For a Weyl group G and an automorphism θ of order 2, the set of involutions and θ-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 = Sn, θ 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 θ.
Extended Symmetric Spaces and θ-twisted Involution GraphsMathematics, Data Science, and Statistics Faculty Research
Journal TitleCommunications in Algebra
Citation InformationJ. B. Collins, Ruth Haas, Aloysius G. Helminck, Jessie Lenarz, Kristine Engel Pelatt, Silvia Saccon & Matthew Welz (2020) Extended symmetric spaces and θ-twisted involution graphs, Communications in Algebra, DOI: 10.1080/00927872.2019.1711106