Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties

By K. Rietsch. 2001

132 citations recorded by the source corpus.

We show that the set of totally positive unipotent lower-triangular Toeplitz matrices in GLn 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 GLn(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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b2e18098384afca574e1bb818ded24c36171a6ac

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In Algebra and Number Theory

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Citations in the corpus

A mirror symmetric construction of qHT∗(G/P)(q) (2005, background): “In the special case of qH(SLn/P ) an ad hoc proof of this relationship between the Schubert classes is given in [31].”
A positive Monk formula in the S1‐equivariant cohomology of type A Peterson varieties (2009, background): “Hessenberg varieties arise in many areas of mathematics, including geometric representation theory [12, 31, 33], numerical analysis [8], mathematical physics [24, 29], combinatorics [10], and algebraic geometry [5, 6, 34, 35].”
The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A (2015, background): “A special case of Hessenberg varieties called the Peterson variety Petn arises in the study of the quantum cohomology of the flag variety [31, 39], and more generally, geometric properties and invariants of many different types of Hessenberg varieties (including in Lie types other than A) have been ”
Quantum cohomology of G/P and homology of affine Grassmannian (2007, result): “which has remained unpublished for a decade; see however [12] and [24] for statements of some of Peterson’s results.”
Exponents for B-stable ideals (2004, background): “ R be the ambient vector space containing the coroot lattice Q”
Quantum Cohomology of Grassmannians (2001, background): “The same idea can be used to prove the quantum Monk’s formula for flag varieties SLn=B [4, 7] and even the quantum Pieri formula for partial flag varieties SLn=P [5, 6].”