Ideals generated by 2-minors, collection of cells and stack polyominoes

Abstract

In this paper we study ideals generated by quite general sets of 2-minors of an $m\times n$-matrix of indeterminates. The sets of 2-minors are defined by collections of cells and include 2-sided ladders. For convex collections of cells it is shown that the attached ideal of 2-minors is a Cohen--Macaulay prime ideal. Primality is also shown for collections of cells whose connected components are row or column convex. Finally the class group of the ring attached to a stack polyomino and its canonical class is computed, and a classification of the Gorenstein stack polyominoes is given.

29 pages, 32 figures

References in corpus (2)

• Special simplices and Gorenstein toric rings
• Ideals generated by adjacent 2-minors

Cited by in corpus (19)

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• On Gröbner bases and Cohen-Macaulay property of closed path polyominoes
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• Linearly related polyominoes
• On Cohen-Macaulay non-prime collections of cells
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• Non-simple polyominoes of Kőnig type and their canonical module
• Nonsimple polyominoes and prime ideals
• Simple Polyominoes are Prime
• Shellable flag simplicial complexes of non-simple polyominoes
• Cohen-Macaulay types of certain edge subrings of bipartite graphs and (generalized) Fuss-Catalan numbers
• Rank Bounded Hibi Subrings for Planar Distributive Lattices
• Toric rings of nonsimple polyominoes
• Toric representations of algebras defined by certain nonsimple polyominoes
• Toric rings, inseparability and rigidity
• Syzygies of Hibi rings