Structure constants of Peterson Schubert calculus

TL;DR

This paper provides explicit formulas for the structure constants in Peterson Schubert calculus across all Lie types, solving a longstanding open problem and deriving new combinatorial formulas.

Contribution

It introduces a positive, type-uniform formula for equivariant structure constants using the Cartan matrix, applicable to all Lie types.

Findings

Explicit formulas for all equivariant structure constants in Peterson Schubert calculus

Solution to an open problem in Lie theory for all types

Derivation of a type-uniform formula for mixed Φ-Eulerian numbers

Abstract

We give an explicit, positive, and type-uniform formula for all equivariant structure constants of the Peterson Schubert calculus in arbitrary Lie types, using only the Cartan matrix of the corresponding root system $\Phi$. This solves an open problem originally asked by Harada--Tymoczko in type A for all Lie types. As an application, we derive a type-uniform formula for the mixed $\Phi$-Eulerian numbers.