Given a finite crystallographic root system $\Phi$ whose Dynkin diagram has a non-trivial automorphism, it yields a new root system $\Phi_{ \tau}$ 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 $\{ \sigma^{ (w)} \mid w \in W\}$ called combinatorial Schubert classes. By Lanini-Zainoulline (2018), a twisted quadratic folding $\Phi \rightsquigarrow \Phi_{ \tau}$ induces an embedding of the respective Coxeter groups $\varepsilon: W_{ \tau} \hookrightarrow W$ and a ring homomorphism $\varepsilon^*: \mathcal{ Z}(\mathcal{ G}) \rightarrow \mathcal{ Z}(\mathcal{ G}_{ \tau})$ between the corresponding structure algebras. This paper studies the $\varepsilon^*$-preimage of Schubert classes and provides a combinatorial criterion for a Schubert class $\sigma^{ (u)}_{ \tau}$ of $\mathcal{ Z}(\mathcal{ G}_{ \tau})$ to admit a Schubert class $\sigma^{ (w)}$ of $\mathcal{ Z}(\mathcal{ G})$ such that the relation $\varepsilon^*(\sigma^{ (w)}) = c \cdot \sigma^{ (u)}_{ \tau}$ holds for some nonzero scalar $c.$
Categories: math.GR