$\LE$-diagrams and totally positive bases inside the nonnegative Grassmannian

TL;DR

This paper explores the structure of totally positive bases within the nonnegative Grassmannian, introducing $ ext{LE}$-diagrams, conjecturing mutation relations, and connecting to cluster algebras for certain cell classes.

Contribution

It defines $ ext{LE}$-diagrams and TP-bases for nonnegative Grassmannian cells, proposes mutation conjectures, and links these structures to cluster algebras in specific cases.

Findings

Conjecture that $ ext{LE}$-diagrams can be mutated to a special Plücker set $ ext{S}$.

Proved the conjecture for weakly-connected cells.

Established connections between lattice-path-matroid cells and cluster algebras.

Abstract

There is a cell decomposition of the nonnegative Grassmannian. For each cell, totally positive bases(TP-bases) is defined as the minimal set of Pl\"ucker variables such that all other nonzero Pl\"ucker variables in the cell can be expressed in those variables in a subtraction-free rational function. This is the generalization of the TP-bases defined for nonnegative part of $GL_k$ defined in \cite{FZ5}. For each cell, we have a $\LE$-diagram and a natural way to label the dots inside the diagram with Pl\"ucker variables. Those set of Pl\"ucker variables form a TP-bases of the cell. Using mutations coming from 3-term Pl\"ucker relation, we conjecture that they can be mutated to a special set of Pl\"ucker variable $\S$. All other nonzero Pl\"ucker variables in the cell will be expressed as a subtraction-free Laurent polynomial in variables of $\S$. We define TP-diagrams to express the transformation procedure in terms of moves on a diagram. We will prove the conjecture for certain class of cells called weakly-connected cells. Then we will study the connection with cluster algebras through lattice-path-matroid cells.