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2020年11月13日(金) 15:00~16:30
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藤田直樹氏(東京大学)
Schubert calculus from polyhedral parametrizations of Demazure crystals
One approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Through an identification of the cohomology ring of the type A full flag variety with the polytope ring of the Gelfand-Tsetlin polytopes, Kiritchenko-Smirnov-Timorin realized each Schubert class as a sum of reduced (dual) Kogan faces. In this talk, we discuss its generalization to other Lie types through the theory of Kashiwara crystal bases. We first explicitly describe string parametrizations of opposite Demazure crystals in general Lie type, which give a natural generalization of reduced dual Kogan faces. We then relate reduced Kogan faces with Demazure crystals in type A through the theory of mitosis operators on reduced pipe dreams. This relation is naturally extended to the case of type C, which leads to the theory of Schubert calculus on symplectic Gelfand-Tsetlin polytopes.