Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties

Abstract

We show that the set of totally positive unipotent lower-triangular Toeplitz matrices in G L n GL_n forms a real semi-algebraic cell of dimension n − 1 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 G L n ( C ) GL_n(\mathbb {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 A case.

Quantum multiplication operators for Lagrangian and orthogonal Grassmannians

“…In this article, we heavily use one of Peterson's results, which states that the quantum cohomology ring of a homogeneous variety is isomorphic with the coordinate ring of so-called Peterson variety corresponding to the homogeneous variety ( [16], [17], [20], [2]). Unlike general homogeneous varieties, for these manifolds together with the Grassmannian, there is a much simpler isomorphic variety that can replace the Peterson variety.…”

Efficient quantum circuits for Toeplitz and Hankel matrices

“…Hankel matrices [19][20][21][22][23][24]. Moreover, any arbitrary matrix can be decomposed into a product of these matrices [16].…”

Untitled

“…, n − 1}, (1-1) where N : ‫ރ‬ n → ‫ރ‬ n denotes the principal nilpotent operator. These varieties have been much studied due to their relation to the quantum cohomology of the flag variety [Kostant 1996;Rietsch 2003]. Thus it is natural to study their topology, including the structure of their (equivariant) cohomology rings.…”