C型Richardson边界与奇维射影空间的Newton-Okounkov退化
Type C Richardson Boundaries and Newton-Okounkov Degenerations of Odd-Dimensional Projective Space
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中文总结 AI 辅助
该研究对比奇维射影空间的两种镜像理论退化图景，计算C型边界与对应Laurent多项式，构造Newton–Okounkov体并验证半群等同性，还给出连接两类边界的秩一权退化。
中文摘要 AI 辅助
我们比较了与奇维射影空间相关的两种镜像理论退化图景。若将\\(\PP^{2n-1}\\)视为环面簇，可得到经典的环面镜像。若将其视为\\(C_n\\)型齐性空间\\(Sp_{2n}/P_1\\)，Rietsch的李理论构造可在对偶侧给出一个超势。我们计算了\\(C_n\\)型边界\\(D_C\\)，同时在对偶侧的Lusztig环面上计算了对应的Laurent多项式。我们构造了Newton–Okounkov体，且在每种情形下，增进度值半群都与对应Newton多面体的极对偶锥的格点半群等同。这给出了依附于同一射影空间的两种环面退化图景。我们还展示了一种秩一权退化，它在保持环境射影空间固定的情况下将\\(D_C\\)与标准环面边界连接起来。
英文摘要
We compare two mirror-theoretic degeneration pictures attached to odd-dimensional projective space. If \(\PP^{2n-1}\) is viewed as a toric variety, one obtains the classical toric mirror. If it is viewed as the type \(C_n\) homogeneous space \(Sp_{2n}/P_1\), 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.
