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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.

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 GKMVarieties/Documentation_GKMVarieties.m2:57:0.