  Hamiltonian Learning is a process of recovering system Hamiltonian from
measurements, which is a fundamental problem in quantum information processing.
In this study, we investigate the problem of learning the symmetric Hamiltonian
from its eigenstate. Inspired by the application of group theory in block
diagonal secular determination, we have derived a method to determine the
number of linearly independent equations about the Hamiltonian unknowns
obtained from an eigenstate. This number corresponds to the degeneracy of the
associated irreducible representation of the Hamiltonian symmetry group. To
illustrate our approach, we examine the XXX Hamiltonian and the XXZ
Hamiltonian. We first determine the Hamiltonian symmetry group, then work out
the decomposition of irreducible representation, which serves as foundation for
analyzing the uniqueness of recovered Hamiltonian. Our numerical findings
consistently align with our theoretical analysis.
