Tensor decompositions on simplicial complexes with invariance

Tensors are ubiquitous in mathematics and the sciences, as they allow to store information in a concise way. Decompositions of tensors may give insights into their structure and complexity. In this work, we develop a new framework for decompositions of tensors, taking into account invariance, positivity and a geometric arrangement of their local spaces. We define an invariant decomposition with indices arranged on a simplicial complex which is explicitly invariant under a group action. We give a constructive proof that this decomposition exists for all invariant tensors, after possibly enriching the simplicial complex. We further define several decompositions certifying positivity, and prove similar existence results, as well as inequalities between the corresponding ranks. Our results generalize results from the theory of tensor networks, used in the study of quantum many-body systems, for example.