Birational sequences for the Grassmannian Gr(3,n)

TL;DR

This paper explores iterated birational sequences for the Grassmannian Gr(3,n), analyzing their associated valuations, initial forms of Plücker relations, and the structure of cones in the tropical Grassmannian, with computational classification for Gr(3,6).

Contribution

It introduces a new class of iterated birational sequences for Gr(3,n) and characterizes their associated cones and valuations, extending previous work on Grassmannian degenerations.

Findings

Initial forms of Plücker relations are binomial.

Cones depend only on the first two indices in some cases.

Classification of cones for Gr(3,6) up to symmetry.

Abstract

Following the ideas of Bossinger and Fang, Fourier, and Littelman, we study iterated sequences for the Grassmannian $\operatorname{Gr} (3, n)$ as a special class of birational sequences. For each iterated sequence $S$, there is a weighting matrix $M_{S}$ corresponding to a valuation on the rational coordinate ring and we show that the initial form of a Pl\"{u}cker relation $\operatorname{in}_{M_S} (R_{I,J} )$ is binomial. We show that, in some cases, the cones $C_S$ in the tropical Grassmannian that satisfy $\operatorname{in}_{M_S} (\mathcal{I}_{3,n}) = \operatorname{in}_{C_S} (\mathcal{I}_{3,n})$ only depend on the first two indices used in each iteration. In the case of $\operatorname{Gr} (3, 6)$, these cones are obtained computationally and are classified up to automorphism induced by the symmetric group $S_6$.