Elimination ideals of Pl\"ucker ideals and algebras with straightening laws

TL;DR

This paper investigates the algebraic structure of the Grassmannian's Pl"ucker ideal, focusing on quadratic relations, and characterizes certain Gorenstein subalgebras with straightening laws.

Contribution

It provides a combinatorial characterization of Gorenstein ASL subalgebras within the Pl"ucker algebra of Grassmannians of lines.

Findings

Quadratic Pl"ucker relations form a Gr"obner basis.

Identification of Gorenstein ASL subalgebras.

Characterization of projections of the Grassmannian of lines.

Abstract

It is well known that the Pl\"ucker ideal defining the Grassmannian is generated by quadratic Pl\"ucker relations. These relations form a reverse lexicographic Gr\"obner basis and endow the Pl\"ucker 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 Pl\"ucker algebra of $\mathrm{Gr}(2,n)$.