Twisted quadratic foldings of root systems and liftings of Schubert classes

TL;DR

This paper explores twisted quadratic foldings of root systems and their impact on Schubert classes within structure algebras, providing criteria for lifting Schubert classes through algebra homomorphisms.

Contribution

It introduces a combinatorial criterion for when Schubert classes in folded root systems can be lifted to original systems via algebra homomorphisms.

Findings

Provides a criterion for lifting Schubert classes

Establishes a connection between classical and Lusztig's foldings

Analyzes the structure algebra embeddings in root system foldings

Abstract

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 $c.$