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Virginia Tech

1. Welch, Amanda Renee. Double Affine Bruhat Order.

Degree: PhD, Mathematics, 2019, Virginia Tech

Given a finite Weyl group W_fin with root system Phi_fin, one can create the affine Weyl group W_aff by taking the semidirect product of the translation group associated to the coroot lattice for Phi_fin, with W_fin. The double affine Weyl semigroup W can be created by using a similar semidirect product where one replaces W_fin with W_aff and the coroot lattice with the Tits cone of W_aff. We classify cocovers and covers of a given element of W with respect to the Bruhat order, specifically when W is associated to a finite root system that is irreducible and simply laced. We show two approaches: one extending the work of Lam and Shimozono, and its strengthening by Milicevic, where cocovers are characterized in the affine case using the quantum Bruhat graph of W_fin, and another, which takes a more geometrical approach by using the length difference set defined by Muthiah and Orr. Advisors/Committee Members: Orr, Daniel D. (committeechair), Mihalcea, Constantin Leonardo (committee member), Shimozono, Mark M. (committee member), Loehr, Nicholas A. (committee member).

Subjects/Keywords: Weyl group; double affine; Bruhat order; coverings; cocoverings

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APA (6th Edition):

Welch, A. R. (2019). Double Affine Bruhat Order. (Doctoral Dissertation). Virginia Tech. Retrieved from

Chicago Manual of Style (16th Edition):

Welch, Amanda Renee. “Double Affine Bruhat Order.” 2019. Doctoral Dissertation, Virginia Tech. Accessed May 25, 2019.

MLA Handbook (7th Edition):

Welch, Amanda Renee. “Double Affine Bruhat Order.” 2019. Web. 25 May 2019.


Welch AR. Double Affine Bruhat Order. [Internet] [Doctoral dissertation]. Virginia Tech; 2019. [cited 2019 May 25]. Available from:

Council of Science Editors:

Welch AR. Double Affine Bruhat Order. [Doctoral Dissertation]. Virginia Tech; 2019. Available from: