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

**11**

votes

**0**answers

228 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. ...

**8**

votes

**0**answers

458 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 ...

**6**

votes

**0**answers

151 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 ...

**6**

votes

**0**answers

90 views

### 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 $A_0 = k^0[x_1, \dots, x_n] = k$ ...

**5**

votes

**0**answers

141 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 ...

**5**

votes

**0**answers

352 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 ...

**2**

votes

**0**answers

42 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
...

**2**

votes

**0**answers

107 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 + ...

**1**

vote

**0**answers

98 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 ...

**1**

vote

**0**answers

436 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 ...

**1**

vote

**0**answers

153 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 ...

**1**

vote

**0**answers

191 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 ...

**0**

votes

**0**answers

133 views

### Symmetric kernel of tensor product

Let $V,W$ be two vector spaces, and let $L_i:V\rightarrow W$, $i=1,\ldots,n$ be $n$ linear maps with disjoint kernels $K_i$ of dimension $1$.
Consider the tensor product of these maps $L_1\otimes ...

**0**

votes

**0**answers

51 views

### What can we say about variational energies here?

Let $V_{ij}^{lk}$ be any $nm \times nm$ real symmetric matrix, $\forall i,j,k,l$
\begin{equation}
V_{ij}^{kl}=V_{ji}^{lk}
\end{equation}
(So for the indices we have $1 \leq k,l \leq m$ and $1 \leq i,j ...

**0**

votes

**0**answers

142 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 $l\in L$ ...

**0**

votes

**0**answers

480 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 ...