Gorenstein ASL Subalgebras in Plücker Algebras

• Gorenstein ASL subalgebras are algebras with straightening laws whose canonical module is isomorphic to the algebra up to a homogeneous shift.
• They are characterized in the quadratic Plücker algebra of Gr(2,n) by interval graph structures with clique overlaps satisfying specific purity conditions.
• Their study reveals deep connections with combinatorial sequences like Catalan and Fibonacci numbers, linking invariant theory and algebraic geometry.

A Gorenstein ASL subalgebra is a subalgebra with straightening laws (ASL) that is also Gorenstein—its canonical module is isomorphic to the algebra itself up to homogeneous shift. Within the context of the quadratic Plücker algebra of the Grassmannian of lines $\mathrm{Gr}(2, n)$, such subalgebras are characterized by their combinatorial properties, in particular the structure of the underlying poset and associated interval graphs. The Gorenstein property imposes strong symmetry on the Hilbert series and yields rich connections to combinatorics, invariant theory, and classical enumerative families such as Catalan and Fibonacci numbers.

1. Algebras with Straightening Laws: Foundations and Properties

An algebra with straightening laws is a standard graded, Noetherian $\mathbb{K}$-algebra $R = \bigoplus_{d \geq 0} R_d$ generated by a collection $\phi: P \to \bigcup_{d>0} R_d$ where $P$ is a finite poset and $\phi$ is injective. $R$ is called an ASL on $P$ over $\mathbb{K}$ if:

• (ASL-1): The set of standard monomials $\phi(\alpha_1)\phi(\alpha_2)\dots\phi(\alpha_k)$ with $\alpha_1 \leq \dots \leq \alpha_k$ forms a $\mathbb{K}$-basis of $R$.
• (ASL-2): For incomparable $\alpha, \beta \in P$, their product $\phi(\alpha)\phi(\beta)$ can be uniquely written as a $\mathbb{K}$-linear combination of standard monomials, each beginning with a strictly smaller poset element than both $\alpha$ and $\beta$.

ASLs include several significant families of algebras, such as Plücker coordinate rings, Stanley-Reisner rings, and the discrete LS algebras. The straightening relations can be realized as the defining relations of $R$, expressed as $R \cong \mathbb{K}[x_\alpha: \alpha \in P]/I$.

2. The Gorenstein Property for ASL Subalgebras

The Gorenstein property, for a standard graded Cohen–Macaulay $\mathbb{K}$-algebra $A = \bigoplus_{d \geq 0} A_d$ of Krull dimension $d$, is determined by the isomorphism $\omega_A \cong A(a)$, with $\omega_A$ the canonical module and $a$ the $a$-invariant. Equivalently, for the Hilbert series

$H_A(t) = \frac{h(t)}{(1-t)^d}$

with $h(t)$ a polynomial, $A$ is Gorenstein if $h(t)$ is palindromic of degree $s$, i.e., $h_i = h_{s-i}$. For ASLs on distributive lattices, combinatorial criteria replace the homological ones: $A$ is Gorenstein if and only if the subposet of join-irreducible elements $J(L)$ is pure of some fixed rank.

3. Classification in the Plücker Algebra of $\mathrm{Gr}(2, n)$

The homogeneous coordinate ring of $\mathrm{Gr}(2, n)$ under the Plücker embedding is given by

$$S = \mathbb{K}[x_i, y_i: 1 \leq i \leq n] / (x_i y_j - x_j y_i =: p_{ij},~1 \leq i < j \leq n)$$

The set $L_n = \{p_{ij}: 1 \leq i < j \leq n\}$, with distributive-lattice order $p_{ij} \leq p_{k\ell}$ iff $i \leq k$ and $j \leq \ell$, underlies the ASL structure via the quadratic Plücker relations

$$Q_{ijkl} := p_{i\ell}p_{jk} - p_{ik}p_{j\ell} + p_{ij}p_{k\ell}$$

for $1 \leq i < j < k < \ell \leq n$ (Borovik et al., 11 Jan 2026). An ASL subalgebra $\mathbb{K}[L] \subset \mathbb{K}[L_n]$ generated by some sublattice $L \subset L_n$ is sought such that $\mathbb{K}[L]$ is Gorenstein.

Every $L \subset L_n$ yields an edge-graph $G_L$ on vertex set $[n]$ with $\{i,j\} \in E(G_L)$ iff $p_{ij} \in L$. The main classification result states:

• $\mathbb{K}[L]$ is Cohen–Macaulay and defined by the elimination ideal of the Plücker ideal iff $G_L$ is an interval graph.
• For $G_L$ an interval graph with maximal cliques $C_1 = [a_1, b_1], \dots, C_s = [a_s, b_s]$, $\mathbb{K}[L]$ is Gorenstein if and only if for each $1 \leq i < s$, $2 \leq |C_i \cap C_{i+1}| \leq 3$.

This condition on clique overlaps translates to the purity of $J(L)$ and enforces the palindromicity of the $h$-vector of $\mathbb{K}[L]$ (Borovik et al., 11 Jan 2026).

4. Gröbner Bases, Elimination, and Purity Criteria

The quadratic Plücker relations $Q_{ijkl}$ form a Gröbner basis for the Plücker ideal $I_{L_n}$ under suitable term orders (Borovik et al., 11 Jan 2026). If $L \subset L_n$ is downward-closed in $\Pi_n$, then the elimination ideal $$I_L = I_{L_n} \cap \mathbb{K}[p_{ij}: p_{ij} \in L]$$ is generated by those $Q_{ijkl}$ for which $p_{i\ell} \in L$. This ensures that all such $\mathbb{K}[L]$ are quadratic ASLs and Cohen–Macaulay.

By the theorem of Hibi–Stanley, $\mathbb{K}[L]$ is Gorenstein if and only if $J(L)$ is pure. In the interval graph context, join-irreducibles correspond to edges on the border of consecutive maximal cliques. The purity condition is equivalent to the condition $2 \leq |C_i \cap C_{i+1}| \leq 3$, ensuring the Gorenstein property (Borovik et al., 11 Jan 2026).

5. Combinatorial Enumerative Aspects and Explicit Examples

For $n=5$, all perfect (maximal dimension $2n-3=7$) compatible sublattices correspond to the 5 interval graphs on 5 vertices with clique intersection sizes 2 or 3. Representative maximal cliques are

Maximal Cliques	Condition on Intersections
$[1,5]$	N/A
$[1,4], [2,5]$	$|C_1 \cap C_2|=3$
$[1,3], [2,5]$	$|C_1 \cap C_2|=2$
$[1,4], [3,5]$	$|C_1 \cap C_2|=2$
$[1,3], [2,4], [3,5]$	$|C_1 \cap C_2|=2$, $|C_2 \cap C_3|=2$

In each case, the Hilbert series is palindromic, indicative of the Gorenstein property (e.g., for $L$ with a single clique $[1,5]$, $H(t) = (1 + 7t + 7t^2 + t^3)/(1-t)^3$, $a=0$).

Perfect compatible sublattices of $L_n$ are in bijection with non-crossing, non-nested arc arrangements on $n$ points, counted by the Catalan number $C_{n-2} = \frac{1}{n-1}\binom{2n-4}{n-2}$. The Gorenstein subalgebras (those satisfying $2 \leq |C_i \cap C_{i+1}| \leq 3$) are enumerated by a Fibonacci-type recursion, yielding a closed form: $$\frac{1}{5} \left[ (5 + 2\sqrt{5})\left(\frac{3+\sqrt{5}}{2}\right)^{n-4} + (5 - 2\sqrt{5})\left(\frac{3-\sqrt{5}}{2}\right)^{n-4} \right]$$ for the number of Gorenstein ASL subalgebras of $\mathrm{Gr}(2, n)$ of maximal Krull dimension $2n-3$ (Borovik et al., 11 Jan 2026).

6. Connections with Invariant Theory and LS Algebras

Discrete LS algebras over totally ordered sets, as established in (Chirivì, 2018), are homogeneous coordinate rings of irreducible projective toric varieties and admit realizations as invariant rings of finite abelian groups acting linearly without pseudo-reflections. The Gorenstein criterion in this context is that $G \subset \mathrm{SL}_{N+1}(\mathbb{C})$, which is equivalent to a certain numerical congruence on associated lcm’s $M_i$ along maximal chains. More generally, the Gorenstein property can thus be tested for general ASL subalgebras by checking this group-theoretic condition after flat degeneration to the discrete case. This connects the palindromicity of Hilbert series and purity of join-irreducible posets in the ASL context to the representation-theoretic structure of the algebra as a ring of invariants (Chirivì, 2018).

7. Combinatorial and Geometric Significance

Gorenstein ASL subalgebras of the Plücker algebra encode Stanley–Reisner rings of certain quasi-forests (interval graphs) with clique complexes consisting of stacked intervals with limited overlaps. Their enumeration via the Catalan and Fibonacci families ties these algebras to a broad spectrum of classical combinatorial structures, including non-crossing arc systems and nested partitions. On the algebraic side, the quadratic generation of elimination ideals via Gröbner bases persists throughout these subalgebras, linking their structure closely to toric ideals of “almost complete” graphs within the circular-arc family. This provides a broad combinatorial framework for the study and classification of Gorenstein ASL subalgebras, connecting concrete elimination and invariant-theoretic constructions with deep enumerative and homological symmetry conditions (Borovik et al., 11 Jan 2026, Chirivì, 2018).