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The LiE software package [vLCL92] deals with modules (or, equivalently, representations) of connected reductive complex Lie groups. The key to this approach is the observation that every finite-dimensional module of such a group consists of a direct sum of irreducible modules. Each of these irreducible modules has exactly one so-called `highest weight'. Thus we have a bijection between representations and sets of highest weights with multiplicities. The latter is called a `decomposition polynomial'.
LiE exploits this bijection and enables the user to do calculations with these decomposition polynomials. Besides converting between highest weights, dominant weights, or all weights, it can compute for example dimensions, compute symmetric, alternating, Littlewood-Richardson or regular tensor products of modules, `branch' a module to subgroups or `collect' to a supergroup, and do many more related operations.
The big advantage of this approach is that one need not construct the modules themselves, but can do all computations with just the decomposition polynomial. Whereas the dimension of a representation could easily go into the tens of thousands, the number of highest weights in a decomposition polynomial is typically much more modest.
The major advantage of LiE is that it was specifically made for doing these computations, and it is therefore extremely fast.
This advantage, however, is at the same time its major disadvantage: besides computations with these decomposition polynomials it is only capable of doing a very small set of other computations.
Furthermore, it has been largely unmaintained since 2000.
Magma, however, is being actively maintained now, and has features on many aspects of discrete mathematics in general and Lie theory in particular.
Moreover, it is very easy for the user to write custom functions to suit the particular task at hand.
These observations inspired us to port the functionality that is in LiE to Magma.
Rather than insert LiE as a black box into Magma, we decided to really port the source code. The main advantage of this approach is the usage of the efficient datatypes and corresponding functions in Magma. The result is a set of functions that works nicely with the other aspects of Magma, thus making it easy to do research involving various fields of mathematics at the same time.
Acknowledgements
Introduction
Virtual Decompositions
Toral Elements
The Bruhat Order
Data Structures
Constructing Representation Decompositions
Properties of Representation Decompositions
Creating New Representation Decompositions from Existing Ones
Operations on Representation Decompositions
Basic Arithmetic with Representation Decompositions
Calculating with Representations
Other Functions for Representation Decompositions
Operations Related to the Symmetric Group
Operations Related to the Weyl Group
Subgroups of Small Rank
Bibliography
DETAILS
Introduction
Virtual Decompositions
Toral Elements
The Bruhat Order
Data Structures
Constructing Representation Decompositions
LieRepresentationDecomposition(R) : RootDtm -> LieRepDec
LieRepresentationDecomposition(R, v) : RootDtm, ModTupFldElt -> LieRepDec
LieRepresentationDecomposition(R, Wt, Mp) : RootDtm, SeqEnum, SeqEnum -> LieRepDec
WeightsAndMultiplicities(D) : LieRepDec -> SeqEnum, SeqEnum
AdjointRepresentationDecomposition(R) : RootDtm -> LieRepDec
Example LieReps_LieRepDecAdjointEx (H92E1)
Properties of Representation Decompositions
D eq E : LieRepDec, LieRepDec -> BoolElt
RootDatum(D) : LieRepDec -> RootDtm
VirtualDecomposition(C) : LieRepDec -> LieRepDec
DecomposeCharacter(C) : LieRepDec -> LieRepDec
RepresentationDimension(D) : LieRepDec -> RngIntElt
DominantCharacter(D) : LieRepDec -> LieRepDec
CasimirValue(R, w) : RootDtm, ModTupRngElt -> FldRatElt
QuantumDimension(R, w) : RootDtm, ModTupRngElt -> SetMulti
Example LieReps_LieRepDecQuantumDimensionEx (H92E2)
Creating New Representation Decompositions from Existing Ones
CopyRepresentation(D) : LieRepDec -> LieRepDec
D + E : LieRepDec, LieRepDec -> BoolElt
D + v : LieRepDec, ModTupRngElt -> BoolElt
D * c : LieRepDec, RngIntElt -> LieRepDec
D / c : LieRepDec, RngIntElt -> LieRepDec
D * E : LieRepDec, LieRepDec -> LieRepDec
SubWeights(D, Q, S) : LieRepDec, SeqEnum, RootDtm -> LieRepDec
PermuteWeights(D, pi, S) : LieRepDec, GrpPermElt, RootDtm -> LieRepDec
Example LieReps_LieRepDecArithmeticEx (H92E3)
Operations on Representation Decompositions
Basic Arithmetic with Representation Decompositions
AddRepresentation( D, v, c) : LieRepDec, ModTupFldElt, RngIntElt ->
D +:= v : LieRepDec, ModTupFldElt ->
AddRepresentation( D, E, c) : LieRepDec, LieRepDec, RngIntElt ->
D +:= E : LieRepDec, LieRepDec ->
D *:= c : LieRepDec, RngIntElt ->
D /:= c : LieRepDec, RngIntElt ->
Calculating with Representations
AlternatingDominant(D, w) : LieRepDec, GrpPermElt -> LieRepDec
AlternatingDominant(D) : LieRepDec, GrpPermElt -> LieRepDec
Example LieReps_LieRepDecAltDomEx (H92E4)
Branch(FromGrp, ToGrp, v, M) : RootDtm, RootDtm, ModTupFldElt, AlgMatElt -> LieRepDec
Branch(ToGrp, D, M) : RootDtm, LieRepDec, AlgMatElt -> LieRepDec
Collect(R, D, M) : RootDtm, LieRepDec, AlgMatElt -> LieRepDec
Example LieReps_LieRepDecBranchCollectEx (H92E5)
Demazure(R, v, w) : RootDtm, ModTupFldElt, GrpPermElt -> LieRepDec
Demazure(R, v) : RootDtm, ModTupFldElt -> LieRepDec
Example LieReps_LieRepDecBranchCollectEx (H92E6)
LittlewoodRichardsonTensor(v, w) : ModTupFldElt, ModTupFldElt -> LieRepDec
Example LieReps_LieRepDecLRTensorEx (H92E7)
AdamsOperator(R, n, v) : RootDtm, RngIntElt, ModTupFldElt -> LieRepDec
SymmetricTensor(R, n, v) : RootDtm, RngIntElt, ModTupFldElt -> LieRepDec
AlternatingTensor(R, n, v) : RootDtm, RngIntElt, ModTupFldElt -> LieRepDec
Plethysm(R, lambda, v) : RootDtm, SeqEnum, ModTupFldElt -> LieRepDec
Spectrum(R, v, t) : RootDtm, ModTupFldElt, SeqEnum -> SeqEnum
Example LieReps_LieRepDecSpectrumEx (H92E8)
Tensor(R, v, w) : RootDtm, ModTupFldElt, ModTupFldElt -> .
TensorPower(R, n, v) : RootDtm, RngIntElt, ModTupFldElt -> LieRepDec
Example LieReps_LieRepDecTensorPowerEx (H92E9)
AlternatingWeylSum(R, v) : RootDtm, ModTupFldElt -> LieRepDec
Other Functions for Representation Decompositions
FundamentalClosure(R, S) : RootDtm, SetEnum -> SetEnum
ClosureLiE(R, S) : RootDtm, SetEnum -> SetEnum
RestrictionMatrix(R, S) : RootDtm, RootDtm -> AlgMatElt
Example LieReps_LieRepDecResMatEx (H92E10)
KLPolynomial(x, y) : GrpPermElt, GrpPermElt -> RngUPolElt
RPolynomial(x, y) : GrpPermElt, GrpPermElt -> RngUPolElt
Example LieReps_LieRepDecPolysEx (H92E11)
Exponents(R) : RootDtm -> SeqEnum
Example LieReps_LieRepDecExponentsEx (H92E12)
ToLiE(D) : LieRepDec -> MonStgElt
FromLiE(R, p) : RootDtm, MonStgElt -> LieRepDec
Example LieReps_LieRepDecToFromLiEEx (H92E13)
Operations Related to the Symmetric Group
ConjugationClassLength(l) : SeqEnum -> RngIntElt
PartitionToWeight(l) : SeqEnum -> SeqEnum
WeightToPartition(v) : SeqEnum -> SeqEnum
TransposePartition(l) : SeqEnum -> SeqEnum
Operations Related to the Weyl Group
BruhatLessOrEqual(x, y) : GrpPermElt, GrpPermElt -> BoolElt
BruhatDescendents(x) : GrpPermElt -> SetEnum
BruhatDescendents(X) : SetEnum -> SetEnum
Example LieReps_LieRepDecBruhatDescendents (H92E14)
TransversalElt(W, x, H) : GrpPermCox, GrpPermElt, GrpPermCox -> GrpPermElt
TransversalElt(W, H, x, J) : GrpPermCox, GrpPermCox, GrpPermElt, GrpPermCox -> GrpPermElt
WeylMatrix(R, w) : RootDtm, GrpPermElt -> AlgMatElt
WeylWordFromAction(W, M) : GrpPermCox, AlgMatElt -> GrpPermElt
Example LieReps_LieRepDecWeylMatrix (H92E15)
Subgroups of Small Rank
LiEMaximalSubgroups() : -> SeqEnum
MaximalSubgroups(G) : MonStgElt -> SeqEnum[MonStgElt]
RestrictionMatrix(G, H) : MonStgElt, MonStgElt -> AlgMatElt
Bibliography
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