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Generalities on GKM graphs

Definition

Note

Although the definition below is written for smooth algebraic GKM spaces, the functionality of this package applies equally well to Hamiltonian GKM space as discussed in the supporting article of this paper. By GKM space we will mean either a GKM variety as defined below, or a Hamiltonian GKM space.

GKM spaces have been introduced in [GKM98]. Many of the combinatorial definitions in this package follow [GZ00]. For the purpose of this package, by GKM variety we mean a smooth projective varieties over $\mathbb{C}$ with an algebraic torus action such that the action has a finite number of fixed points and a finite number of 1-dimensional orbits.

The GKM graph associated to a torus $T$ acting on a GKM variety $X$ is the following datum:

  • A graph having the fixed points as vertices, such that two vertices are connected by an unoriented) edge if there is a 1-dimensional orbit passing through the two fixed points.
  • An axial function $\mathrm{w}\colon E \rightarrow M$ from the set of oriented edges of the graph to the weight lattice $M$ of $T$. (By oriented edge we mean an unoriented edge of the graph plus a choice of orientation.)

In this package, the codomain $M$ of the axial function can be a free $\mathbb{Z}$-module or a free $\mathbb{Q}$-module. Since the GKM graph of a GKM variety is always regular (with the valency of every vertex being the complex dimension of the space), this package assumes that GKM graphs are regular.

Famous examples of GKM varities include projective space, (generalised/partial) flag varieties, smooth Schubert varieties, and smooth toric varieties, see Standard Constructions.

This package represents GKM varieties purely by their GKM graphs. For some applications, the additional datum of a GKM connection is necessary, see Connections.

Note

We have added support for non-compact GKM spaces, which arise for example from quasi-projective algebraic GKM spaces or as total space of GKM vector bundles over GKM spaces. On the level of GKM graphs, this means that standalone flags (sometimes called semi-infinite edges) are allowed:

  • Each vertex $p$ of the GKM graph has a set of flags. These are given by the $T$-invariant linear subspaces of $T_pX$.
  • The axial function assigns to each flag the $T$-weight of that linear subspace.
  • Two flags at different vertices form an edge if and only if they correspond to tangent spaces of the same 1-dimensional orbit.
  • Every edge consists of precisely two flags.

Index