Parametrizations of flag varieties

By Robert J. Marsh, K. Rietsch. 2003

126 citations recorded by the source corpus.

For the flag variety G/B of a reductive algebraic group G we define and describe explicitly a certain (set-theoretical) cross-section φ : G/B → G. The definition of φ depends only on a choice of reduced expression for the longest element w0 in the Weyl group W . It assigns to any gB a representative g ∈ G together with a factorization into simple root subgroups and simple reflections. The cross-section φ is continuous along the components of Deodhar’s decomposition of G/B [6]. We introduce a generalization of the Chamber Ansatz of [2] and give formulas for the factors of g = φ(gB). These results are then applied to parametrize explicitly the components of the totally nonnegative part of the flag variety (G/B)≥0 defined by Lusztig [10], giving a new proof of Lusztig’s conjectured cell decomposition of (G/B)≥0 . We also give minimal sets of inequalities describing these cells.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/934d16897d670d86494fcfca1974dbaba122f054

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Regularity theorem for totally nonnegative flag varieties (2019, methodology): “In the latter case we require w(j−1) < w(j), unlike [MR04].”
Families of rationally simply connected varieties over surfaces and torsors for semisimple groups (2008, methodology): “We review some of the notation of [ MR04 , Section 2]. Let G be a connected reductive algebraic group over an algebraically closed field k. Choose a maximal torus T ⊂ G and let T ⊂ B be a Borel subgroup of G containing T. Let B − be the opposite Borel subgroup, so that T = B ∩ B − . Let U (resp.”
Total positivity, Grassmannians, and networks (2006, methodology): “Figure 19.3. A failed -condition implies a failed MR -condition”
Matching polytopes, toric geometry, and the totally non-negative Grassmannian (2009, background): “Other combinatorial aspects of (Grk,n)≥0, and more generally of (G/P)≥0, were investigated by Marsh and Rietsch [ 12 ], Rietsch [16], and the third author [22, 23].”
A mirror symmetric construction of qHT∗(G/P)(q) (2005, methodology): “Note only that our conventions differ from [28] in that B+ and B− are interchanged.”
Positroid varieties and cluster algebras (2019, methodology): “We describe a parametrization of this torus following [MR04].”
Total positivity for cominuscule Grassmannians (2007, background): “In this subsection we give background on distinguished and positive distinguished subexpressions; for more details, see [6] and [11].”