Combinatorics of semi-toric degenerations of Schubert varieties in type \(C\) (Q6581850)

Context: A goal of Schubert calculus is to compute the structure constants of the cohomology ring of a flag variety with respect to the basis consisting of Schubert classes. An approach is to realize Schubert classes as concrete combinatorial objects (for example, Schubert polynomials). Using the polytope ring of the Gelfand-Tsetlin polytopes, Kiritchenko-Smirnov-Timorin realized each Schubert class as a sum of reduced Kogan faces in [\textit{V. A. Kirichenko} et al., Russ. Math. Surv. 67, No. 4, 685--719 (2012; Zbl 1258.14055); translation from Usp. Mat. Nauk 67, No. 4, 89--128 (2012)]. Using a semitoric degeneration of a Schubert variety, the first author introduced a generalization of reduced Kogan faces to symplectic Gelfand-Tsetlin polytopes and extended the result of Kiritchenko-Smirnov-Timorin to the case of type C in [\textit{N. Fujita}, Adv. Math. 397, Article ID 108201, 42 p. (2022; Zbl 1494.14054)].\N\NResult: The purpose of the present paper is to develop combinatorics of this type C model of Schubert classes. Precisely, the authors introduce a combinatorial model to this type C generalization using a kind of pipe dream with self-crossings (Theorem 1). Then they prove that the type C generalization can be constructed by skew-mitosis operators (Theorem 2).\N\NKey: The idea is to generalize the combinatorial constructions of the set of reduced pipe dreams in type A to the set of skew pipe dreams in type C.\N\NStructure: In Sect. 2, they introduce a path model to skew pipe dreams. In Sect. 3, they recall the definitions of ladder moves and mitosis operators for skew pipe dreams. In Sect. 4 and 5, they prove Theorems 1 and 2.