Quantum Cohomology of Minuscule Homogeneous Spaces

Abstract

We study the quantum cohomology of (co)minuscule homogeneous varieties under a unified perspective. We show that three points Gromov-Witten invariants can always be interpreted as classical intersection numbers on auxiliary homogeneous varieties. Our main combinatorial tools are certain quivers, in terms of which we obtain a quantum Chevalley formula and a higher quantum Poincaré duality. In particular we compute the quantum cohomology of the two exceptional minuscule homogeneous varieties.

The quantum cohomology of flag varieties and the periodicity of the Schubert structure constants

“…Then c λ,µ,ν = c λt,µt,νt . Example 1.17 The first non-trivial example is the equality of the degree of σ 2,1,2,1 · σ 2,1,2,1 · σ 1,2,2,1 (in partition notation, σ 1 · σ 1 · σ 2 ) in G (2,4) and the degree of σ 2,1,2,1,2,2,1,1 · σ 2,1,2,1,2,2,1,1 · σ 1,2,2,1,2,2,1,1 (in partition notation, σ 3,2 · σ 3,2 · σ 4,2 ) in G (4,8). This equality holds because both of these degrees are equal to the GromovWitten invariant I F (2,4;8), (2,4) …”

“…Theorem 1. 4 If there exists a non-zero, three-pointed Gromov-Witten invariant of the two-step flag variety F(k 1 , k 2 ; n) of degree (d 1 , d 2 ), then the following inequalities hold.…”

Projected Richardson varieties and affine Schubert varieties

“…[20,Section 8]. For the classical types the explicit description can presumably be recovered from [7], and for other cominuscule types see [8]. Thus the coefficients π Thus (x , y ) (x, y) in Q J .…”

On the Cohomology of Some Exceptional Symmetric Spaces

“…As such, E III is also called the fourth Severi variety, a complex 16-dimensional projective variety in CP 26 , characterized as one of the four smooth projective varieties of small critical codimension in their ambient CP N , and that are unable to fill it through their secant and tangent lines [28]. The projective model of E VII is instead known as the Freudenthal variety, a complex 27-dimensional projective variety in CP 55 , considered for example in the sequel of papers [7]. Next, among the listed symmetric spaces, the five Wolf spaces E II = E 6 SU(6) · Sp (1) , E VI = E 7 Spin(12) · Sp (1) , E IX = E 8 E 7 · Sp (1) ,…”