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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

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.

Quiver Grassmannians, cyclic quiver, equivariant cohomology, GKM theory
Mathematical Subject Classification
Primary: 16G20
Secondary: 14L30, 14M15
Received: 4 January 2021
Revised: 27 October 2022
Accepted: 13 May 2023
Published: 8 October 2023
Martina Lanini
Dipartimento di Matematica
Università di Roma Tor Vergata
Alexander Pütz
Institute of Mathematics
University Paderborn

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