The isomorphism problem for Schubert varieties

TL;DR

This paper provides a practical criterion to determine when two Schubert varieties, associated with Kac-Moody groups, are isomorphic, based on their Weyl group elements and cohomology rings.

Contribution

It introduces a new criterion for Schubert variety isomorphism that depends on Cartan matrices and reduced words, linking geometric and algebraic properties.

Findings

A practical criterion for Schubert variety isomorphism

Isomorphism characterized by cohomology ring isomorphism

Applicable to varieties from different flag varieties

Abstract

Schubert varieties in the full flag variety of Kac-Moody type are indexed by elements of the corresponding Weyl group. We give a practical criterion for when two such Schubert varieties (from potentially different flag varieties) are isomorphic, in terms of the Cartan matrix and reduced words for the indexing Weyl group elements. As a corollary, we show that two such Schubert varieties are isomorphic if and only if there is an isomorphism between their integral cohomology rings that preserves the Schubert basis.