Hochschild cohomology of generalised Grassmannians

BELMANS, Pieter; Smirnov, Maxim

doi:10.4171/DM/912

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Hochschild cohomology of generalised Grassmannians

BELMANS, Pieter; Smirnov, Maxim

2023 • In Documenta Mathematica, 28 (1), p. 11 - 53

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https://hdl.handle.net/10993/57980

DOI
10.4171/DM/912

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Keywords :

Bott vanishing; Hochschild cohomology; Hochschild–Kostant–Rosenberg decomposition; homogeneous spaces; partial flag varieties; Mathematics (all)

Abstract :

[en] We compute the Hochschild–Kostant–Rosenberg decomposition of the Hochschild co-homology of generalised Grassmannians, i.e., partial flag varieties associated to maximal parabolic subgroups in a simple algebraic group, in terms of representation-theoretic data. We explain how the decomposition is concentrated in global sections for the (co)minuscule and (co)adjoint generalised Grassmannians, and conjecture that for (almost) all other cases the same vanishing of the higher cohomology does not hold. Our methods give an explicit partial description of the Gerstenhaber algebra structure for the Hochschild cohomology of cominuscule and adjoint generalised Grass-mannians. We also consider the case of adjoint partial flag varieties in type A, which are associated to certain submaximal parabolic subgroups.

Disciplines :

Mathematics

Author, co-author :

BELMANS, Pieter ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

Smirnov, Maxim; Instititut für Mathematik, Universität Augsburg, Augsburg, Germany

External co-authors :

yes

Language :

English

Title :

Hochschild cohomology of generalised Grassmannians

Publication date :

2023

Journal title :

Documenta Mathematica

ISSN :

1431-0635

eISSN :

1431-0643

Publisher :

EMS Press

Volume :

Issue :

Pages :

11 - 53

Peer reviewed :

Peer Reviewed verified by ORBi

Funding text :

The first author acknowledges the support of the FWO (Research Foundation— Flanders). The second author was partially supported by the Deutsche Forschungsge-meinschaft (DFG, German Research Foundation) – Projektnummer 448537907. We want to thank the Max Planck Institute for Mathematics for the pleasant working conditions during the start of this project, and its high performance computing infrastructure.Funding. The first author acknowledges the support of the FWO (Research Foundation— Flanders). The second author was partially supported by the Deutsche Forschungsge-meinschaft (DFG, German Research Foundation) – Projektnummer 448537907. We want to thank the Max Planck Institute for Mathematics for the pleasant working conditions during the start of this project, and its high performance computing infrastructure.

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