Matter symmetries in supersymmetric standard models from non-invertible selection rules

We discuss phenomenological implications of non-invertible selection rules in the framework of the supersymmetric standard model. We find that a remnant Z₂ symmetry of fusion algebras which holds at all-loop order plays the role of R-parity, forbidding baryon and lepton violating operators. In addition, a combination of Standard Model gauge symmetry and the non-invertible selection rules lead to baryon triality and proton hexality that can protect the proton from decay. In general, our finding selection rules can not realize the same results as conventional ZN symmetries in the supersymmetric standard model. We also clarify the assignments of matter fields under the fusion algebras that are consistent with SU(4) × SU(2) × SU(2), SU(5), and SO(10) grand unified theories.