Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties

2001/12/03 by Konstanze Rietsch, Rietsch, Konstanze · 9 citations
Mathematics · #14N15 #14N35 #15A48 #20G20 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA) #math.AG #math.CO #math.QA #msc:14N15 #msc:14N35 #msc:15A48 #msc:20G20

paper · pdf · doi:10.48550/arxiv.math/0112024

arxiv created 2001/12/03 · arxiv updated 2009/11/30

Abstract

We show that the set of totally positive unipotent lower-triangular Toeplitz matrices in GL_n form a real semi-algebraic cell of dimension n-1. Furthermore we prove a natural cell decomposition for its closure. The proof uses properties of the quantum cohomology rings of the partial flag varieties of GL_n(\C) relying in particular on the positivity of the structure constants, which are enumerative Gromov--Witten invariants. We also give a characterization of total positivity for Toeplitz matrices in terms of the (quantum) Schubert classes. This work builds on some results of Dale Peterson's which we explain with proofs in the type A case.

Cited by

• Inverse limits of various posets
• Coordinate rings of regular semisimple Hessenberg varieties and cohomology rings of regular nilpotent Hessenberg varieties
• Totally nonnegative Peterson variety and strongly dominant weight polytope
• Markov chains on Weyl groups from the geometry of the flag variety
• Totally positive Toeplitz matrices: classical and modern
• Entrywise preservers of sign regularity
• Structure constants of Peterson Schubert calculus
• An anticanonical perspective on G/P Schubert varieties
• Unfolding of equivariant F-bundles and application to the mirror symmetry of flag varieties

Related

• Totally positive Toeplitz matrices: classical and modern
• Tropical Toeplitz matrices and parametrisations
• Quantum Cohomology of Grassmannians and Total Positivity
• Quantum cohomology of partial flag manifolds
• Positivity of minuscule quantum K-theory