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.

Authors

Version

This documentation describes version 0.1 of GKMVarieties.

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 GKMVarieties/Documentation_GKMVarieties.m2:57:0.