Higher Bruhat orders of types B and C

TL;DR

This paper extends the concept of higher Bruhat orders to types B and C, building on type A theory and geometric interpretations involving symmetric cubillages of cyclic zonotopes.

Contribution

It introduces new versions of higher Bruhat orders for types B and C, expanding the geometric and combinatorial framework beyond type A.

Findings

Defined higher Bruhat orders for types B and C

Connected these orders to symmetric cubillages of cyclic zonotopes

Extended geometric interpretations of Bruhat orders

Abstract

We propose versions of higher Bruhat orders for types $B$ and $C$. This is based on a theory of higher Bruhat orders of type~A and their geometric interpretations (due to Manin--Shekhtman, Voevodskii--Kapranov, and Ziegler), and on our study of the so-called symmetric cubillages of cyclic zonotopes.