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HighestWeights :: HighestWeights

HighestWeights -- decompose free resolutions and graded modules with a semisimple Lie group action

Description

This package provides tools to study the representation theoretic structure of equivariant free resolutions and graded modules with the action of a semisimple Lie group. The methods of this package allow one to consider the free modules in an equivariant resolution, or the graded components of a module, as representations of a semisimple Lie group by means of their weights and to obtain their decomposition into highest weight representations.

This package implements an algorithm introduced in Galetto - Propagating weights of tori along free resolutions. The methods of this package are meant to be used in characteristic zero.

The following links contain some sample computations carried out using this package. The first and second example are discussed in more detail, so we recommend reading through them first.

  • Example 1 -- The coordinate ring of the Grassmannian
  • Example 2 -- The Buchsbaum-Rim complex
  • Example 3 -- A multigraded Eagon-Northcott complex
  • Example 4 -- The Eisenbud-Fl√łystad-Weyman complex
  • Example 5 -- The singular locus of a symplectic invariant
  • Example 6 -- The coordinate ring of the spinor variety
  • Example 7 -- With the exceptional group G2

Author

Certification a gold star

Version 0.6.5 of this package was accepted for publication in volume 7 of the journal The Journal of Software for Algebra and Geometry on 5 June 2015, in the article Free resolutions and modules with a semisimple Lie group action. That version can be obtained from the journal or from the Macaulay2 source code repository, http://github.com/Macaulay2/M2/blob/master/M2/Macaulay2/packages/HighestWeights.m2, commit number a434adb94f76f9be38131f87745867b0d7925405.

Version

This documentation describes version 0.6.5 of HighestWeights.

Source code

The source code from which this documentation is derived is in the file HighestWeights.m2. The auxiliary files accompanying it are in the directory HighestWeights/.

Exports

  • Functions and commands
  • Symbols
    • Forward -- propagate weights from domain to codomain
    • GroupActing -- stores the Dynkin type of the group acting on a ring
    • LeadingTermTest -- check the columns of the input matrix for repeated leading terms
    • LieWeights -- stores the (Lie theoretic) weights of the variables of a ring
    • MinimalityTest -- check that the input map is minimal
    • Range -- decompose only part of a complex