Hecke Type Algebras, Triple Schubert Calculus and Their Applications to Equivariant Positivity

Abstract

This thesis introduces a triple variable framework for Schubert calculus that generalizes traditional single and double versions. The primary focus of this work is the computation of structure constants for the equivariant cohomology of flag varieties. First, I provide a new proof of Triple Pieri rules, which offer explicit formulas for the product of certain geometric classes. These rules are derived using skew divided difference operators and identities within Hecke-type algebras. Second, I develop pipe puzzles, a visual and combinatorial model for computing these structure constants. This model is specifically designed for permutations with separated descents. It generalizes earlier structures, including bumpless pipe dreams and the puzzle model of Knutson and Zinn-Justin. In the proof of this model, I utilize lattice models and the Yang--Baxter equation. I also provide the code used to compute R-matrices for specific representations. A key finding of this research is that triple Schubert structure constants satisfy two types of recurrence relations. This suggests that although the triple framework is broader than its predecessors, its underlying proofs can be simplified. Third, I present the major geometric contribution of this work: a new interpretation of triple variables as equivariant parameters within higher-dimensional flag varieties. This perspective allows for the proof of equivariant positivity conjectures originally proposed by Samuel and Kirillov. The proof relies on a refined version of Graham's positivity theorem regarding effective cycles.

Finally, there are several potential applications of these results. The Hecke-type algebra identities may extend to generalized cohomology theories. The simplified proof for puzzle rules can likely be applied to other combinatorial models. Additionally, the R-matrix code can be adapted for further representations. The geometric interpretation offers a path to explaining positivity in K-theory and beyond.