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.

翻译：拼图是解释格拉斯曼流形Littlewood-Richardson系数的通用组合工具。本文提出拼图理想的概念，其簇与拼图的平铺一一对应，并构建了一个代数框架来构造拼图理想，该框架适用于格拉斯曼流形的Knutson-Tao-Woodward拼图及其$T$-等变与$K$-理论变体。针对单侧自由的拼图，我们提出自由侧拼图理想，其簇与自由侧拼图的平铺一一对应，且自由侧拼图理想的消去理想包含了关于自由侧格拉斯曼流形结构常数的所有信息。引入这些拼图理想除了具有基础代数意义外，更具备通过求解定义多项式系统来寻找格拉斯曼流形拼图所有平铺的计算可行性，本文通过计算Gröbner基的示例拼图验证了该方法的有效性。