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

2009 ◽  
Vol 346 (2) ◽  
pp. 419-447 ◽  
Author(s):  
Izzet Coskun
Keyword(s):  
Quantum Cohomology ◽  
Flag Varieties ◽  
Structure Constants

scholarly journals The Quantum Cohomology Ring of Flag Varieties

1999 ◽  
Vol 351 (7) ◽  
pp. 2695-2729 ◽  
Author(s):  
Ionuţ Ciocan-Fontanine
Keyword(s):  
Cohomology Ring ◽  
Quantum Cohomology ◽  
Flag Varieties

scholarly journals Classical Aspects of Quantum Cohomology of Generalized Flag Varieties

2011 ◽  
Vol 2012 (16) ◽  
pp. 3706-3722 ◽  
Author(s):  
Naichung Conan Leung ◽  
Changzheng Li
Keyword(s):  
Quantum Cohomology ◽  
Flag Varieties

scholarly journals Residual categories for (co)adjoint Grassmannians in classical types

2021 ◽  
Vol 157 (6) ◽  
pp. 1172-1206
Author(s):  
Alexander Kuznetsov ◽  
Maxim Smirnov
Keyword(s):  
Automorphism Group ◽  
Cohomology Ring ◽  
Quantum Cohomology ◽  
Algebraic Groups ◽  
Fano Variety ◽  
Flag Varieties ◽  
Picard Number ◽  
Coherent Sheaves ◽  
Small Quantum ◽  
Exceptional Collection

In our previous paper we suggested a conjecture relating the structure of the small quantum cohomology ring of a smooth Fano variety of Picard number 1 to the structure of its derived category of coherent sheaves. Here we generalize this conjecture, make it more precise, and support it by the examples of (co)adjoint homogeneous varieties of simple algebraic groups of Dynkin types $\mathrm {A}_n$ and $\mathrm {D}_n$ , that is, flag varieties $\operatorname {Fl}(1,n;n+1)$ and isotropic orthogonal Grassmannians $\operatorname {OG}(2,2n)$ ; in particular, we construct on each of those an exceptional collection invariant with respect to the entire automorphism group. For $\operatorname {OG}(2,2n)$ this is the first exceptional collection proved to be full.

scholarly journals GAUSS–MANIN SYSTEM AND THE VIRTUAL STRUCTURE CONSTANTS

2002 ◽  
Vol 13 (05) ◽  
pp. 445-477 ◽  
Author(s):  
MASAO JINZENJI
Keyword(s):  
Differential Equations ◽  
General Type ◽  
Quantum Cohomology ◽  
Rational Curves ◽  
Previous Article ◽  
Projective Hypersurface ◽  
Structure Constants

In this paper, we discuss some applications of Givental's differential equations to enumerative problems on rational curves in projective hypersurfaces. Using this method, we prove some of the conjectures on the structure constants of quantum cohomology of projective hypersurfaces, proposed in our previous article. Moreover, we clarify the correspondence between the virtual structure constants and Givental's differential equations when the projective hypersurface is Calabi–Yau or general type.

scholarly journals Chevalley-Monk and Giambelli formulas for Peterson Varieties

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Elizabeth Drellich
Keyword(s):  
Cohomology Ring ◽  
Quantum Cohomology ◽  
Type A ◽  
Flag Variety ◽  
Flag Varieties ◽  
Dimensional Torus ◽  
One Dimensional ◽  
International Audience ◽  
Schubert Classes ◽  
Partial Flag Varieties

International audience A Peterson variety is a subvariety of the flag variety $G/B$ defined by certain linear conditions. Peterson varieties appear in the construction of the quantum cohomology of partial flag varieties and in applications to the Toda flows. Each Peterson variety has a one-dimensional torus $S^1$ acting on it. We give a basis of Peterson Schubert classes for $H_{S^1}^*(Pet)$ and identify the ring generators. In type A Harada-Tymoczko gave a positive Monk formula, and Bayegan-Harada gave Giambelli's formula for multiplication in the cohomology ring. This paper gives a Chevalley-Monk rule and Giambelli's formula for all Lie types.

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