arXiv subjects

Artem Anisimov

Publications and source records attributed to Artem Anisimov.

3 recordsLinked to original sources

arXiv2012

On existence of double coset varieties

Let $G$ be a complex affine algebraic group and $H, F \subset G$ be closed subgroups. The homogeneous space $G / H$ can be equipped with structure of a smooth quasiprojective variety. The situation is different for double coset varieties $\dcosets{F}{G}{H}$. In this paper we give examples showing that the variety $\dcosets{F}{G}{H}$ does not necessarily exist. We also address the question of existence of $\dcosets{F}{G}{H}$ in the category of constructible spaces and show that under sufficiently general assumptions $\dcosets{F}{G}{H}$ does exist as a constructible space.

math.AG↗

arXiv2011

Spherical subgroups and double coset varieties

Let $\GroupG$ be a connected reductive algebraic group, $\GroupH \subsetneq \GroupG$ a reductive subgroup and $\GroupT \subset \GroupG$ a maximal torus. It is well known that if charactersitic of the ground field is zero, then the homogeneous space $\GroupG/\GroupH$ is a smooth affine variety, but never an affine space. The situation changes when one passes to double coset varieties $\dcosets{\GroupF}{\GroupG}{\GroupH}$. In this paper we consider the case of $\GroupG$ classical and $\GroupH$ connected spherical and prove that either the double coset variety $\dcosets{\GroupT}{\GroupG}{\GroupH}$ is singular, or it is an affine space. We also list all pairs $\GroupH \subset \GroupG$ such that $\dcosets{\GroupT}{\GroupG}{\GroupH}$ is an affine space.

math.AG↗

arXiv2011

On stability of diagonal actions and tensor invariants

For a connected simply connected semisimple algebraic group $G$ we prove existence of invariant tensors in tensor powers of rational $G$-modules and establish relations between existence of such invariant tensors and stability of diagonal actions of $G$ on affine algebraic varieties.

math.RT↗