Richardson varieties, projected Richardson varieties and positroid varieties

Abstract

Background 1.1. Symmetric groups and related combinatorics 1.2. Algebraic geometry and group notation 1.3. Grassmannians, Flag varieties, Plücker coordinates 1.4. Bruhat decomposition, Schubert cells, Schubert varieties 1.5. Richardson varieties 1.6. Projected Richardson varieties 1.7. Positroid varieties 2. The Plücker algebra and homogeneous coordinate rings of Richardson varieties 2.1. Classical theory of the Plücker algebra 2.2. The ideal of the Richardson versus the sum of the ideals of the Schuberts 2.3. Standard monomial theory and semistandard tableaux 2.4. The simplicial complex of standard monomials 2.5. Gröbner degeneration of Matrix Schubert and Matrix Richardson varieties 2.6. Gelfand-Tsetlin degeneration of the Plücker algebra 2.7. Gelfand-Tsetlin degeneration of the coordinate ring of the Richardson variety 2.8. Frobenius splitting and its consequences 3. The Bott-Samelson varieties and brick varieties 3.1. Bott-Samelson varieties 3.2. Matrix products formulas for open Bott-Samelson varieties 3.3. Maps from Bott-Samelsons to Schubert cells 3.4. Brick varieties and Richardsons 4. The Deodhar decomposition 4.1. The Deodhar pieces 4.2. Matrix product formulas for Deodhar pieces 4.3. The Deodhar pieces do not form a stratification 4.4. Unipeak and univalley words 5. Total positivity 5.1. Totally nonnegative subspaces and flags 5.2. Cell complexes and total positivity 5.3. Positivity in partial flag manifolds 6. Positroids 6.1. What follows from the general theory 6.2. Affine permutations 6.3. Cyclic rank matrices 6.4. Grassmann necklaces 6.5. The cohomology class of a positroid variety 7. Plabic graphs 7.1. Plabic graphs and the boundary measurement map with positive real weights 7.2. Zig-zag paths 7.3. The twist and its consequences 8. Acknowledgments References