# Tagged Questions

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

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### A bound on the number of bilinear functions needed in order to obtain the minmax

For $n\in\mathbb N$, let $\Delta(n)=\{x\in\mathbb R^n:x_i\geq 0, \sum_ix_i=1\}$ be the set of probability vectors in $\mathbb R^n$. Is there a function $m:\mathbb N\to\mathbb N$ such that for any ...
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### About a particular definition of a Hessian of a function of tuples of matrices

Say I have a function $L : (W_1,..,W_{H+1}) \rightarrow \mathbb{R}$ i.e it takes a tuple of $n$ matrices of different dimensions and computes a number from them. Then I see being defined a ...
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### Exterior Powers of finite abelian group

Let $A$ be a finite $\mathbb{Z}$-module (i.e., a finite abelian group). My question is: for what $n\in \mathbb{Z}^{n\geq 2}$ the map \begin{align} \alpha_{n}:\bigwedge^nA&\to A^{\otimes n}\\ a_1\...
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### Tensor matricizations and their decompositions

Suppose we have a 4-index tensor $t_{ijkl}$ (all 4 dimensions are equal size). We can make a matrix out of it by taking first and last two indexes as new indexes: $t_{ijkl} \rightarrow M_{ij, kl}$. ...
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### 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 ...
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### Is there an algebraic way to characterise the ordinary integral flags?

Fix a vector space $V$ and an integer $1\leq n<\dim V$. If $\mathcal{I}\subseteq\Lambda^\bullet V^*$ is an ideal, I denote by $\mathcal{I}^i:=\mathcal{I}\cap\Lambda^iV^*$ its $i^\textrm{th}$ ...
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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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### Standard rational functions from matrices

In linear algebra we get introduced to standard polynomials that are associated to matrices such as characteristic polynomials and determinants. What are some of the standard rational functions that ...
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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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### 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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### 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 ...
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If $A$ is a finite-dimensional algebra over a field $k$, the usual norm $N_{A/k}: A \to k$ maps $a$ to the determinant of the $k$-linear endomorphism of $A$ given by $x \mapsto ax$. For $A \in \textbf{... 4answers 1k views ### Why there is a relation among the second-order minors of a symmetric$4\times 4$matrix? A$4\times 4$symmetric matrix $$\left( \begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{12} & a_{22} & a_{23} & a_{24} \\ a_{13} & a_{23} & a_{33} & ... 0answers 120 views ### Metric(s) on Grassmann Manifold and Plucker Embedding I'm working on a numerical optimization problem that naturally lives on the Grassmann Manifold Gr_N(\mathbb{C^M}), however the objective function is defined on the alternating algebra given by the ... 1answer 145 views ### Classification of 3-forms in dimension 7 I'm looking for a classification of 3-forms over a real vector space of dimension 7 as for the 3-forms in dimension 6. References on the latter case are R. Bryant On the geometry of almost ... 0answers 289 views ### Bunnity of multilinear maps Is there a way to compute the following nullity of multilinear maps? As it is different from any nullity I know of, I call it bunnity after myself:-)) If it already has a name, it be nice to know it. ... 0answers 44 views ### A tensor equation related to an invariant of a diffeomorphism Let M be an n-dimensional differentiable manifold, f : U \rightarrow V a diffeomorphism between open neighbourhoods U, V of M with f(x)=x for some x \in U, and let R, S, T be ... 0answers 128 views ### Real-rooted polynomials and higher rank matrices For A and B being matrices of the same dimension and B being rank 1, one knows that det(A+tB) is a linear polynomial in t \in \mathbb{R}. Hence by Taylor series it follows that det(A + tB)... 1answer 107 views ### Lifting quadratic forms on the cotangent bundle to higher level forms Backround In several complex variables, an essential tool is Hormander's machinery for solving the \overline{\partial} problem with L^2 estimates. If \alpha is a (p,q+1) form on a domain \... 1answer 261 views ### Can the method of small moments prove a bound on the norms of random trilinear forms? If F(v_1,\dots,v_k) is a k-linear form on \mathbb R^n, the norm I want to consider is$$ ||F|| = \sup \frac{ F(v_1,\dots, v_k)}{\prod_{i=1}^k \left|\left|v_i\right|\right|} $$where the vector ... 1answer 68 views ### Image of skew-symmetric bilinear map which is never zero on linearly independent vectors I hope this is not too elementary. Let B: V\ \times V \to W be a skew-symmetric bilinear map where V, W are finite dimensional real vector spaces. Assume that B (u, v) is never zero for ... 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 ... 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\... 1answer 351 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 \... 1answer 390 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. ... 1answer 189 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 ... 3answers 246 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 ... 2answers 575 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 lowered,... 3answers 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 \phi((a_{... 1answer 168 views ### Eigenvalue problem with quadratic constraints \circ Consider the following eigenvalue problem :$$Ax=\lambda x \hspace{0.5cm} (1)$$where matrice A \in \mathbb{R}_{n \times n} is a positive semi-definite with eigenvectors x = (x_{1},x_{2},..... 3answers 762 views ### A basis of the symmetric power consisting of powers I have asked this question on math.se, but did not get an answer - I was quite surprised because I thought that lots of people must have though about this before: Let V be a complex vector space ... 1answer 125 views ### Number of linear independent equations Is there any general rule to find the number of linearly independent equations such that$$L_i(T_{\mu\nu},\partial_\eta T_{\mu\nu},\partial_\omega\partial_\eta T_{\mu\nu},...)=0$$where L_i is a ... 0answers 155 views ### When a hyperplane of symmetric forms is determined by a quadric hypersurface? Let L be a 2D real vector space, L^* its dual, and \{V,\omega\} the symplectic space with V=L\oplus L^* and \omega unambiguously defined by \omega(l,\lambda):=\lambda(l), for all l\in L ... 2answers 304 views ### An expression with an alternating trilinear form, written in terms of the determinant and a symmetric bilinear form I am trying to understand a line from MacLachlan/Reid's The Arithmetic of Hyperbolic 3-Manifolds (it's in 3.4 if you have the book), that seems it should be elementary but I can't seem to find where ... 4answers 1k views ### Generalization of the Polarisation Formula for Symmetric Bilinear Forms to Symmetric multilinear Forms Hello, Given a symmetric bilinear form f:V\times V \to K , where V is a vectorspace and K is an appropriate field, define the quadratic form q:V \to K as q(v):= f(v,v). The Polarisation ... 3answers 740 views ### A NICE necessary and sufficient condition on positive semi-definiteness of a matrix with a special structure! Let$$ 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 & \... 0answers 586 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 ... 0answers 153 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 ... 1answer 241 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 ... 1answer 166 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 ... 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 ... 0answers 155 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{...
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 ...