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# Admissible Sequences for Twisted Involutions in Weyl Groups

Published:2011-04-25
Printed: Dec 2011
• Ruth Haas,
Department of Mathematics, Smith College, Northampton, MA 01063, USA
• Aloysius G. Helminck,
Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA
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## Abstract

Let $W$ be a Weyl group, $\Sigma$ a set of simple reflections in $W$ related to a basis $\Delta$ for the root system $\Phi$ associated with $W$ and $\theta$ an involution such that $\theta(\Delta) = \Delta$. We show that the set of $\theta$-twisted involutions in $W$, $\mathcal{I}_{\theta} = \{w\in W \mid \theta(w) = w^{-1}\}$ is in one to one correspondence with the set of regular involutions $\mathcal{I}_{\operatorname{Id}}$. The elements of $\mathcal{I}_{\theta}$ are characterized by sequences in $\Sigma$ which induce an ordering called the Richardson-Springer Poset. In particular, for $\Phi$ irreducible, the ascending Richardson-Springer Poset of $\mathcal{I}_{\theta}$, for nontrivial $\theta$ is identical to the descending Richardson-Springer Poset of $\mathcal{I}_{\operatorname{Id}}$.
 MSC Classifications: 20G15 - Linear algebraic groups over arbitrary fields 20G20 - Linear algebraic groups over the reals, the complexes, the quaternions 22E15 - General properties and structure of real Lie groups 22E46 - Semisimple Lie groups and their representations 43A85 - Analysis on homogeneous spaces

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