Twisted Quadratic Foldings of Root Systems and Combinatorial Schubert Calculus

Abstract

This thesis builds on the connection of two widely studied objects in the literature, that is, foldings of finite root systems and the structure algebras of moment graphs associated with finite root systems. Given a finite crystallographic root system Φ whose Dynkin diagram has a nontrivial 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 type H₄ starting from an automorphism of the root lattice of type E₈. Lanini-Zainoulline (2018) developed the notion of a twisted quadratic folding of a root system, which describes both the classical foldings and Lusztig's folding on the same footing. Our second key object of study is the structure algebra Z(G) of the moment graph G associated with a finite root system and its reflection group W. The structure algebra Z(G) 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. Each Schubert class σ⁽ʷ⁾ is an S-valued function on W, whose value is explicitly known for any finite reflection group W. Lanini-Zainoulline (2018) showed that a twisted quadratic folding Φ -> Φτ induces an embedding of the respective Coxeter groups ε : Wτ → W and a ring homomorphism ε∗ : Z(G) → Z(Gτ) between the corresponding structure algebras. This thesis investigates how the induced map ε* relates the Schubert classes of the original structure algebra Z(G) to those of the folded structure algebra Z(Gτ). In particular, we will provide a combinatorial criterion for a Schubert class στ⁽ᵘ⁾ of Z(Gτ) to admit a Schubert class σ⁽ʷ⁾ of Z(G) such that the relation ε*(σ⁽ʷ⁾) = c · στ⁽ᵘ⁾ holds for some nonzero scalar c. We will also prove that ε* is surjective after an appropriate extension of the coefficient ring.