(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 …

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 = …

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 com …

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 …

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" $\wedge$ o …

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 …

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 …

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 qu …

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 …

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 thi …

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_ …

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 variab …

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 $\la …

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 basi …

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-sym …