GKM-theory for torus actions on cyclic quiver Grassmannians

Lanini, Martina; Pütz, Alexander

doi:10.2140/ant.2023.17.2055

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GKM-theory for torus actions on cyclic quiver Grassmannians

Martina Lanini and Alexander Pütz

Vol. 17 (2023), No. 12, 2055–2096

DOI: 10.2140/ant.2023.17.2055

Abstract

We define and investigate algebraic torus actions on quiver Grassmannians for nilpotent representations of the equioriented cycle. Examples of such varieties are type $A$ flag varieties, their linear degenerations and finite-dimensional approximations of both the affine flag variety and affine Grassmannian for ${GL}_{n}$ . We show that these quiver Grassmannians equipped with our specific torus action are GKM-varieties and that their moment graph admits a combinatorial description in terms of the coefficient quiver of the underlying quiver representations. By adapting to our setting results by Gonzales, we are able to prove that moment graph techniques can be applied to construct module bases for the equivariant cohomology of the quiver Grassmannians listed above.

Keywords

Quiver Grassmannians, cyclic quiver, equivariant cohomology, GKM theory

Mathematical Subject Classification

Primary: 16G20

Secondary: 14L30, 14M15

Milestones

Received: 4 January 2021

Revised: 27 October 2022

Accepted: 13 May 2023

Published: 8 October 2023

Authors

Martina Lanini
Dipartimento di Matematica Università di Roma Tor Vergata Rome Italy

Alexander Pütz
Institute of Mathematics University Paderborn Germany

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Vol. 17, No. 12, 2023