# GKMVarieties -- computations with GKM varieties and moment graphs

## Description

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 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.

## Contributors

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

## Version

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/.

## Exports

• 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 .