The isomorphism problem for Grassmannian Schubert varieties

TL;DR

This paper characterizes when Schubert and Richardson varieties in different Grassmannians are isomorphic, linking these isomorphisms to properties of Young diagrams and posets, and extends known results to more general cases.

Contribution

It establishes a precise criterion for isomorphism of Schubert and Richardson varieties across different Grassmannians based on diagram and poset properties, including a conjecture for the converse.

Findings

Schubert varieties are isomorphic iff their Young diagrams are transposition equivalent.

Richardson varieties are isomorphic iff their skew diagrams are semi-isomorphic as posets.

Conjecture: the converse holds for Richardson varieties.

Abstract

We prove that Schubert varieties in potentially different Grassmannians are isomorphic as varieties if and only if their corresponding Young diagrams are identical up to a transposition. We also discuss a generalization of this result to Grassmannian Richardson varieties. In particular, we prove that Richardson varieties in potentially different Grassmannians are isomorphic as varieties if their corresponding skew diagrams are semi-isomorphic as posets, and we conjecture the converse. Here, two posets are said to be semi-isomorphic if there is a bijection between their sets of connected components such that the corresponding components are either isomorphic or opposite.