Abstract
Classification of Coxeter groups with finitely many elements of $\protect \mathbf{a}$-value 2
Highlights
This paper concerns Lusztig’s a-function on Coxeter groups
The “a = n” result of Shi from Proposition 3.13, that is, the result that a(w) = n(w) for every fully commutative element w in a Weyl or affine Weyl group, is a very powerful tool used in this paper
We extended Shi’s result to star reducible Coxeter groups in Proposition 3.15, and we believe it would be very interesting to know whether the result can be further generalized to arbitrary Coxeter groups
Summary
This paper concerns Lusztig’s a-function on Coxeter groups. The a-function was first defined for finite Weyl groups via their Hecke algebras by Lusztig in [20]; subsequently, the definition was extended to affine Weyl groups in [21] and to arbitrary Coxeter groups in [22]. A key fact we shall use is that each element of a-value 2 in a Coxeter group must be fully commutative in the sense of Stembridge (see Section 3.2). For each of these subgraphs, we will construct an infinite family of fully commutative elements that we call “witnesses” and verify that they have a-value 2. The problems include generalizing the aforementioned result of Shi to arbitrary Coxeter groups, enumerating elements of a-value 2 in a(2)-finite Coxeter groups, and the classification of a(3)-finite Coxeter groups
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
