  Frustrated magnetic systems can host highly interesting phases known as
classical spin liquids (CSLs), which feature {extensive} ground state
degeneracy and lack long-range magnetic order. Recently, Yan and Benton et al.
proposed a classification scheme of CSLs in the large-$\mathcal{N}$ (soft spin)
limit [arXiv.2305.00155], [arXiv:2305.19189]. This scheme classifies CSLs into
two categories: the algebraic CSLs and the fragile topological CSLs, each with
their own correlation properties, low energy effective description, and finer
classification frameworks. In this work, we further develop the classification
scheme by considering the role of crystalline symmetry. We present a
mathematical framework for computing the band representation of the flat bands
in the spectrum of these CSLs, which extends beyond the conventional
representation analysis. It allows one to determine whether the algebraic CSLs,
which features gapless points on their bottom flat bands, are protected by
symmetry or not. It also provides more information on the finer classifications
of algebraic and fragile topological CSLs. We demonstrate this framework via
concrete examples and showcase its power by constructing a pinch-line algebraic
CSL protected by symmetry.
