Elimination ideals of Plücker ideals and algebras with straightening laws

TL;DR

The paper studies quadratically generated projections of the Grassmannian $\,\mathrm{Gr}(2,n)\,$ and characterizes when subalgebras ${\mathcal{R}}_{\mathbb{K}}[L]$ of the Plücker coordinate ring are algebras with straightening laws (ASL) with elimination ideals. It translates the problem into the distributive lattice $L_n$ and its sublattices, linking ASL structure to compatibility (i.e., $L_n\setminus L$ being a poset ideal of $\Pi_n$) and showing that the defining ideal $I_L$ is the elimination ideal of $I_{L_n}$ with $I_L=(Q_{ijk\ell}: 1\le i<j<k<\ell\le n,\, p_{i\ell}\in L)$. By associating sublattices to interval graphs, the authors provide a graph-theoretic criterion for the Gorenstein property, hinging on the intersections of consecutive maximal cliques, and they enumerate perfect compatible sublattices via Catalan numbers. An appendix offers a second Gröbner basis containing cubics, yielding a Catalan interpretation through nested arc arrangements. Overall, the work connects Grassmannian coordinate rings, ASL theory, and interval-graph combinatorics to produce explicit Gorenstein criteria and enumerations for a broad class of subalgebras.

Abstract

It is well known that the Plücker ideal defining the Grassmannian is generated by quadratic Plücker relations. These relations form a reverse lexicographic Gröbner basis and endow the Plücker algebra with the structure of an algebra with straightening laws (ASL). In this paper, we study quadratically generated projections of the Grassmannian of lines $\mathrm{Gr}(2,n)$. We then combinatorially characterize the Gorenstein ASL subalgebras of the Plücker algebra of $\mathrm{Gr}(2,n)$.

Figures (8)

• Figure 1: The Hasse diagram of the distributive lattice $L_5$.
• Figure 2: The Hasse diagram of the poset $\Pi_5$.
• Figure 3: The Hasse diagram of a perfect compatible sublattice of $L_7$. The fundamental chain is highlighted in bold.
• Figure 4: Perfect compatible sublattices of the lattice $L_5$.
• Figure 5: Two isomorphic graphs: an interval graph (left) and a non-interval graph (right).

Theorems & Definitions (43)

• Example 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 2.4
• Theorem 2.5
• proof
• Definition 2.6
• Example 2.7