LieLink is a Mathematica interface for LiE, a computer package for Lie algebra computations. LieLink makes it possible to call LiE functions directly from Mathematica, enabling Mathematica to compute e.g.
- Tensor products of representations,
- Symmetric or alternating tensor powers of representations,
- Branchings of representations to subalgebras,
and much more. See the LiE manual for all its capabilities. Here is an short LieLink example that computes the tensor product of two fundamental representation of the Lie algebra A2:
<<LieLink`
SetDefaultAlgebra["A2"]
LieTensor[{1, 0}, {1, 0}]
(* => {0,1} + {2,0} *)
For a list of all supported LiE functions, see the last section. LieLink requires at least Mathematica 7.
LieLink can be installed automatically by entering the following command in Mathematica:
Import["https://raw.github.com/teake/LieLink/master/Install.m"]
Download the latest release https://github.com/teake/LieLink/releases, unzip it, and move the LieLink directory to the Mathematica applications directory. The location of this directory can be found by typing the command
FileNameJoin @ {$UserBaseDirectory, "Applications"}
into Mathematica.
LieLink needs a compiled version of LiE to run. It comes bundled with LiE versions for OS X, Windows, 32-bit and 64-bit Linux, all precompiled on x86 architecture. If none of these version work on your computer, you need to compile LiE yourself.
Download the compile-only version of LiE from http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/, unzip it, and compile the GAP version by running
make noreadline
make Liegap.exe
from the command line. Note that OS X users will first need to replace the makefile with this file. Compilation on Windows is best done with MinGW in an MSYS shell.
After compilation, you need to tell LieLink where the compiled version of LiE can be found. This can be done before loading the package as follows:
LieLink`$LieDirectory = "path/to/directoryofcompiledLiE";
<<LieLink`
Once installed, LieLink can be loaded in Mathematica by typing
<<LieLink`
All LiE functions have new Mathematica-like names (see the list in the
last section).
For example, the LiE function tensor is called LieTensor in LieLink:
LieTensor[{1, 1}, {1, 1}, "A2"]
(* => {0, 0} + {0, 3} + 2 {1, 1} + {2, 2} + {3, 0} *)
The Mathematica names of LiE functions can be looked up with the LieFunction
command:
LieFunction["tensor"]
(* => LieTensor *)
Besides using the new Mathematica-like names of LiE functions, it is also possible to directly call LiE with a textual query:
LieQuery["dim(A2)"]
(* => "8" *)
Unlike the new short-hand functions like LieTensor, LieQuery doesn't parse its
output to Mathematica syntax but keeps it in plain-text.
Groups are entered as strings in LieLink. For example, the equivalent of the LiE
command dim(A2) is now Dim["A2"].
LieLink also introduces a new Mathematica type called LieTerm, which is used for
terms in Laurent polynomials. LieTerm[1,2,3] prints as {1,2,3} with red brackets,
and is the equivalent of writing 1X[1,2,3] in LiE. LieTerm can also be used
as input:
SymmetricTensorPower[2, LieTerm[1,1], "A2"]
(* => LieTerm[0, 0] + LieTerm[1, 1] + LieTerm[2, 2] *)
Currently LieLink supports two system parameters of LiE, namely maxnodes and maxobjects.
They can be set by changing the Mathematica variables $MaxNodes and $MaxObjects
respectively, which should always be integers. Their default values are
$MaxNodes = 9999;
$MaxObjects = 99999;
Below is a list of all new Mathematica-like names for LiE functions in LieLink.
This list is stored in the Mathematica variable LieFunctionTable.
Adams -> Adams, LR_tensor -> LittlewoodRichardson,
adjoint -> AdjointRepresentation, max_sub -> MaximalSubgroups,
alt_dom -> AlternatingDominant, n_comp -> NumberOfSimpleComponents,
alt_tensor -> AlternatingTensorPower, next_part -> NextPartition,
alt_W_sum -> AlternatingWeylSum, next_perm -> NextPermutation,
block_mat -> BlockdiagonalMatrix, next_tabl -> NextTableau,
branch -> Branch, norm -> RootNorm,
Bruhat_desc -> BruhatDescendant, n_tabl -> NumberOfTableaux,
Bruhat_leq -> BruhatLeq, n_vars -> NumberOfVariables,
canonical -> CanonicalWeylWord, orbit -> GroupOrbit,
Cartan -> CartanMatrix, plethysm -> Plethysm,
Cartan -> CartanProduct, pos_roots -> PositiveRoots,
Cartan_type -> CartanType, p_tensor -> TensorPower,
center -> GroupCenter, print_tab -> PrintTableau,
cent_roots -> CentralizingRoots, reduce -> WeylReduce,
centr_type -> CentralizerType, reflection -> Reflection,
class_ord -> ConjugacyClassOrder, res_mat -> RestrictionMatrix,
closure -> Closure, row_index -> RowIndex,
collect -> InverseBranch, R_poly -> RPolynomial,
contragr -> Contragradient, r_reduce -> RightWeylReduce,
decomp -> Decomposition, RS -> RobinsonSchensted,
degree -> PolynomialDegree, shape -> TableauShape,
Demazure -> Demazure, sign_part -> PartitionSign,
det_Cartan -> DetCartan, spectrum -> ToralSpectrum,
diagram -> DynkinDiagram, support -> Support,
dim -> Dim, sym_char -> SymmetricCharacter,
dom_char -> DominantCharacter, sym_orbit -> SymmetricOrbit,
dominant -> Dominant, sym_tensor -> SymmetricTensorPower,
dom_weights -> DominantWeights, tableaux -> TableauxOfPartition,
exponents -> Exponents, tensor -> LieTensor,
filter_dom -> FilterDominant, to_part -> ToPartition,
from_part -> FromPartition, trans_part -> TransposePartition,
fundam -> FundamentalRoots, unique -> CanonicalMatrix,
high_root -> HighestRoot, v_decomp -> VirtualDecomposition,
i_Cartan -> InverseCartan, W_action -> WeylAction,
inprod -> InnerProduct, W_orbit -> WeylOrbit,
KL_poly -> KazhdanLusztigPolynomial, W_orbit_graph -> WeylOrbitGraph,
Lie_code -> LieCode, W_orbit_size -> WeylOrbitSize,
Lie_group -> LieGroup, W_order -> WeylOrder,
Lie_rank -> LieRank, W_rt_action -> WeylRootAction,
long_word -> LongestWord, W_rt_orbit -> WeylRootOrbit,
l_reduce -> LeftWeylReduce, W_word -> WeylWord
lr_reduce -> LeftRightWeylReduce,