Balanced notation for $τ$-rigid pairs

Abstract

Over any finite dimensional algebra $Λ$ over any field $K$, we propose a new notation for $τ$-rigid pairs: We represent these as ``balanced pairs'' $\frac AB$, We show that many formulas in $τ$-tilting theory are simplified in the balanced notation. For example, the Jasso category and its dual, the torsion class and torsion-free class associated to any $τ$-tilting pair. Other dual concepts, such as signed and cosigned exceptional sequences are also easier to compare using the balanced notation. Balanced notation is also visually appealing because it matches the wall labels for the semi-invariant picture in the hereditary case and almost matches the wall labels in the general case. We also show that, in the functorially finite case, the formulas for torsion classes given by Igusa and Maresca [11] agree with the standard formulas of Adachi, Iyama and Reiten [1] when expressed in balanced notation.