Abstract:
The higher Bruhat orders are partial orders introduced by Manin and Schechtman in 1989. The first higher Bruhat order is isomorphic to the well-known weak Bruhat order on the symmetric group on \$1, 2, \ldots, n\$. The second higher Bruhat order is a partial order on commutation classes of reduced expressions for the longest permutation. In the general case, these partial orders can equivalently be described using hyperplane arrangements, oriented matroids, and zonotopal tilings. The higher Bruhat orders have applications to areas such as Bott-Samelson varieties and Steenrod algebras. Generalizing the higher Bruhat orders to other settings has been an area of recent interest. Following work of Ben Elias and Daniel Hothem, we address the problem of extending the notion of higher Bruhat orders to all intervals in the affine symmetric groups. We prove our construction holds for all symmetric group intervals. We will discuss a conjectural generalization for the affine case along with current results pertaining to commutativity classes of reduced words in the affine setting. Finally, we explore connections to machine learning (ML) techniques in mathematical research and discuss our construction of the Algebraic Combinatorics Dataset Repository for the ML community as well as two lessons learned about using ML for mathematical discovery.
This talk is based on joint work with Sara Billey, Kevin Liu and on joint work with Sara Billey, Davis Brown, Jesse He, Helen Jenne, Henry Kvinge, and Mark Raugas.
Note: This talk begins with a pre-seminar (aimed at graduate students) at 3:30–4:00. The main talk starts at 4:10.
Join Zoom Meeting: https://washington.zoom.us/j/
Meeting ID: 915 4733 5974