Chain algebras of finite distributive lattices

We introduce a family of toric algebras defined by maximal chains of a finite distributive lattice. Applying results on stable set polytopes, we conclude that every such algebra is normal and Cohen–Macaulay, and give an interpretation of its Krull dimension in terms of the combinatorics of the under...

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Bibliographic Details
Published in:Journal of algebraic combinatorics 2024, Vol.59 (2), p.473-494
Main Authors: Gasanova, Oleksandra, Nicklasson, Lisa
Format: Article
Language:English
Subjects:
Algebra
Combinatorial analysis
Combinatorics
Computer Science
Convex and Discrete Geometry
Group Theory and Generalizations
Hilbert series
Koszul algebra
Lattices
Lattices (mathematics)
Mathematics
Mathematics and Statistics
Order
Ordered Algebraic Structures
Polytopes
Toric ideal
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Summary:We introduce a family of toric algebras defined by maximal chains of a finite distributive lattice. Applying results on stable set polytopes, we conclude that every such algebra is normal and Cohen–Macaulay, and give an interpretation of its Krull dimension in terms of the combinatorics of the underlying lattice. When the lattice is planar, we show that the corresponding chain algebra is generated by a sortable set of monomials and is isomorphic to a Hibi ring of another finite distributive lattice. As a consequence, it has a defining toric ideal with a quadratic Gröbner basis, and its h -vector counts ascents in certain standard Young tableaux. If instead the lattice has dimension n > 2 , we show that the defining ideal has minimal generators of degree at least n .
ISSN:0925-9899
1572-9192
1572-9192
DOI:10.1007/s10801-023-01294-8