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# 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 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.

## See also

- Example: generalized flag varieties
- Example: smooth toric varieties

## Authors

- Chris Eur <chriseur@stanford.edu>
- Ritvik Ramkumar <ritvik@math.berkeley.edu>

## Version

This documentation describes version 0.1 of GKMVarieties, released August 1, 2020.

## Citation

If you have used this package in your research, please cite it as follows:

    @misc{GKMVarietiesSource,
      title = {{GKMVarieties: computations with GKM manifolds and moment graphs. Version~0.1}},
      author = {Chris Eur and Ritvik Ramkumar},
      howpublished = {A \emph{Macaulay2} package available at
        "https://github.com/chrisweur/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
  - `affineToricRing(List)` -- see affineToricRing -- computes the toric ring associated to a monomial map
  - `affineToricRing(Matrix)` -- see affineToricRing -- computes the toric ring associated to a monomial map
  - `ampleKClass(GKMVariety)` -- see ampleKClass -- the class of an ample line bundle
  - `ampleKClass(GKMVariety,KClass)` -- see ampleKClass -- the class of an ample line bundle
  - bases(FlagMatroid) -- compute the bases of a flag matroid
  - `bruhatOrder(GKMVariety)` -- see bruhatOrder -- computes the Bruhat order on a generalized flag variety
  - `cellOrder(GKMVariety)` -- see cellOrder -- the poset of a stratification of a GKM variety
  - `cellOrder(MomentGraph)` -- see cellOrder -- the poset of a stratification of a GKM variety
  - cellOrder(MomentGraph,Poset) -- define a cell order on a moment graph
  - `charts(GKMVariety)` -- see charts -- outputs the torus-invariant affine charts of a GKM variety
  - `charts(GKMVariety,List)` -- see charts -- outputs the torus-invariant affine charts of a GKM variety
  - compose(EquivariantMap,EquivariantMap) -- computes the composition of two equivariant morphisms
  - `diagonalMap(GKMVariety)` -- see diagonalMap -- constructs the diagonal morphism
  - EquivariantMap ** EquivariantMap -- computes the product of two equivariant morphisms
  - euler(KClass) -- computes the equivariant Euler characteristic of an equivariant K-class
  - `flagGeomTuttePolynomial(FlagMatroid)` -- see flagGeomTuttePolynomial -- computes the flag-geometric Tutte polynomial of flag matroids
  - `flagMap(GKMVariety,GKMVariety)` -- see flagMap -- creates equivariant maps between generalized flag varieties
  - `flagMatroid(List)` -- see flagMatroid -- construct a flag matroid
  - `flagMatroid(Matrix,List)` -- see flagMatroid -- construct a flag matroid
  - `generalizedFlagVariety(String,ZZ,List)` -- see generalizedFlagVariety -- makes a generalized flag variety as a GKM variety
  - `generalizedFlagVariety(String,ZZ,List,Ring)` -- see generalizedFlagVariety -- makes a generalized flag variety as a GKM variety
  - `generalizedSchubertVariety(GKMVariety,Thing)` -- see generalizedSchubertVariety -- create a generalized Schubert variety
  - GKMVariety ** GKMVariety -- product of GKM varieties
  - isWellDefined(FlagMatroid) -- check if a flag matroid is well-defined
  - isWellDefined(KClass) -- whether the input is a well-defined equivariant K-class
  - KClass * KClass -- computes the product of two equivariant K-classes
  - KClass + KClass -- computes the sum of two equivariant K-classes
  - KClass ^ ZZ -- computes powers of an equivariant K-classes
  - latticePoints(FlagMatroid) -- lattice points of a base polytope of a flag matroid
  - `lieType(GKMVariety)` -- see lieType -- outputs the Lie type of a generalized flag variety
  - `makeCharacterRing(ZZ)` -- see makeCharacterRing -- constructs the character ring of a torus
  - `makeGKMVariety(KClass)` -- see makeGKMVariety -- constructs a GKM variety
  - `makeGKMVariety(List,List,Ring)` -- see makeGKMVariety -- constructs a GKM variety
  - `makeGKMVariety(List,Ring)` -- see makeGKMVariety -- constructs a GKM variety
  - `makeGKMVariety(MomentGraph)` -- see makeGKMVariety -- constructs a GKM variety
  - `makeGKMVariety(MomentGraph,Ring)` -- see makeGKMVariety -- constructs a GKM variety
  - `makeGKMVariety(NormalToricVariety)` -- see makeGKMVariety -- constructs a GKM variety
  - `makeGKMVariety(NormalToricVariety,Ring)` -- see makeGKMVariety -- constructs a GKM variety
  - `makeKClass(GKMVariety,List)` -- see makeKClass -- constructs an equivariant K-class
  - makeKClass(GKMVariety,FlagMatroid) -- the equivariant K-class of a flag matroid
  - makeKClass(GKMVariety,ToricDivisor) -- create the KClass from a ToricDivisor
  - map(GKMVariety,GKMVariety,List) -- creates a EquivariantMap
  - `momentGraph(List,HashTable,Ring)` -- see momentGraph -- creates a moment graph
  - MomentGraph ** MomentGraph -- the product of two moment graphs
  - momentGraph(GKMVariety) -- view the moment graph of a GKM variety
  - momentGraph(GKMVariety,MomentGraph) -- define a moment graph for a GKM variety
  - normalToricVariety(GKMVariety) -- converts a GKM variety back into a toric variety
  - `orbitClosure(GKMVariety,Matrix)` -- see orbitClosure -- computes the equivariant K-class of a torus orbit closure of a point in a generalized flag variety
  - `projectiveSpace(ZZ)` -- see projectiveSpace -- constructs projective space as a GKM variety
  - `projectiveSpace(ZZ,Ring)` -- see projectiveSpace -- constructs projective space as a GKM variety
  - pullback(EquivariantMap) -- computes the pullback map of equivariant K-classes of an equivariant map
  - `pushforward(EquivariantMap)` -- see pushforward -- computes the pushforward map of equivariant K-classes of an equivariant map
  - `setIndicator(Set,ZZ)` -- see setIndicator -- computes the signed indicator vector of an admissible set
  - `trivialKClass(GKMVariety)` -- see trivialKClass -- the equivariant K-class of the structure sheaf
  - underlyingGraph(MomentGraph) -- the underlying (undirected) graph of a moment graph
- Symbols

## For the programmer

The object GKMVarieties is a package, defined in GKMVarieties.m2, with auxiliary files in GKMVarieties/.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GKMVarieties/Documentation_GKMVarieties.m2:57:0.