Extended symmetric spaces and θ-twisted involution graphs

Abstract

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 = S_n, 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.

Keywords:

• Algebraic combinatorics
• symmetric spaces

2010 MATHEMATICS SUBJECT CLASSIFCATION:

• 05E15
• 05E25
• 20F55
• 05C25

Acknowledgements

This work was initiated as part of the program Collaborative Research: AIM & ICERM Research Experiences for Undergraduate Faculty (REUF) NSF award no. DMS-1620073. We thank the organizers of the program, the AIM staff, and NSF for their support.

Additional information

Funding

This article was funded by National Science Foundation.