Representation ring of Levi subgroups versus cohomology ring of flag varieties

We study the equivariant oriented cohomology ring $\mathtt{h}_{T}(G/P)$ of partial flag varieties using the moment map approach. We define the right Hecke action on this cohomology ring, and then prove that the respective Bott–Samelson classes in $\mathtt{h}_{T}(G/P)$ can be obtained by applying this action to the fundamental class of the identity point, hence generalizing previously known results of Chow groups by Brion, Knutson, Peterson, Tymoczko and others. Our main result concerns the equivariant oriented cohomology theory $\mathfrak{h}$ corresponding to the 2-parameter Todd genus. We give a new interpretation of Deodhar’s parabolic Kazhdan–Lusztig basis, i.e., we realize it as some cohomology classes (the parabolic Kazhdan–Lusztig (KL) Schubert classes) in $\mathfrak{h}_{T}(G/P)$. We make a positivity conjecture, and a conjecture about the relationship of such classes with smoothness of Schubert varieties. We also prove the latter in several special cases.

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Extended quadratic algebra and a model of the equivariant cohomology ring of flag varieties

St Petersburg Mathematical Journal ◽

10.1090/s1061-0022-2011-01151-3 ◽

2011 ◽

Vol 22 (3) ◽

pp. 447-447

Author(s):

A. N. Kirillov ◽

T. Maeno

Keyword(s):

Equivariant Cohomology ◽

Cohomology Ring ◽

Quadratic Algebra ◽

Flag Varieties

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Residual categories for (co)adjoint Grassmannians in classical types

Compositio Mathematica ◽

10.1112/s0010437x21007090 ◽

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.

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Chevalley-Monk and Giambelli formulas for Peterson Varieties

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2449 ◽

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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