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# 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-Fløystad-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

- Federico Galetto <galetto.federico@gmail.com>

## Certification

Version 0.6.5 of this package was accepted for publication in volume 7 of 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 (DOI: 10.2140/jsag.2015.7.9). That version can be obtained from the journal.

## Version

This documentation describes version 0.6.5 of HighestWeights, released July 03, 2015.

## Citation

If you have used this package in your research, please cite it as follows:

    @misc{HighestWeightsSource,
      title = {{HighestWeights: A \emph{Macaulay2} package. Version~0.6.5}},
      author = {Federico Galetto},
      howpublished = {A \emph{Macaulay2} package available at
        \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
    }

    @article{HighestWeightsArticle,
      title = {{Free resolutions and modules with a semisimple Lie group action}},
      author = {Federico Galetto},
      journal = {The Journal of Software for Algebra and Geometry},
      volume = {7},
      year = {2015},
    }

## Exports

- Functions and commands
  - decomposeWeightsList -- decompose a list of weights into highest weights
  - getWeights -- retrieve the (Lie theoretic) weight of a monomial
  - highestWeightsDecomposition -- irreducible decomposition of a complex, ring, ideal or module
  - propagateWeights -- propagate (Lie theoretic) weights along equivariant maps
  - setWeights -- attach (Lie theoretic) weights to the variables of a ring
- Methods
  - getWeights(RingElement) -- retrieve the (Lie theoretic) weight of a monomial
  - `highestWeightsDecomposition(Complex)` -- see highestWeightsDecomposition(Complex,ZZ,List) -- decompose an equivariant complex of graded free modules
  - highestWeightsDecomposition(Complex,ZZ,List) -- decompose an equivariant complex of graded free modules
  - highestWeightsDecomposition(Ideal,List) -- decompose an ideal with a semisimple Lie group action
  - `highestWeightsDecomposition(Ideal,ZZ)` -- see highestWeightsDecomposition(Ideal,List) -- decompose an ideal with a semisimple Lie group action
  - `highestWeightsDecomposition(Ideal,ZZ,ZZ)` -- see highestWeightsDecomposition(Ideal,List) -- decompose an ideal with a semisimple Lie group action
  - highestWeightsDecomposition(Module,List,List) -- decompose a module with a semisimple Lie group action
  - `highestWeightsDecomposition(Module,ZZ,List)` -- see highestWeightsDecomposition(Module,List,List) -- decompose a module with a semisimple Lie group action
  - `highestWeightsDecomposition(Module,ZZ,ZZ,List)` -- see highestWeightsDecomposition(Module,List,List) -- decompose a module with a semisimple Lie group action
  - highestWeightsDecomposition(Ring,List) -- decompose a ring with a semisimple Lie group action
  - `highestWeightsDecomposition(Ring,ZZ)` -- see highestWeightsDecomposition(Ring,List) -- decompose a ring with a semisimple Lie group action
  - `highestWeightsDecomposition(Ring,ZZ,ZZ)` -- see highestWeightsDecomposition(Ring,List) -- decompose a ring with a semisimple Lie group action
  - propagateWeights(Matrix,List) -- propagate (Lie theoretic) weights along an equivariant map of graded free modules
  - setWeights(PolynomialRing,DynkinType,List) -- attach (Lie theoretic) weights to the variables of a ring
- 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

## For the programmer

The object HighestWeights is a package, defined in HighestWeights.m2, with auxiliary files in HighestWeights/.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/HighestWeights/doc.m2:55:0.