### D.13.3 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.13.3.1 isAface |  | Checks whether the given face is an a-face.
> D.13.3.2 afaces |  | Returns a list of intvecs that correspond to the set of all a-faces, optionally for given list of simplex faces.
> D.13.3.3 fullDimImages |  | Finds the afaces which have a full-dimensional projection.
> D.13.3.4 minimalAfaces |  | compute the minimal a-faces among the a-faces with full dimensional projection.
> D.13.3.5 orbitCones |  | Returns the list of all orbit cones.
> D.13.3.6 GITcone |  | Returns the GIT-cone containing the given weight vector.
> D.13.3.7 GITfan |  | Compute GIT-fan.
> D.13.3.8 GITfanFromOrbitCones |  | Compute GIT-fan from orbit cones.
> D.13.3.9 GITfanParallel |  | Compute GIT-fan in parallel from orbit cones.
> D.13.3.10 GKZfan |  | Returns the GKZ-fan of the matrix Q.
> D.13.3.11 movingCone |  | Compute the moving cone.
> D.13.3.12 computeAfaceOrbits |  | Compute orbits of a-faces under a permutation group action.
> D.13.3.13 minimalAfaceOrbits |  | Find the minimal a-face orbits.
> D.13.3.14 orbitConeOrbits |  | Project the a-face orbits to orbit cone orbits.
> D.13.3.15 minimalOrbitConeOrbits |  | Find the minimal orbit cone orbits.
> D.13.3.16 intersectOrbitsWithMovingCone |  | Intersect orbit cone orbits with moving cone.
> D.13.3.17 groupActionOnQImage |  | Determine the induced group action in the target of the grading matrix.
> D.13.3.18 groupActionOnHashes |  | Determine the induced group action on the set of orbit cones.
> D.13.3.19 storeActionOnOrbitConeIndices |  | Write the group action on the set of orbit cones to a file.
> D.13.3.20 permutationFromIntvec |  | Create a permutation from an intvec of images.
> D.13.3.21 permutationToIntvec |  | Return the intvec of images.
> D.13.3.22 evaluateProduct |  | Evaluate a list of products of group elements in terms of a given representation of the elements as permutations.
> D.13.3.23 GITfanSymmetric |  | Compute GIT-fan from orbit cones by determining a minimal representing set for the orbits of maximal dimensional GIT-cones.
> D.13.3.24 GITfanParallelSymmetric |  | Compute GIT-fan in parallel from orbit cones by determining a minimal representing set for the orbits of maximal dimensional GIT-cones.
> D.13.3.25 bigintToBinary |  | Convert bigint into a sparse binary representation specifying the indices of the one-entries
> D.13.3.26 binaryToBigint |  | Convert sparse binary representation specifying the indices of the one-entries to bigint
> D.13.3.27 applyPermutationToIntvec |  | Apply permutation to a set of integers represented as an intvec
> D.13.3.28 hashToCone |  | Convert a bigint hash to a GIT-cone
> D.13.3.29 hashesToFan |  |
> D.13.3.30 gitCone |  | Returns the GIT-cone around the given weight vector w

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