Puzzle Ideals for Grassmannians

TL;DR

This paper introduces puzzle ideals as algebraic objects that correspond to puzzle tilings for Grassmannians, providing a new algebraic framework and computational methods to study their structure constants.

Contribution

It proposes the concept of puzzle ideals and side-free puzzle ideals, linking combinatorial puzzle tilings to algebraic varieties and enabling computational approaches.

Findings

Puzzle ideals correspond one-to-one with puzzle tilings.

The algebraic framework applies to Knutson-Tao-Woodward puzzles and variants.

Computational methods like Gr"obner bases facilitate enumeration of puzzle tilings.

Abstract

Puzzles are a versatile combinatorial tool to interpret the Littlewood-Richardson coefficients for Grassmannians. In this paper, we propose the concept of puzzle ideals whose varieties one-one correspond to the tilings of puzzles and present an algebraic framework to construct the puzzle ideals which works with the Knutson-Tao-Woodward puzzle and its $T$-equivariant and $K$-theoretic variants for Grassmannians. For puzzles for which one side is free, we propose the side-free puzzle ideals whose varieties one-one correspond to the tilings of side-free puzzles, and the elimination ideals of the side-free puzzle ideals contain all the information of the structure constants for Grassmannians with respect to the free side. Besides the underlying algebraic importance of the introduction of these puzzle ideals is the computational feasibility to find all the tilings of the puzzles for Grassmannians by solving the defining polynomial systems, demonstrated with illustrative puzzles via computation of Gr\"obner bases.