On two notions of total positivity for partial flag varieties

TL;DR

This paper compares two notions of total positivity for partial flag varieties, showing they coincide only when the flag dimensions are consecutive, and explores their cell decompositions and symmetries.

Contribution

It establishes the precise conditions under which Lusztig's and Postnikov's positivity notions agree for partial flag varieties, and analyzes their cell and matroid decompositions.

Findings

Positivity notions agree iff the dimensions are consecutive.

Cell decomposition matches matroid decomposition under the same condition.

Identifies compatibility of positivity with cyclic group actions.

Abstract

Given integers $1 \le k_1 < \cdots < k_l \le n-1$, let $\text{Fl}_{k_1,\dots,k_l;n}$ denote the type $A$ partial flag variety consisting of all chains of subspaces $(V_{k_1}\subset\cdots\subset V_{k_l})$ inside $\mathbb{R}^n$, where each $V_k$ has dimension $k$. Lusztig (1994, 1998) introduced the totally positive part $\text{Fl}_{k_1,\dots,k_l;n}^{>0}$ as the subset of partial flags which can be represented by a totally positive $n\times n$ matrix, and defined the totally nonnegative part $\text{Fl}_{k_1,\dots,k_l;n}^{\ge 0}$ as the closure of $\text{Fl}_{k_1,\dots,k_l;n}^{>0}$. On the other hand, following Postnikov (2007), we define $\text{Fl}_{k_1,\dots,k_l;n}^{\Delta>0}$ and $\text{Fl}_{k_1,\dots,k_l;n}^{\Delta\ge 0}$ as the subsets of $\text{Fl}_{k_1,\dots,k_l;n}$ where all Pl\"{u}cker coordinates are positive and nonnegative, respectively. It follows from the definitions that Lusztig's total positivity implies Pl\"{u}cker positivity, and it is natural to ask when these two notions of positivity agree. Rietsch (2009) proved that they agree in the case of the Grassmannian $\text{Fl}_{k;n}$, and Chevalier (2011) showed that the two notions are distinct for $\text{Fl}_{1,3;4}$. We show that in general, the two notions agree if and only if $k_1, \dots, k_l$ are consecutive integers. We give an elementary proof of this result (including for the case of Grassmannians) based on classical results in linear algebra and the theory of total positivity. We also show that the cell decomposition of $\text{Fl}_{k_1,\dots,k_l;n}^{\ge 0}$ coincides with its matroid decomposition if and only if $k_1,\dots,k_l$ are consecutive integers, which was previously only known for complete flag varieties, Grassmannians, and $\text{Fl}_{1,3;4}$. Finally, we determine which notions of positivity are compatible with a natural action of the cyclic group of order $n$ that rotates the index set.