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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, 20G20, 22E15, 22E46, 43A85 show english descriptions Linear algebraic groups over arbitrary fields
Linear algebraic groups over the reals, the complexes, the quaternions
General properties and structure of real Lie groups
Semisimple Lie groups and their representations
Analysis on homogeneous spaces
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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