2021/05/26 by Serizawa, Maiko
#05E16 #14M15 #20F55 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2105.12601
Given a finite crystallographic root system Φ whose Dynkin diagram has a non-trivial automorphism, it yields a new root system Φτ by a so-called classical folding. On the other hand, Lusztig's folding (1983) folds the root system of type E8 to H4 starting from an automorphism of the root lattice of type E8. 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 Z(G) of the moment graph G associated with a finite root system and its reflection group W is an algebra over a certain polynomial ring S, whose underlying module is free with a distinguished basis \σ(w) | w ∈ W\ called combinatorial Schubert classes. By Lanini-Zainoulline (2018), a twisted quadratic folding Φ\rightsquigarrow Φτ induces an embedding of the respective Coxeter groups ε: Wτ \hookrightarrow W and a ring homomorphism ε^*: Z(G) → Z(Gτ) between the corresponding structure algebras. This paper studies the ε^*-preimage of Schubert classes and provides a combinatorial criterion for a Schubert class σ(u)τ of Z(Gτ) to admit a Schubert class σ(w) of Z(G) such that the relation ε^*(σ(w)) = c ⋅ σ(u)τ holds for some nonzero scalar c.