Extended symmetric spaces and \(\theta\)-twisted involution graphs. (English) Zbl 1441.05242
Summary: For a Weyl group \(G\) and an automorphism \(\theta\) of order 2, the set of involutions and \(\theta\)-twisted involutions can be generated by considering actions by basis elements, creating a poset structure on the elements. Haas and Helminck showed that there is a relationship between these sets and their Bruhat posets. We extend that result by considering other bases and automorphisms. We show for \(G=S_n,\theta\) an involution, and any basis consisting of transpositions, the extended symmetric space is generated by a similar algorithm. Moreover, there is an isomorphism of the poset graphs for certain bases and \(\theta\).
MSC:
| 05E16 | Combinatorial aspects of groups and algebras |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
| 06A07 | Combinatorics of partially ordered sets |
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