On Enumerating and Generalizing Higher Bruhat Orders with Connections to Machine Learning

Abstract

The higher Bruhat orders B(n, k) were introduced by Manin–Schechtman to study discriminantal hyperplane arrangements and subsequently studied by Ziegler, who connected B(n, k) to oriented matroids. In this thesis, we study three topics: the enumeration of B(n, k), a generalization of B(n, k) to identity intervals in the affine symmetric group, and connectionsto machine learning techniques in mathematical research. Our work on the enumeration of B(n, k) improves upon Balko’s asymptotic lower and upper bounds on |B(n, k)| by a factor exponential in k, gives a proof of Ziegler’s formula for |B(n, n - 3)|, and relates B(n, n - 4) to totally symmetric plane partitions. Our work on the generalization of B(n, k) to identity intervals in the affine symmetric group answers Elias’s conjecture on the acyclicity of directed braid graphs on commutation classes of reduced expressions of an affine permutation in the affirmative. This generalization was inspired by the work of Elias-Hothem in generalizing B(n, k) to identity intervals in the ordinary symmetric group. Our work on exploring machine learning (ML) techniques in mathematical research involves releasing the Algebraic Combinatorics Dataset Repository to the ML community and led to two lessons learned about using ML for mathematical discovery.