On the quantum cohomology of adjoint varieties

Abstract

We study the quantum cohomology of quasi-minuscule and quasi-cominuscule homogeneous spaces. The product of any two Schubert cells does not involve powers of the quantum parameter higher than 2. With the help of the quantum to classical principle we give presentations of the quantum cohomology algebras. These algebras are semi-simple for adjoint non coadjoint varieties and some properties of the induced strange duality are shown.

Quantum multiplication operators for Lagrangian and orthogonal Grassmannians

“…We refer to §2 of [5] for the notion of minuscule or cominuscule homogeneous varieties of Proposition 3.4. Note that OG(n) is minuscule and LG(n) is cominuscule (see §2 of [5]).…”

“…Note that OG(n) is minuscule and LG(n) is cominuscule (see §2 of [5]). Thus, the rings qH * (OG(n)) q=1 and qH * (LG(n)) q=1 are semisimple.…”

Root-theoretic Young diagrams and Schubert calculus: Planarity and the adjoint varieties

“…Note that use of RYDs, even for (co)adjoint varieties, is not mandatory: there is a different way to index their Schubert varieties, see [6]. For isotropic Grassmannians of classical type, [18,19] uses another way, and [4] yet another.…”

Parabolic Kazhdan–Lusztig polynomials for quasi-minuscule quotients

“…In this work we study the quasi-minuscule quotients that are not minuscule. These quotients (also known as (co)-adjoint quotients) have been classified (see, e.g., [6]) and there are three infinite families and four exceptional ones. Using the standard notation for the classification of the finite Coxeter systems, the non-trivial (co)-adjoint quotients are: (A n , S \ {s 1 , s n }), (B n , S \ {s n−2 }), (D n , S\{s n−2 }), (E 6 , S\{s 0 }), (E 7 , S\{s 1 }), (E 8 , S\{s 7 }), and (F 4 , S\{s 4 }), where we number the generators as in [1] (see Appendix A1 and Exercises 20,21,22,23 in Chapter 8, and also below).…”