Symmetry Analysis of Compact Tetraquark States and Implications for the Fully Charmed Candidates $X(6600)$, $X(6900)$, and $X(7100)$
Motivated by recent experimental observations, we investigate the $J^P$ distribution of low-energy compact tetraquark states using symmetry analysis based on inherent nodal structures. Assuming tetrahedral and square configurations for the $qq\bar q\bar q$ system, we derive the allowed orbital structures from the restricted representations of $S_4$ onto $S_2\times S_2$ for $L\leq3$. The accessible-state distribution is particularly prominent in the $J^P=2^+$, $2^-$, and $3^-$ sectors, with the $2^+$ sector showing the strongest low-energy preference. We further find that the symmetry-driven distribution is qualitatively similar to that of the three-flavor four-quark system, and that the dominant $J^P=2^+$ pattern persists under phenomenological weightings inspired by chromomagnetic interaction (CMI) considerations. These results suggest that the low-lying compact tetraquark spectrum is primarily constrained by symmetry, while the detailed distribution exhibits sensitivity to dynamical weightings. Applying this framework to the fully charmed candidates $X(6600)$, $X(6900)$, and $X(7100)$, we find that their observed $J^{PC}=2^{++}$ quantum numbers are consistent with a low-lying compact tetraquark interpretation. The present analysis identifies the relative ordering of the $1^-$ and $1^+$ states as a sensitive channel, suggesting a direction for future non-perturbative investigations.