Stratifying the space of barcodes using Coxeter complexes

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