Tangent Cones of Schubert Varieties for $A_n$ of lower rank

TL;DR

This paper computes tangent cones at the identity of Schubert varieties in type A for ranks up to four and proposes conjectures for higher ranks, advancing understanding of their local geometric structure.

Contribution

It provides explicit calculations for low-rank cases and formulates conjectures for general rank, filling gaps in the geometric understanding of Schubert varieties.

Findings

Tangent cones explicitly calculated for ranks up to four.

Several conjectures proposed for arbitrary rank $n$.

Enhanced understanding of local geometry of Schubert varieties.

Abstract

We calculate the tangent cones at unity of Schubert varieties for $A_n$, where $n$ is less or equal to four. We state several conjectures for an arbitrary $n$.