Symmetric Amplituhedra I: Spinor Helicity

For every pair of integers $0\le k\le N,$ and and every subgroup of $D_N\times {\mathbb{Z}}_2$, we construct the nonnegative symmetric Grassmannian, which is the subspace of the nonnegative Grassmannian fixed under that group, where the dihedral part acts on columns (with a sign correction to keep nonnegativity) and the ${\mathbb{Z}}_2$ factor exchanges a space with its orthocomplement (again, with a sign correction). This construction generalizes the earlier constructions of \cite{Karpman2018,Fraser2020,Shevchenko2025}. We study the geometry of this space. Then, when $k$ is further restricted to the range $[2,N-2]$ we also define a symmetric amplituhedron. We analyze its basic properties and its conjectural BCFW decomposition. The ABJM and reflected Lagrangian amplituhedra are the special cases corresponding to the groups $\{1\}\times {\mathbb{Z}}_2,$ and the group generated by a reflection combined with the exchange,