nLab Grassmann necklace

Definition

A $(k,n)$-Grassmann necklace $I = (I_1, I_2,\ldots, I_n)$ is an $n$-tuple of $k$-element subsets of $[n]$ such that for each $a\in[n]$,

(1) $I_{a+1} = I_a if a\notin I_a$

(2) $I_{a+1} = I_a \backslash \{a\}\cup \{b\}$ for some $b\in I_a$ if $a\in I_a$

with subcripts taken modulo $n$.

Properties

Grassmann necklaces parametrize totally nonnegative points in Grassmannian.

Literature

Introduced in

• Alexander Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764

category: combinatorics