Isomorphism of Bott–Samelson varieties across Weyl groups

Determine necessary and sufficient conditions under which two Bott–Samelson varieties associated to words in possibly different Weyl groups are isomorphic as algebraic varieties. Formulate the isomorphism criterion explicitly in terms of the indexing words and the corresponding Coxeter/Weyl group data.

Background

The paper proves that Bott–Samelson rings admit a uniform quadratic presentation and are purely combinatorial invariants determined by Coxeter data of the words, including Koszulness and the full Kähler package. Motivated by the combinatorial invariance result (Corollary 3.7 in the paper’s numbering), the authors pose an isomorphism classification problem for Bott–Samelson varieties analogous to the known isomorphism criteria for Schubert varieties.

In particular, Richmond and Slofstra established an isomorphism criterion for Schubert varieties in full flag varieties of Kac–Moody type in terms of Cartan matrices and reduced words. The authors ask for an analogous classification for Bott–Samelson varieties associated with words drawn from potentially different Weyl groups, spurred by their ring-level combinatorial invariance.