Mathematical Research Letters

Volume 23 (2016)

Number 4

Geometric mitosis

Pages: 1071 – 1097

DOI: http://dx.doi.org/10.4310/MRL.2016.v23.n4.a5

Author

Valentina Kiritchenko (Lab. of Algebraic Geometry and Faculty of Mathematics, National Research Univ., Higher School of Economics, Moscow, Russia; and Institute for Information Transmission Problems, Moscow, Russia)

Abstract

We describe an elementary convex geometric algorithm for realizing Schubert cycles in complete flag varieties by unions of faces of polytopes. For $GL_n$ and Gelfand–Zetlin polytopes, combinatorics of this algorithm coincides with that of the mitosis on pipe dreams introduced by Knutson and Miller. For $Sp_4$ and a Newton–Okounkov polytope of the symplectic flag variety, the algorithm yields a new combinatorial rule that extends to $Sp_{2n}$.

Keywords

Demazure operator, flag variety, Newton–Okounkov polytope, Schubert calculus

2010 Mathematics Subject Classification

05E10, 14M15, 52B20

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