Type C Richardson Boundaries and Newton-Okounkov Degenerations of Odd-Dimensional Projective Space

We compare two mirror-theoretic degeneration pictures attached to odd-dimensional projective space. If ${\text{PP}}^{2n-1}$ is viewed as a toric variety, one obtains the classical toric mirror. If it is viewed as the type $C_n$ homogeneous space $S{p}_{2n}\mathrm{/}{P}_1$, Rietsch’s Lie-theoretical construction gives a superpotential on the dual side. We compute the type $C_n$ boundary $D_C$, while on the dual-side Lusztig torus we compute the corresponding Laurent polynomial. We construct Newton–Okounkov bodies, and in each case the degree-augmented value semigroup is identified with the lattice-point semigroup of the cone over the polar dual of the corresponding Newton polytope. This gives two toric degeneration pictures attached to the same projective space. We also exhibit a rank-one weight degeneration connecting $D_C$ with the standard toric boundary while keeping the ambient projective space fixed.