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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

Authors

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

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/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/GKMVarieties/Documentation_GKMVarieties.m2:57:0.