Richardson volume models for skew Schur and skew Schur $P/Q$-functions

Abstract

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 \[ \mathcal N(s_{\lambda/\mu}),\qquad \mathcal N(P_{\lambda/\mu}),\qquad \mathcal N(Q_{\lambda/\mu}) \] are realizable volume polynomials. This settles the skew-Schur and Schur-$P$ Lorentzian conjectures of Huh--Matherne--M\'esz\'aros--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 $Q_{\lambda/\mu}$. 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 $2^{\ell(\lambda)-\ell(\mu)}$ and $2^{|\lambda|-|\mu|-\ell(\lambda)+\ell(\mu)}$, and a regular complete-intersection bridge explains the difference between the two projective-bundle shifts.