Extending the AB-Index for Braid and Reflection Arrangements

Explore the extended ab-index of hyperplane arrangements in this mathematical lecture that delves into advanced combinatorial and algebraic structures. Learn about the connections between the extended ab-index and the cd-index, as well as its relationship to the ab-index of face posets. Discover how this mathematical tool evaluates to the coarse flag Hilbert-Poincaré series and the Chow polynomial, providing insights into geometric and algebraic properties of arrangements. Examine a companion statistic for descents on permutations that generates the extended ab-index, offering a new perspective on classical combinatorial objects. Investigate the challenging open problem of finding analogous companion statistics for reflection arrangements that could generalize left-to-right maxima on permutations to finite reflection groups. The presentation draws from collaborative research with Lorenzo Vecchi, Galen Dorpalen-Barry, Elena Hoster, and Josh Maglione, representing cutting-edge developments in algebraic combinatorics and the theory of hyperplane arrangements.