In this paper, we introduce the signed barcode, a new visual representation of the global structure of the rank invariant of a multi-parameter persistence module or, more generally, of a poset representation. Like its unsigned counterpart in one-parameter persistence, the signed barcode decomposes the rank invariant as a $\Z$-linear combination of rank invariants of indicator modules supported on segments in the poset. We develop the theory behind these decompositions, both for the usual rank invariant and for its generalizations, showing under what conditions they exist and are unique. We also show that, like its unsigned counterpart, the signed barcode reflects in part the algebraic structure of the module: specifically, it derives from the terms in the minimal rank-exact resolution of the module, i.e., its minimal projective resolution relative to the class of short exact sequences on which the rank invariant is additive. To complete the picture, we show some experimental results that illustrate the contribution of the signed barcode in the exploration of multi-parameter persistence modules.

翻译：本文引入符号条形码，这是一种用于表示多参数持久性模（或更一般地，偏序集表示）的秩不变量的全局结构的新可视化方法。与单参数持久性中的无符号条形码类似，符号条形码将秩不变量分解为偏序集中支撑于线段上的指示模的秩不变量的$\Z$线性组合。我们建立了这些分解背后的理论体系，涵盖常规秩不变量及其推广形式，阐明了它们存在且唯一的条件。我们还证明，与无符号条形码类似，符号条形码部分反映了模的代数结构：具体而言，它源于模的极小秩正合分解中的项，即相对于秩不变量可加性成立的短正合序列类而言的极小投射分解。最后，我们展示了一些实验结果，用以说明符号条形码在多参数持久性模探索中的贡献。