V \to K$ is homogeneous polynomial of degree $n$ if there exist the symmetric $n$-linear form $f\colon V^{\times n}\to K$ such ...

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**0**answers

410 views

### On a tentative generalization of the Schmidt decomposition

Background
I am a PhD student in Physics and I am currently developing quite refined computer codes that allow to simulate many-body quantum systems living on a lattice. The difficulty resides in ...

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votes

**2**answers

419 views

### Tensor and Hom objects for finite flat group schemes

Is the category of finite flat group schemes equipped with "tensor products" and Hom-objects, encoding bilinear maps? I'm aware that the Cartier dual is $Hom(\mathbb{G}, \mathbb{G}_m)$, and want to ...

**5**

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**1**answer

635 views

### Co-ends as a trace operation on profunctors

The n-lab site on profunctors (http://ncatlab.org/nlab/show/profunctor) describes profunctor composition as using a co-end to "trace out" the connecting variable:
$F\circ G := \int^{d\in D} F(-, d) ...