Searcharxiv⌕ Search

arXiv subjects

Nathaniel Schwartz

Publications and source records attributed to Nathaniel Schwartz.

5 recordsLinked to original sources
arXiv2020

On involutions of type $\mathrm{O}(\mathrm{q},k)$ over a field of characteristic two

In this article we study the involutions of $\mathrm{O}(V,\mathrm{q})$, an orthogonal group for a vector space $V$ with quadratic form $\mathrm{q}$ over a field of characteristic 2. The classification proceeds by discussing conjugacy classes of involutions arising as a product of transvections, involutions with respect to a hyperbolic space, and involutions acting nontrivially in the radical of $V$. We achieve a complete classification of the conjugacy classes of involutions when the quadratic space $(V,\mathrm{q})$ is non-defective, and conclude with a discussion of the defective case.

math.GR↗
arXiv2015

$k$-involutions of $\text{SL}(n,k)$ over Fields of Characteristic 2

Symmetric $k$-varieties generalize Riemannian sym\-me\-tric spaces to reductive groups defined over arbitrary fields. For most perfect fields, it is known that symmetric $k$-varieties are in one-to-one correspondence with isomorphy classes of $k$-involutions. Therefore, it is useful to have representatives of each isomorphy class in order to describe the $k$-varieties. Here we give matrix representatives for each isomorphy class of $k$-involutions of $\text{SL}(n,k)$ in the case that $k$ is any field of characteristic 2; we also describe fixed point groups of each type of involution.

math.GR↗
arXiv2013

On involutions and generalized symmetric spaces of dicyclic groups

Let $G=\Dc_{n}$ be the dicyclic group of order $4n$. Let $φ$ be an automorphism of $G$ of order $k$. We describe $φ$ and the generalized symmetric space $Q$ of $G$ associated with $φ$. When $φ$ is an involution, we describe its fixed point group $H=G^φ$ along with the $H$-orbits and $G$-orbits of $Q$ corresponding to the action of $φ$-twisted conjugation.

math.GR↗