Mathematical Research Letters

Volume 23 (2016)

Number 4

Geometric mitosis

Pages: 1071 – 1097



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)


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 $Sp_{2n}$.


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

2010 Mathematics Subject Classification

05E10, 14M15, 52B20

Accepted 5 February 2016

Published 16 September 2016