$\LE$-diagrams and totally positive bases inside the nonnegative Grassmannian
| dc.creator | OH, Suho | |
| dc.date | 2008-09-04 | |
| dc.date.accessioned | 2026-07-07T10:00:43Z | |
| dc.date.available | 2026-07-07T10:00:43Z | |
| dc.description | There is a cell decomposition of the nonnegative Grassmannian. For each cell, totally positive bases(TP-bases) is defined as the minimal set of Plücker variables such that all other nonzero Plücker 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 Plücker variables. Those set of Plücker variables form a TP-bases of the cell. Using mutations coming from 3-term Plücker relation, we conjecture that they can be mutated to a special set of Plücker variable $§$. All other nonzero Plücker variables in the cell will be expressed as a subtraction-free Laurent polynomial in variables of $§$. 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. | |
| dc.description | 37 pages, 36 figures | |
| dc.identifier | https://arxiv.org/abs/0809.0871 | |
| dc.identifier | http://arxiv.org/abs/0809.0871 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/168401 | |
| dc.subject | Combinatorics | |
| dc.title | $\LE$-diagrams and totally positive bases inside the nonnegative Grassmannian | |
| dc.type | text |