Staircase diagrams and the enumeration of smooth Schubert varieties

Staircase diagrams and enumeration of smooth Schubert varieties

Combinatorics in Schubert varieties and Specht modules

This thesis consists of two parts. Both parts are devoted to finding links between geometric/algebraic objects and combinatorial objects. In the first part of the thesis, we link Schubert varieties in the full flag variety with hyperplane arrangements. Schubert varieties are parameterized by elements of the Weyl group. For each element of the Weyl group, we construct certain hyperplane arrangem...

Smoothness of Schubert Varieties via Patterns in Root Systems

The aim of this article is to present a smoothness criterion for Schubert varieties in generalized flag manifolds G/B in terms of patterns in root systems. We generalize Lakshmibai-Sandhya’s well-known result that says that a Schubert variety in SL(n)/B is smooth if and only if the corresponding permutation avoids the patterns 3412 and 4231. Our criterion is formulated uniformly in general Lie ...

Smoothness of Schubert Varieties via Patterns in Root Subsystems

The aim of this article is to present a smoothness criterion for Schubert varieties in generalized flag manifolds G/B in terms of patterns in root systems. We generalize Lakshmibai-Sandhya’s well-known result that says that a Schubert variety in SL(n)/B is smooth if and only if the corresponding permutation avoids the patterns 3412 and 4231. Our criterion is formulated uniformly in general Lie ...

Smooth and palindromic Schubert varieties in affine Grassmannians

We completely determine the smooth and palindromic Schubert varieties in affine Grassmannians, in all Lie types. We show that an affine Schubert variety is smooth if and only if it is a closed parabolic orbit. In particular, there are only finitely many smooth affine Schubert varieties in a given Lie type. An affine Schubert variety is palindromic if and only if it is a closed parabolic orbit, ...