Tambara–Yamagami (TY) categories form one of the simplest families of non-pointed fusion categories. Despite their elementary description, these categories play a central role in mathematical physics. Studying generalizations of TY fusion rings can then provide a controlled framework to explore the new phenomena that arise when one moves beyond the classical TY setting. In this talk, I will focus on the generalization of the TY fusion ring, proposed by Jordan–Larson. One of the most useful tools to study fusion rings is the classification of their non-negative integer matrix (NIM-) representations. NIM-reps can be used as an effective method to detect algebra objects in the fusion category underlying the fusion ring. In this talk, we will compute and classify the irreducible NIM-reps of both proposed extensions as well as detect candidate algebra objects associated to these NIM-reps. This work is joint with Agustina Czenky, Emily McGovern, Monique Müller, and Ana Ros Camacho.