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.

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
- 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.