If $V$ is given to be a vector space that is not finite-dimensional, it doesn't seem to be possible to exhibit an explicit non-zero linear functional on $V$ without further informa …

Let $A$ be an $m\times n$-matrix and $B$ an $n \times m$-matrix over the same field. Consider the matrices $C=AB$ and $D=BA$. It is probably well known (and not difficult to show) …

Background I would like to know if there is some slick machinery to solve the following representation-theoretic problem. Let $\left(V,\langle-,-\rangle\right)$ be a finite-dimen …

Hi there, Please tell me if I should divide these into individual questions next time. Short intro: Spherical Harmonics are a nice collection of functions. They are orthogonal a …

Consider a connected symplectic manifold $(M, \omega)$ of dimension $m=2n$. A few preliminary reminders (mostly to fix the notation): A vector field $X$ is symplectic if its flow p …

There is a nice tool in representation theory, the Howe duality, which as I know works for certain pairs of classical Lie algebras (the reference to the complete list of Howe dual …