On two notions of total positivity for partial flag varieties

Abstract

Given integers 1≤k1<⋯<kl≤n−1, let Flk1,…,kl;n denote the type A partial flag variety consisting of all chains of subspaces (Vk1⊂⋯⊂Vkl) inside Rn, where each Vk has dimension k. Lusztig (1994, 1998) introduced the totally positive part Flk1,…,kl;n>0 as the subset of partial flags which can be represented by a totally positive n×n matrix, and defined the totally nonnegative part Flk1,…,kl;n≥0 as the closure of Flk1,…,kl;n>0. On the other hand, following Postnikov (2007), we define Flk1,…,kl;nΔ>0 and Flk1,…,kl;nΔ≥0 as the subsets of Flk1,…,kl;n where all Plücker coordinates are positive and nonnegative, respectively. It follows from the definitions that Lusztig's total positivity implies Plücker 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 Flk;n, and Chevalier (2011) showed that the two notions are distinct for Fl1,3;4. We show that in general, the two notions agree if and only if k1,…,kl 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 Flk1,…,kl;n≥0 coincides with its matroid decomposition if and only if k1,…,kl are consecutive integers, which was previously only known for complete flag varieties, Grassmannians, and Fl1,3;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.