Divisor labelling of staircase diagrams and fiber bundle structures on Schubert varieties

Abstract

Let Gr(𝑟, 𝑛) denote the Grassmannian of 𝑟-dimensional subspaces of the 𝑛-dimensional vector space ℂⁿ over the field of complex numbers. Each Gr(𝑟, 𝑛) contains a unique codimension 1 Schubert subvariety called the Schubert divisor of the Grassmannian. In this project, we will discuss the correspondence between the set of permutations avoiding the patterns 3412, 52341, 52431, and 53241, and the set of Schubert varieties in the complete flag variety which are iterated fiber bundles of Grassmannians or Grassmannian Schubert divisors. Using this geometrical structure, we calculate the generating function that enumerates the permutations avoiding these patterns.