Stratifying the space of barcodes using Coxeter complexes

Embeddings of the space of barcodes in Euclidean spaces are unstable due to the permutation of the bars of a barcode. We use tools from geometric group theory to produce a stratification of the space ℬ_n of barcodes with n bars that takes into account these permutations. This gives insights in the combinatorial structure of ℬ_n . The top-dimensional strata are indexed by permutations associated to barcodes as defined by Kanari, Garin and Hess. More generally, the strata correspond to marked double cosets of parabolic subgroups of the symmetric group Sym_ n . This subdivides ℬ_n into regions that consist of barcodes with the same averages and standard deviations of birth and death times and the same permutation type. We obtain coordinates that form a new invariant of barcodes, extending the one of Kanari–Garin–Hess. This description also gives rise to metrics on ℬ_n that coincide with modified versions of the bottleneck and Wasserstein metrics.