Balanced Notation for τ-Rigid Pairs

We introduce a new notation for τ-rigid pairs called “balanced pairs”. The notation AB allows for simplification of some formulas, for example, the Jasso category and its dual. One of the most compelling formulas is that in the hereditary case, the bricks that label the “covering” and “flooring” walls for the g-vector of AB are precisely A and B respectively. In the general case, we give a precise formula for these wall labels. We also define “lower” and “upper” chambers for g(AB), and we prove that these correspond to generalized Bongartz and co-Bongartz completions of the balanced pair AB.