Polyominoes and Knutson ideals

TL;DR

This paper investigates the radicality and primality of polyomino ideals by exploring Knutson ideals, demonstrating that certain classes of polyominoes have these properties and providing explicit Gr"{o}bner bases.

Contribution

It introduces the study of Knutson ideals within polyominoes and proves that several classes of polyomino ideals are Knutson, radical, and prime, with computed Gr"{o}bner bases.

Findings

Polyomino ideals of closed path, weakly closed path, simple thin, and ladder polyominoes are Knutson.

Certain thin polyominoes have Knutson ideals, which are radical.

Explicit Gr"{o}bner bases are computed for these classes.

Abstract

In this article, we study two fundamental questions on polyomino ideals which are radicality and primality. In order to study the question of radicality, we initiate the study of Knutson ideals among polyominoes. Knutson ideals were introduced by Conca and Varbaro after the work of Knutson on compatibly split ideals. Knutson ideals are known to have nice properties, for example, they are well behaved with Gr\"{o}bner bases, and it has square-free initial ideals; hence they are radical. We show that polyomino ideals associated with closed path, weakly closed path, simple thin, and ladder polyominoes are Knutson. We also show that polyomino ideals associated with a class of thin polyominoes are Knutson; hence they are radical. In fact, we show that these polyomino ideals are prime and the reduced Gr\"{o}bner basis is computed. Furthermore, we prove that under a certain condition, if a parallelogram polyomino is extracted from another parallelogram polyomino, the resulting collection of cells is Knutson. We also compute their Gr\"{o}bner basis.