Portia Anderson "Symmetries of Knutson-Tao puzzles with hexagonal and parallelogram-shaped boundary"
Channel: Schubert Seminar
Date: 2022-12-12

Title: Edge-swapping symmetries of Knutson-Tao puzzles with hexagonal and parallelogram-shaped boundary


Abstract: Classically, Schubert calculus is about computing structure constants in the cohomology ring of the Grassmannian by taking transverse intersections of Schubert varieties. Knutson-Tao puzzles are triangular combinatorial gadgets that compute the same constants. Thus Schubert calculus forms a bridge between the worlds of combinatorics and geometry, allowing the two sides to converse in often interesting ways. Symmetries that are readily observed in puzzles can sometimes illuminate more obscure geometric phenomena, or vice versa. We will explore some of this interplay by now considering puzzles with more than three boundary edges, which we endow with a geometric interpretation. In pursuit of a geometric understanding of their rotational symmetry, we discover and prove stronger combinatorial statements on edge-swapping symmetries of parallelogram-shaped puzzles computing equivariant cohomology. We will also discuss expansion of this work to puzzles with hexagonal boundary.