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GKMVarieties -- computations with GKM varieties and moment graphs


A GKM variety is a variety $X$, often assumed to be smooth and complete, with an action of an algebraic torus $T$ satisfying the following conditions: (i) $X$ is equivariantly formal with respect to the action of $T$, (ii) $X$ has finitely many $T$-fixed points, and (iii) $X$ has finitely many one-dimensional $T$-orbits. The data of the zero and one dimensional $T$-orbits of $X$ define the moment graph of $X$, with which one can carry out $T$-equivariant cohomology and $T$-equivariant $K$-theory computations via the method of localization. This package provides methods for these computations in Macaulay2.

For mathematical background see:

  • [BM01] T. Braden and R. MacPherson. From moment graphs to intersection cohomology. Math. Ann. 321 (2001), 533-551.
  • [BGH02] E. Bolker, V. Guillemin, and T. Holm. How is a graph like a manifold? arXiv:math/0206103.
  • [CDMS18] A. Cameron, R. Dinu, M. Michalek, and T. Seynnaeve. Flag matroids: algebra and geometry. arXiv:1811.00272.
  • [DES20] R. Dinu, C. Eur, and T. Seynnaeve. K-theoretic Tutte polynomials of morphisms of matroids. arXiv:math/2004.00112.
  • [FS12] A. Fink and S. Speyer. K-classes for matroids and equivariant localization. Duke Math. J. 161 (2012), no. 14, 2699-2723.
  • [GKM98] M. Goresky, R. Kottwitz, and R. MacPherson. Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131 (1998), no. 1, 25-83.
  • [RK03] I. Rosu. Equivariant K-theory and equivariant cohomology. With an Appendix by I. Rosu and A. Knutson. Math. Z. 243 (2003), 423-448.
  • [Tym05] J. Tymoczko. An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson. Contemp. Math. 388 (2005), 169-188.
  • [VV03] G. Vezzosi and A. Vistoli. Higher algebraic K-theory for actions of diagonalizable groups. Invent. Math. 153 (2003), no. 1, 1–44.


The following people have contributed code, improved existing code, or enhanced the documentation: Tim Seynnaeve.

See also



This documentation describes version 0.1 of GKMVarieties.

Source code

The source code from which this documentation is derived is in the file GKMVarieties.m2. The auxiliary files accompanying it are in the directory GKMVarieties/.


  • Types
    • EquivariantMap -- the class of all equivariant morphisms between GKM varieties
    • FlagMatroid -- the class of all flag matroids
    • GKMVariety -- the class of all GKM varieties
    • KClass -- the class of all equivariant K-classes
    • MomentGraph -- the class of all moment graphs
  • Functions and commands
    • affineToricRing -- computes the toric ring associated to a monomial map
    • ampleKClass -- the class of an ample line bundle
    • bruhatOrder -- computes the Bruhat order on a generalized flag variety
    • cellOrder -- the poset of a stratification of a GKM variety
    • charts -- outputs the torus-invariant affine charts of a GKM variety
    • diagonalMap -- constructs the diagonal morphism
    • flagGeomTuttePolynomial -- computes the flag-geometric Tutte polynomial of flag matroids
    • flagMap -- creates equivariant maps between generalized flag varieties
    • flagMatroid -- construct a flag matroid
    • generalizedFlagVariety -- makes a generalized flag variety as a GKM variety
    • generalizedSchubertVariety -- create a generalized Schubert variety
    • lieType -- outputs the Lie type of a generalized flag variety
    • makeCharacterRing -- constructs the character ring of a torus
    • makeGKMVariety -- constructs a GKM variety
    • makeKClass -- constructs an equivariant K-class
    • momentGraph -- creates a moment graph
    • orbitClosure -- computes the equivariant K-class of a torus orbit closure of a point in a generalized flag variety
    • projectiveSpace -- constructs projective space as a GKM variety
    • pushforward -- computes the pushforward map of equivariant K-classes of an equivariant map
    • setIndicator -- computes the signed indicator vector of an admissible set
    • trivialKClass -- the equivariant K-class of the structure sheaf
  • Methods
  • Symbols

For the programmer

The object GKMVarieties is a package.