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

learn more… | top users | synonyms

10
votes
3answers
496 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. ...
2
votes
1answer
132 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. ...
1
vote
0answers
192 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 \...
5
votes
4answers
1k 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 ...
7
votes
1answer
314 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 \...
4
votes
3answers
3k 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 $\mathbb{R}^n$ ...
6
votes
2answers
348 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 ...
4
votes
2answers
215 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 ...
3
votes
1answer
774 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 ...
0
votes
0answers
526 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
1answer
302 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 ...
8
votes
1answer
1k 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
votes
1answer
825 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 ...
11
votes
0answers
494 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 ...
4
votes
2answers
497 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 ...
6
votes
1answer
774 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) \...