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Khovanskii bases of subalgebras arising from finite distributive lattices

Akihiro Higashitani, Koji Matsushita, Koichiro Tani

TL;DR

This work classifies when the generating set $\mathcal{F}_{L(P)}$ forms a Khovanskii basis for the subalgebra $\mathcal{R}(L(P))$ arising from a finite distributive lattice $L(P)$. It proves an equivalence: for $P$ irreducible under ordinal sum, $\mathcal{F}_{L(P)}$ is a Khovanskii basis for a compatible monomial order if and only if $P$ is $\{(2+2),(1+1+1)\}$-free and equivalently $L(P)$ is a generalized snake poset. The criterion uses the co-comparability graph $G_{L(P)}$ and a subduction test on primitive even closed walks to decide the basis property, with explicit positive (Plücker-algebra) and negative (divisor lattice of $36$, boolean lattice) examples. The results tie Khovanskii theory to toric/Plücker algebras and to combinatorial poset structure, delivering a complete combinatorial classification in terms of generalized snake posets and their forbidden patterns via Birkhoff's representation.

Abstract

The notion of Khovanskii bases was introduced by Kaveh and Manon. It is a generalization of the notion of SAGBI bases for a subalgebra of polynomials. The notion of SAGBI bases was introduced by Robbiano and Sweedler as an analogue of Gröbner bases in the context of subalgebras. A Hibi ideal is an ideal of a polynomial ring that arises from a distributive lattice. For the development of an analogy of the theory of Hibi ideals and Gröbner bases within the framework of subalgebras, in this paper, we investigate when the set of the polynomials associated with a distributive lattice forms a Khovanskii basis of the subalgebras it generates. We characterize such distributive lattices and their underlying posets. In particular, generalized snake posets and $\{(2+2),(1+1+1)\}$-free posets appear as the characterization.

Khovanskii bases of subalgebras arising from finite distributive lattices

TL;DR

This work classifies when the generating set forms a Khovanskii basis for the subalgebra arising from a finite distributive lattice . It proves an equivalence: for irreducible under ordinal sum, is a Khovanskii basis for a compatible monomial order if and only if is -free and equivalently is a generalized snake poset. The criterion uses the co-comparability graph and a subduction test on primitive even closed walks to decide the basis property, with explicit positive (Plücker-algebra) and negative (divisor lattice of , boolean lattice) examples. The results tie Khovanskii theory to toric/Plücker algebras and to combinatorial poset structure, delivering a complete combinatorial classification in terms of generalized snake posets and their forbidden patterns via Birkhoff's representation.

Abstract

The notion of Khovanskii bases was introduced by Kaveh and Manon. It is a generalization of the notion of SAGBI bases for a subalgebra of polynomials. The notion of SAGBI bases was introduced by Robbiano and Sweedler as an analogue of Gröbner bases in the context of subalgebras. A Hibi ideal is an ideal of a polynomial ring that arises from a distributive lattice. For the development of an analogy of the theory of Hibi ideals and Gröbner bases within the framework of subalgebras, in this paper, we investigate when the set of the polynomials associated with a distributive lattice forms a Khovanskii basis of the subalgebras it generates. We characterize such distributive lattices and their underlying posets. In particular, generalized snake posets and -free posets appear as the characterization.
Paper Structure (4 sections, 10 theorems, 22 equations, 12 figures, 1 algorithm)

This paper contains 4 sections, 10 theorems, 22 equations, 12 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Application of Khovanskii basis criterion
  4. generalized snake posets and $\{(2+2), (1+1+1)\}$-free posets

Key Result

Theorem 1.1

Let $P$ be a finite poset and assume that $P$ is irreducible with respect to ordinal sum. Then the following conditions are equivalent:

Figures (12)

  • Figure 1: A poset "$(2+2)$"
  • Figure 2: A poset "$(1+1+1)$"
  • Figure 3: $L(P)$
  • Figure 4: $G_{L(P)}$
  • Figure 5: $L(P)$
  • ...and 7 more figures

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Proposition 2.2: See SturmfelsLectureNote
  • Lemma 2.3: OhsugiHibi1999
  • Lemma 2.4: See BinomialIdeals
  • Definition 2.5
  • Proposition 2.6: See, e.g., MillerSturmfels
  • Lemma 3.1
  • proof
  • ...and 17 more