Admissible Sequences for Twisted Involutions in Weyl Groups

Haas, Ruth ; Helminck, Aloysius G.

Canadian mathematical bulletin, Tome 54 (2011) no. 4, pp. 663-675

Résumé

Let $W$ be a Weyl group, $\sum $ 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 $W$ , ${{\mathcal{J}}_{\theta }}\,=\,\{w\,\in \,W\,|\,\theta (w)\,=\,{{w}^{-1}}\}$ is in one to one correspondence with the set of regular involutions ${{\mathcal{J}}_{\text{ID}}}$ . The elements of ${{\mathcal{J}}_{\theta }}$ are characterized by sequences in $\sum $ which induce an ordering called the Richardson–Springer Poset. In particular, for $\Phi $ irreducible, the ascending Richardson–Springer Poset of ${{\mathcal{J}}_{\theta }}$ , for nontrivial $\theta $ is identical to the descending Richardson–Springer Poset of ${{\mathcal{J}}_{\text{ID}}}$ .

Détail

DOI : 10.4153/CMB-2011-075-1

Mots-clés : 20G15, 20G20, 22E15, 22E46, 43A85

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