1. Interval positroid varieties 1 1.1. Schubert varieties 1 1.2. Schubert calculus 2 1.3. First positivity result 3 1.4. Interval rank varieties 5 2. Vakil’s Littlewood-Richardson rule 7 2.1. Combinatorial shifting 7 2.2. Geometric shifting 7 2.3. Vakil’s degeneration order 9 2.4. Partial puzzles 10 3. Equivariant and Kextensions 11 3.1. K-homology 11 3.2. K-cohomology 12 3.3. Equivariant K-the...

The aim of this article is to link Schubert varieties in the flag manifold with hyperplane arrangements. For a permutation, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial of the corresponding Schubert variety if and only if the Schubert variety is smooth. We give an explicit com...

We recall the root game, introduced in [P], which gives a fairly powerful sufficient condition for non-vanishing of Schubert calculus on a generalised flag manifold G/B. We show that it gives a necessary and sufficient rule for nonvanishing of Schubert calculus on Grassmannians. In particular, a LittlewoodRichardson number is non-zero if and only if it is possible to win the corresponding root ...

Bott-Samelson varieties were originally defined as desingularizations of Schubert varieties and were used to describe the geometry of Schubert varieties. In particular, the cohomology of some line bundles on Bott-Samelson varieties were used to prove that Schubert varieties are normal, Cohen-Macaulay and with rational singularities (see for example [BK05]). In this paper, we will be interested ...

An important invariant of a singular point on an algebraic variety X is its multiplicity : the normalized leading coefficient of the Hilbert polynomial of the local ring. The main result of the present note is an explicit determinantal formula for the multiplicities of points on Schubert varieties in Grassmannians. This is a simplification of a formula obtained in [5]. More recently, the recurr...