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 $b\in [n]$ if $a\in I_a$

with subscripts taken modulo $n$.

Note that we allow $b=a$ in case (2), but we cannot have $b \in I_\{a\}\backslash \{a\}$, or else I_{a+1} would not have $k$ elements.

Properties

• Grassmann necklaces parameterize totally nonnegative points in Grassmannian.

• Grassmann necklaces are in bijection with decorated permutations.

Literature

Introduced in

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

category: combinatorics