斜Schur函数与斜Schur P/Q函数的Richardson体积模型
Richardson volume models for skew Schur and skew Schur $P/Q$-functions
AI总结：
该研究将斜Schur函数、斜Schur P/Q函数对应到Richardson簇的体积多项式，解决相关洛伦兹猜想并推广结果，还推导了多种不等式与多面体相关结论，补充了特征2下的态射结果。
AI中文摘要：
我们将普通斜Schur多项式和斜Schur P函数分别识别为普通Grassmann流形和Lagrangian Grassmann流形中Richardson簇上的最高次全陈相交多项式。随后得到，𝒩(s_{λ/μ})、𝒩(P_{λ/μ})、𝒩(Q_{λ/μ})是可实现体积多项式。这解决了Huh–Matherne–Mészáros–St. Dizier提出的斜Schur和Schur-P洛伦兹猜想，并将后者推广到任意斜P/Q函数。该构造可扩展到与普通Grassmann流形和Lagrangian Grassmann流形的任意不可约子簇相关联的循环变换。其可实现体积解释产生了普通和移位表型重数的反向Khovanskii–Teissier不等式与洛伦兹Hodge–Riemann不等式；精确的普通斜Schur支撑置换体及其极值系数；已知的直移位支撑多面体及其顶点系数；任意斜P/Q函数的隐式精确置换体；以及加权聚合、协方差和两行Littlewood–Richardson结果。Zhenpeng Wang撰写的附录构造了对偶D型旋量循环变换和Q_{λ/μ}的直接Richardson实现。在特征2下，兼容的非常特殊的同态诱导了环境Lagrangian模型与旋量模型之间的有限平根基态射。它们对相应Richardson簇的限制的次数为2^{ℓ(λ)-ℓ(μ)}和2^{|λ|-|μ|-ℓ(λ)+ℓ(μ)}，正则完全交桥解释了两个射影丛位移之间的差异。
英文摘要：
We identify ordinary skew Schur polynomials and skew Schur $P$-functions as top-degree total-Chern intersection polynomials on Richardson varieties in ordinary and Lagrangian Grassmannians. We then obtain that \[ \mathcal N(s_{λ/μ}),\qquad \mathcal N(P_{λ/μ}),\qquad \mathcal N(Q_{λ/μ}) \] are realizable volume polynomials. This settles the skew-Schur and Schur-$P$ Lorentzian conjectures of Huh--Matherne--Mészáros--St.~Dizier and strengthens the latter to arbitrary skew $P/Q$-functions. The constructions extend to cycle transforms attached to arbitrary irreducible subvarieties of ordinary and Lagrangian Grassmannians. Their realizable-volume interpretation yields reverse Khovanskii--Teissier and Lorentzian Hodge--Riemann inequalities for ordinary and shifted tableau multiplicities; exact ordinary skew-Schur support permutahedra and extremal coefficients; the known straight shifted support polytopes with their vertex coefficients; implicit exact permutahedra for arbitrary skew $P/Q$-functions; and weighted-aggregation, covariance, and two-row Littlewood--Richardson consequences. An appendix by Zhenpeng Wang constructs the dual type~$D$ spinor cycle transform and a direct Richardson realization of $Q_{λ/μ}$. In characteristic two, compatible very special isogenies induce finite flat radicial morphisms between the ambient Lagrangian and spinor models. Their restrictions to the corresponding Richardson varieties have degrees $2^{\ell(λ)-\ell(μ)}$ and $2^{|λ|-|μ|-\ell(λ)+\ell(μ)}$, and a regular complete-intersection bridge explains the difference between the two projective-bundle shifts.
