Chain algebras of finite distributive lattices

Abstract

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 $n$.

Figures (6)

• Figure 1: A poset $P$ and the lattice ${\mathcal{J}}(P)$
• Figure 2: Illustration of step 1 in the proof of \ref{['thm:dim']}
• Figure 3: An induced cycle of length $12$ in the Boolean lattice ${\mathcal{B}}_5$.
• Figure 4: A poset $P$, the lattice $L={\mathcal{J}}(P)$ embedded into a grid, and a pair of its incomparable maximal chains
• Figure 5: A poset $P'$ is obtained by imposing strict increase along rows and columns

Theorems & Definitions (38)

• Theorem 1.1: Birkhoff
• Theorem 1.2: Stanley
• Theorem 1.3: Sturmfels
• Remark 1.4
• Definition 2.1
• Example 2.2
• Proposition 2.3
• proof
• Theorem 2.4
• proof