A goal of central importance in the study of hierarchical models for object recognition -- and indeed the visual cortex -- is that of understanding quantitatively the trade-off between invariance and selectivity, and how invariance and discrimination properties contribute towards providing an improved representation useful for learning from data. In this work we provide a general group-theoretic framework for characterizing and understanding invariance in a family of hierarchical models. We show that by taking an algebraic perspective, one can provide a concise set of conditions which must be met to establish invariance, as well as a constructive prescription for meeting those conditions. Analyses in specific cases of particular relevance to computer vision and text processing are given, yielding insight into how and when invariance can be achieved. We find that the minimal sets of transformations intrinsic to the hierarchical model needed to support a particular invariance can be clearly described, thereby encouraging efficient computational implementations. ; United States. Defense Advanced Research Projects Agency (Contract FA8650-06-C-7632)