Singular

D.15.18 gitfan_lib

Library:

gitfan.lib

Purpose:

Compute GIT-fans.

Authors:

Janko Boehm, boehm at mathematik.uni-kl.de
Simon Keicher, keicher at mail.mathematik.uni-tuebingen.de
Yue Ren, ren at mathematik.uni-kl.de

Overview:

This library allows you to calculate GIT-fans, torus orbits and GKZ-fans.

In provides features to make use of symmetries of the torus action under consideration.

The main procedure is GITfan which can be directly applied to an ideal and a grading matrix encoding the torus action, and returns a fan, the associated GIT-fan. We also provide various procedures implementing substeps of the algorithm to deal with large computations.

The library uses the package 'gfanlib' by Anders N. Jensen.

For notation, background, and algorithms see [BKR16].

Functions produce debug output if printlevel is positive.

Elements of the symmetric group Sn of type permutation can be created by the function permutationFromIntvec.
The images of 1,...,n can be obtained by permutationToIntvec. Composition of permutations can be done by the *-Operator, also powers can be computed in the usual way.

References:

[BKR16] J. Boehm, S. Keicher, Y. Ren: Computing GIT-Fans with Symmetry and the Mori Chamber Decomposition of M06bar, https://arxiv.org/abs/1603.09241

Types:

permutation; Permutation in map representation.

Procedures:

D.15.18.1 isAface		Checks whether the given face is an a-face.
D.15.18.2 afaces		Returns a list of intvecs that correspond to the set of all a-faces, optionally for given list of simplex faces.
D.15.18.3 fullDimImages		Finds the afaces which have a full-dimensional projection.
D.15.18.4 minimalAfaces		compute the minimal a-faces among the a-faces with full dimensional projection.
D.15.18.5 orbitCones		Returns the list of all orbit cones.
D.15.18.6 GITcone		Returns the GIT-cone containing the given weight vector.
D.15.18.7 GITfan		Compute GIT-fan.
D.15.18.8 GITfanFromOrbitCones		Compute GIT-fan from orbit cones.
D.15.18.9 GITfanParallel		Compute GIT-fan in parallel from orbit cones.
D.15.18.10 GKZfan		Returns the GKZ-fan of the matrix Q.
D.15.18.11 movingCone		Compute the moving cone.
D.15.18.12 computeAfaceOrbits		Compute orbits of a-faces under a permutation group action.
D.15.18.13 minimalAfaceOrbits		Find the minimal a-face orbits.
D.15.18.14 orbitConeOrbits		Project the a-face orbits to orbit cone orbits.
D.15.18.15 minimalOrbitConeOrbits		Find the minimal orbit cone orbits.
D.15.18.16 intersectOrbitsWithMovingCone		Intersect orbit cone orbits with moving cone.
D.15.18.17 groupActionOnQImage		Determine the induced group action in the target of the grading matrix.
D.15.18.18 groupActionOnHashes		Determine the induced group action on the set of orbit cones.
D.15.18.19 storeActionOnOrbitConeIndices		Write the group action on the set of orbit cones to a file.
D.15.18.20 permutationFromIntvec		Create a permutation from an intvec of images.
D.15.18.21 permutationToIntvec		Return the intvec of images.
D.15.18.22 evaluateProduct		Evaluate a list of products of group elements in terms of a given representation of the elements as permutations.
D.15.18.23 GITfanSymmetric		Compute GIT-fan from orbit cones by determining a minimal representing set for the orbits of maximal dimensional GIT-cones.
D.15.18.24 GITfanParallelSymmetric		Compute GIT-fan in parallel from orbit cones by determining a minimal representing set for the orbits of maximal dimensional GIT-cones.
D.15.18.25 bigintToBinary		Convert bigint into a sparse binary represenation specifying the indices of the one-entries
D.15.18.26 binaryToBigint		Convert sparse binary represenation specifying the indices of the one-entries to bigint
D.15.18.27 applyPermutationToIntvec		Apply permutation to a set of integers represented as an intvec
D.15.18.28 hashToCone		Convert a bigint hash to a GIT-cone
D.15.18.29 hashesToFan
D.15.18.30 gitCone		Returns the GIT-cone around the given weight vector w