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Algorithms for twisted involutions in Weyl groups. (English) Zbl 1254.20033

Summary: Let \((W,\Sigma)\) be a finite Coxeter system, and \(\theta\) an involution such that \(\theta(\Delta)=\Delta\), where \(\Delta\) is a basis for the root system \(\Phi\) associated with \(W\), and \(\mathcal I_\theta=\{w\in W\mid\theta(w)=w^{-1}\}\) the set of \(\theta\)-twisted involutions in \(W\). The elements of \(\mathcal I_\theta\) can be characterized by sequences in \(\Sigma\) which induce an ordering called the Richardson-Spinger Bruhat poset. The main algorithm of this paper computes this poset. Algorithms for finding conjugacy classes, the closure of an element and special cases are also given. A basic analysis of the complexity of the main algorithm and its variations is discussed, as well experience with implementation.

MSC:

20F55 Reflection and Coxeter groups (group-theoretic aspects)
68W30 Symbolic computation and algebraic computation
20-04 Software, source code, etc. for problems pertaining to group theory
06A06 Partial orders, general
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