Konstanze Rietsch

Parametrizations of flag varieties

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.

Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties

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 Petersons which we explain with proofs in the type A case.

Quantum cohomology rings of Grassmannians and total positivity

We give a proof of a result of D. Peterson’s identifying the quantum cohomology ring of a Grassmannian with the reduced coordinate ring of a certain subvariety of GLn. The totally positive part of this subvariety is then constructed and we give closed formulas for the values of the Schubert basis elements on the totally positive points. We then use the developed methods to give a new proof of a formula of Vafa and Intriligator and Bertram for the structure constants (Gromov–Witten invariants). Finally, we use the positivity of these Gromov–Witten invariants to prove certain inequalities for Schur polynomials at roots of unity.

A Mirror Construction for the Totally Nonnegative Part of the Peterson Variety

We explain how A. Giventals mirror symmetric family to the type A flag variety and its proposed generalization to partial flag varieties by Batyrev, Ciocan-Fontanine, Kim and van Straten relate to the Peterson variety Y in SL_n/B. We then use this theory to describe the totally nonnegative part of Y.

A comparison of Landau-Ginzburg models for odd dimensional Quadrics

In [Rie08], the second author dened a Landau-Ginzburg model for homogeneous spaces G=P , as a regular function on an ane subvariety of the Langlands dual group. In this paper, we reformulate this LG model in the case of the odd-dimensional quadric Q2m 1 as a regular function Wt on the complement X of a particular anticanonical divisor in the projective space P 2m = P(H (Q2m 1;C) ). In fact, we express Wt in

Errata to “Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties”

Remark 4.3. If P is the parabolic subgroup, thenGj is a well-defined (regular) function on the Bruhat cell BwPB/B precisely in the case m ∈ I = {n1, . . . , nk}. Proof of Theorem 4.2. (1) is proved in [33]. See also Lemma 2.3 in [35]. We will deduce (2) very explicitly from the ASK presentation. We begin by defining a particular system of coordinates on the affine space BwPB/B. For indexing purposes introduce sets Ω,Ω1,Ω2 defined by Ω := {(r,m) ∈ Z | n1 ≤ r < n, and 1 ≤ m ≤ nl if nl ≤ r < nl+1}, Ω1 := {(nl,m) ∈ Z | where l ∈ {1, . . . , k} and 1 ≤ m ≤ nl}, and Ω2 := Ω \ Ω1. Consider the polynomial rings C[Ω] := C[g m; (r,m) ∈ Ω] and C[Ωi] := C[g m; (r,m) ∈ Ωi] for i = 1, 2. We have n× (nl+1 −nl)-matrices U (l) Ω1 over C[Ω1] defined by

The infinitesimal cone of a totally positive semigroup

Given a complex reductive linear algebraic group split over R with a fixed pinning, it is shown that all elements of the Lie algebra g infinitesimal to the totally positive subsemigroup G>( of G lie in the totally positive cone g>o C g.

TOTAL POSITIVITY, FLAG VARIETIES AND QUANTUM COHOMOLOGY

AbstractThe aim of this contribution is to give an introduction to some aspects of quantum cohomology rings of flag varieties especially explaining their relationship with total positivity. The results outlined here are written up in greater generality elsewhere1. In this paper we give a simplified exposition focusing on the variety of complete flags. We explain how the quantum cohomology ring for the full flag variety can be used to study totally positive uni-triangular Toeplitz matrices. The special case considered here contains most of the main ideas but avoids some of the technical difficulties which come up in the more general setting of partial flag varieties and total nonnegativity.