# Tagged Questions

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

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### Bunnity of multilinear maps

Is there a way to compute the following nullity of multilinear maps? As it is different from any nullity I know of, I call it bunnity after myself:-)) If it already has a name, it be nice to know it. ...
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### 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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### 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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### Augmentation ideal is finitely generated if and only if $A$ is finitely generated as a $k$-algebra?

Let $A \subset k[x_1, \dots, x_n]$ be a subalgebra, which is also a graded subspace $A = \oplus_{i \ge 0} A_i$. One can write $A = A_0 \oplus A_{> 0}$ where we have $A_0 = k^0[x_1, \dots, x_n] = k$ ...
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### 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 ...
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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 $\... 1answer 833 views ### Exact sequences of bundles on Grassmannians We're looking for a large set of exact sequences of vector bundles on Grassmannians. Here's the set up:$V$and$Q$are complex vector spaces of dimensions$d$and$r$respectively$(d\geq r)$, and ... 2answers 206 views ### generalizations of Vandermonde matrix to high dimensions Let$x_1,x_2,\cdots,x_n\in\mathbb{R} $or$\mathbb{C}$. By the non-degeneracy of Vandermonde matrix the maps $$f: \mathbb{R}\longrightarrow\mathbb{R}^n,$$ $$x\longmapsto (1,x,x^2,\cdots,x^{n-... 0answers 177 views ### When is a polynomial ring free over a graded subalgebra? Keep the setting of my previous question and let I := k[x_1, \dots, x_n] \cdot A_{>0} be an ideal of the algebra k[x_1, \dots, x_n] generated by the set A_{>0}. It is clear that I is a ... 2answers 677 views ### basics of classification of trilinear forms (when is it non-discrete) Consider tri-linear forms, \{A_{ijk}\} where i=1,..,n_1, j=1,..,n_2, k=1,..n_3, over a field of zero characteristic, up to the equivalence A\to (U_1,U_2,U_3)(A), by three matrices. What is ... 1answer 315 views ### Question about decomposition of exterior product In their paper "New lower bounds for the border rank of matrix multiplication", Landsberg and Ottaviani make use of the fact that$$\tag{$\dagger$} {\textstyle\bigwedge}^p(V\otimes W) \cong \... 3answers 423 views ### Is there a rank for higher degree homogeneous forms analogous to that of quadratic forms? Given a quadratic form$Q(x_1, ..., x_n)$, there is a natural notion of rank defined by looking at the rank of the unique symmetric matrix associated to the quadratic form, i.e. we consider the ... 1answer 287 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$T\...
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 ...