Twisted quadratic foldings of root systems and liftings of Schubert classes
Given a finite crystallographic root system $Φ$ whose Dynkin diagram has a non-trivial automorphism, it yields a new root system $Φ_τ$ by a so-called classical folding. On the other hand, Lusztig's folding (1983) folds the root system of type $E_8$ to $H_4$ starting from an automorphism of the root lattice of type $E_8.$ The notion of a twisted quadratic folding of a root system was introduced by Lanini-Zainoulline (2018) to describe both the classical foldings and Lusztig's folding on the same footing. The structure algebra $\mathcal{Z}(\mathcal{G})$ of the moment graph $\mathcal{G}$ associated with a finite root system and its reflection group $W$ is an algebra over a certain polynomial ring $\mathcal{S},$ whose underlying module is free with a distinguished basis $\{σ^{(w)} \mid w \in W\}$ called combinatorial Schubert classes. By Lanini-Zainoulline (2018), a twisted quadratic folding $Φ\rightsquigarrow Φ_τ$ induces an embedding of the respective Coxeter groups $\varepsilon: W_τ \hookrightarrow W$ and a ring homomorphism $\varepsilon^*: \mathcal{Z}(\mathcal{G}) \rightarrow \mathcal{Z}(\mathcal{G}_τ)$ between the corresponding structure algebras. This paper studies the $\varepsilon^*$-preimage of Schubert classes and provides a combinatorial criterion for a Schubert class $σ^{(u)}_τ$ of $\mathcal{Z}(\mathcal{G}_τ)$ to admit a Schubert class $σ^{(w)}$ of $\mathcal{Z}(\mathcal{G})$ such that the relation $\varepsilon^*(σ^{(w)}) = c \cdot σ^{(u)}_τ$ holds for some nonzero scalar $c.$