arXiv subjects

Aloysius G. Helminck

Publications and source records attributed to Aloysius G. Helminck.

4 recordsLinked to original sources

arXiv2015

Isomorphy Classes of $k$-Involutions of $\text{SO}(n, k,β)$, $n > 2$

A first characterization of the isomorphism classes of $k$-involutions for any reductive algebraic group defined over a perfect field was given in \cite{Helm2000} using $3$ invariants. In \cite{HWD04,Helm-Wu2002} a full classification of all $k$-involutions on $\text{SL}(n,k)$ for $k$ algebraically closed, the real numbers, the $p$-adic numbers or a finite field was provided. In this paper, we find analogous results to develop a detailed characterization of the $k$-involutions of $\text{SO}(n,k,β)$, where $β$ is any non-degenerate symmetric bilinear form and $k$ is any field not of characteristic $2$. We use these results to classify the isomorphy classes of $k$-involutions of $\text{SO}(n, k,β)$ for some bilinear forms and some fields $k$.

math.RT↗

arXiv2014

Isomorphy Classes of Involutions of $\text{SP}(2n, k)$, $n>2$

A first characterization of the isomorphism classes of $k$-involutions for any reductive algebraic groups defined over a perfect field was given by Helminck in 2000 using $3$ invariants. In 2004, Helminck, Wu, and Dometrius gave a full classification of all involutions on $\text{SL}(n,k)$ for $k$ algebraically closed, the real numbers, the $p$-adic numbers or a finite field was provided. In this paper, we build on these results to develop a detailed characterization of the involutions of $\text{SP}(2n, k)$. We use these results to classify the isomorphy classes of involutions of $\text{SP}(2n, k)$ where $k$ is any field not of characteristic 2.

math.RT↗

arXiv2012

On the Structure of Involutions and Symmetric Spaces of Dihedral Groups

We initiate the study of analogues of symmetric spaces for the family of finite dihedral groups. In particular, we investigate the structure of the automorphism group, characterize the involutions of the automorphism group, and determine the fixed-group and symmetric space of each automorphism.

math.GR↗

arXiv2009

Real double coset spaces and their invariants

Let G be a real form of a complex reductive group. Suppose that we are given involutions σand θof G. Let H=G^σdenote the fixed group of σand let K=G^θdenote the fixed group of θ. We are interested in calculating the double coset space H\backslash G/K. We use moment map and invariant theoretic techniques to calculate the double cosets, especially the ones that are closed. One salient point of our results is a stratification of a quotient of a compact torus over which the closed double cosets fiber as a collection of trivial bundles.

math.RT↗