Closure relations for totally nonnegative cells in G/P

By K. Rietsch. 2005

68 citations recorded by the source corpus.

The totally nonnegative part of a partial flag variety G/P is known to have a decomposition into semi-algebraic cells. We show that the closure of a cell is again a union of cells and give a combinatorial description of the closure relations. The totally nonnegative cells are defined by intersecting the totally nonnegative part with a certain stratification of G/P defined by Lusztig. We also verify the same closure relations for these strata.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c223fa5d64d2ab43eef68f4e8d9e1b79acd9f28c

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Citations in the corpus

The twist for positroid varieties (2016, background): “The varieties in question had been studied earlier by Lusztig [19], Rietsch [31] and others, but the connection to Postnikov’s theory was not made in that earlier work.”
Bruhat Interval Polytopes (2014, background): “;0 and a ∈ R>0, we have χ′ i(t)−1a > 0. Rietsch also showed that the closure of each cell of (G/PJ)≥0 is a union of cells, and described when one cell of (G/PJ)≥0 lies in the closure of another [Rie06]. Using this description, it is easy to determine the set of 0-cells contained in the closure”
Positroid varieties: juggling and geometry (2009, methodology): “roid varieties form a stratiﬁcation of the Grassmannian. Thus for f∈ Bound(k,n)we have Πf= a f′≥f ˚Π f′ =XJ 1 ∩χ(XJ 2 )∩···∩χn−1(X Jn). where (J1,J2,...,Jn)∈ Jugg(k,n)corresponds to f. Proof. Rietsch [Rie06] described the closure relations of Lusztig’s stratiﬁcation of partial ﬂag varieties; see als”
A Formula for Plücker Coordinates Associated with a Planar Network (2008, unknown): “C O ] 1 5 A pr 2 00”
Regularity theorem for totally nonnegative flag varieties (2019, unknown): “This includes the shellability of Q due to Williams [40], and some topological results [31, 21] on Richardson varieties.”
Projections of Richardson Varieties (2010, methodology): “For the other inclusion, using Rietsch’s statement for w ∈ WP and Lemmata 2.3, 3.1, and 2.4, we may assume that the result holds for x ≤P y and prove it for u ≤P w, where u = xs and w = ys are length-additive.”
Cluster structures on strata of flag varieties (2014, background): “s of the complex strata of (5) with (XK)≥0 give a cell decomposition of (XK)≥0, and this was proved by Rietsch [R1]. This cell decomposition was later studied in detail by Marsh-Rietsch [MR], Rietsch [R2], and Rietsch-Williams [RW ]. On the other hand, Goodearl and Yakimov [GY1] have given a new int”