Quilts and T-Systems Original Research Article

B3, the 3-string braid group, has an action on pairs of elements of a group G which is of importance in the study of Moonshine and its generalizations. We call the orbits of this action T-systems, and in particular we wish to study T-systems for finite groups. In this paper, we show that certain diagrams, called quilts, completely represent the structure of T-systems. More importantly, one can draw the quilt of even a fairly sizeable T-system from easily obtainable group-theoretic information, using what we call the multiplication principle and the patch theorem. We give several examples of this process.