[_____]
MULTILINEAR ALGEBRA
Acknowledgements
Introduction
Overview
Verbose Printing
Tensors
Creating Tensors
Black-box Tensors
Tensors with Structure Constant Sequences
Bilinear Tensors
Tensors from Algebraic Objects
New Tensors from Old
Operations with Tensors
Elementary Operations
General Properties
Tensors As Multilinear Maps
Operations with Bilinear Maps
Manipulating Tensor Data
Invariants of Tensors
Standard Invariants
Exporting Tensors
Tensor Spaces
Constructions of Tensor and Cotensor Spaces
Universal Tensor Spaces
Universal Cotensor Spaces
Some Standard Constructions
Operations on Tensor Spaces
Membership and Comparison with Tensor Spaces
Tensor Spaces as Modules
Properties of Tensor Spaces
Tensor Categories
Constructing Tensor Categories
Operations on Tensor Categories
Categorical Operations
Categorical Operations on Tensors
Categorical Operations on Tensor Spaces
Homotopisms
Constructions of Homotopisms
Basic Operations with Homotopisms
Basic Properties of Homotopisms
Linear Invariants of Tensors
Invariants for Bilinear Tensors
Invariants of General Multilinear Maps
Some Extended Examples
Distinguishing Groups
Simplifying Automorphism Group Computations
Bibliography
Introduction
Overview
Verbose Printing
Example Multilinear_VerbosePrinting (H63E1)
T : Magma;
Example Multilinear_PrintToString (H63E2)
Tensors
Creating Tensors
Black-box Tensors
Tensor(S, F) : SeqEnum, UserProgram -> TenSpcElt, List
Example Multilinear_BBTensorsFrame (H63E3)
Example Multilinear_BBCrossProduct (H63E4)
Tensor(D, C, F) : SeqEnum, Any, UserProgram -> TenSpcElt, List
Example Multilinear_BBTripleProduct (H63E5)
Tensors with Structure Constant Sequences
Tensor(D, S) : [RngIntElt], SeqEnum -> TenSpcElt
Example Multilinear_SCTensors (H63E6)
StructureConstants(T) : TenSpcElt -> SeqEnum
Assign(T, ind, k) : TenSpcElt, [RngIntElt], Any -> TenSpcElt
Example Multilinear_SCFromBBTensors (H63E7)
Example Multilinear_SCStored (H63E8)
Bilinear Tensors
Tensor(M, s, t) : [Mtrx], RngIntElt, RngIntElt -> TenSpcElt
Example Multilinear_SymplecticForm (H63E9)
AsMatrices(T, s, t) : TenSpcElt, RngIntElt, RngIntElt -> SeqEnum
Example Multilinear_TrilinearAsMats (H63E10)
Tensors from Algebraic Objects
Tensor(A): Alg -> TenSpcElt, Map
Example Multilinear_D4LieAlgebra (H63E11)
Tensor(Q) : RngUPolRes -> TenSpcElt, Map
Example Multilinear_WittAlgebra (H63E12)
CommutatorTensor(A) : Alg -> TenSpcElt, Map
Example Multilinear_CommutatorFromAlgebra (H63E13)
Example Multilinear_MatrixJordanAlgebra (H63E14)
AssociatorTensor(A) : Alg -> TenSpcElt, Map
Example Multilinear_AssociatorFromAlgebra (H63E15)
pCentralTensor(G, p, s, t) : Grp, RngIntElt, RngIntElt, RngIntElt -> TenSpcElt, List
Example Multilinear_TensorPGroup (H63E16)
MatrixTensor(K, S) : Fld, [RngIntElt] -> TenSpcElt, List
Polarisation(f) : MPolElt -> TenSpcElt, MPolElt
Example Multilinear_TensorPolarization (H63E17)
New Tensors from Old
AlternatingTensor(T) : TenSpcElt -> TenSpcElt
AntisymmetricTensor(T) : TenSpcElt -> TenSpcElt
SymmetricTensor(T) : TenSpcElt -> TenSpcElt
Example Multilinear_AlternatingTensor (H63E18)
Example Multilinear_MakeSymmetric (H63E19)
Shuffle(T, g) : TenSpcElt, GrpPermElt -> TenSpcElt
Example Multilinear_ShuffleToTranspose (H63E20)
Example Multilinear_Shuffling (H63E21)
TensorProduct(t, s) : TenSpcElt, TenSpcElt -> TenSpcElt
Operations with Tensors
Elementary Operations
S + T : TenSpcElt, TenSpcElt -> TenSpcElt
Example Multilinear_ModuleOperations (H63E22)
AssociatedForm(T) : TenSpcElt -> TenSpcElt
Compress(T) : TenSpcElt -> TenSpcElt
Example Multilinear_CompressAssocForm (H63E23)
General Properties
Parent(T) : TenSpcElt -> TenSpc
Domain(T) : TenSpcElt -> List
Codomain(T) : TenSpcElt -> Any
Valence(T) : TenSpcElt -> RngIntElt
Frame(T) : TenSpcElt -> List
BaseRing(T) : TenSpcElt -> Rng
Example Multilinear_BasicProps (H63E24)
TensorCategory(T) : TenSpcElt -> TenCat
ChangeTensorCategory(T, C) : TenSpcElt, TenCat -> TenSpcElt
IsCovariant(T) : TenSpcElt -> BoolElt
Example Multilinear_TensorCatProps (H63E25)
NondegenerateTensor(T) : TenSpcElt -> TenSpcElt, Hmtp
IsNondegenerate(T) : TenSpcElt -> BoolElt
Example Multilinear_Nondegeneracy (H63E26)
Image(T) : TenSpcElt -> ModTupRng
FullyNondegenerateTensor(T) : TenSpcElt -> TenSpcElt, Hmtp
IsFullyNondegenerate(T) : TenSpcElt -> BoolElt
Example Multilinear_FullyNondegenerate (H63E27)
IsAlternating(T) : TenSpcElt -> BoolElt
IsAntisymmetric(T) : TenSpcElt -> BoolElt
IsSymmetric(T) : TenSpcElt -> BoolElt
Example Multilinear_SymmetricPolar (H63E28)
Tensors As Multilinear Maps
x @ T : Tup, TenSpcElt -> Any
Example Multilinear_MultiMapEval (H63E29)
T * f : TenSpcElt, Map -> TenSpcElt
S eq T : TenSpcElt, TenSpcElt -> BoolElt
Example Multilinear_TensorComp (H63E30)
Operations with Bilinear Maps
x * T : Any, TenSpcElt -> Any
x * T : Any, TenSpc -> Any
Example Multilinear_BimapInfix (H63E31)
x * y : BmpUElt, BmpVElt -> Any
LeftDomain(B) : TenSpcElt -> BmpU
RightDomain(B) : TenSpcElt -> BmpV
IsCoercible(U,x) : BmpU, Any -> BoolElt, BmpUElt
Example Multilinear_BimapProduct (H63E32)
Parent(x) : BmpUElt -> BmpU
Parent(X) : BmpU -> TenSpcElt
u1 eq u2 : BmpUElt, BmpUElt -> BoolElt
v1 eq v2 : BmpUElt, BmpUElt -> BoolElt
U1 eq U2 : BmpU, BmpU -> BoolElt
V1 eq V2 : BmpV, BmpV -> BoolElt
Example Multilinear_BimapProduct2 (H63E33)
Manipulating Tensor Data
Slice(T, grid) : TenSpcElt, [SetEnum] -> SeqEnum
Example Multilinear_TensorSlicing (H63E34)
SliceAsMatrices(T, grid, a, b) : TenSpcElt, [SetEnum], RngIntElt, RngIntElt -> SeqEnum
Example Multilinear_SliceAsMatrices (H63E35)
Foliation(T, i) : TenSpcElt, RngIntElt -> Mtrx
Example Multilinear_ExfoliateFoliation (H63E36)
AsTensorSpace(T, i) : TenSpcElt, RngIntElt -> TenSpc, Mtrx
AsCotensorSpace(T) : TenSpcElt -> TenSpc, Mtrx
Example Multilinear_TensorsToSpaces (H63E37)
AsTensor(S) : TenSpc -> TenSpcElt
Example Multilinear_SpacesToTensors (H63E38)
Invariants of Tensors
Induce(X, i) : AlgMat, RngIntElt -> Map, AlgMat
Example Multilinear_Inducing (H63E39)
DerivedFrom(~X, t, C, RC : parameters) : Any, TenSpcElt, RngIntElt, {RngIntElt} ->
Standard Invariants
Radical(T, s) : TenSpcElt, RngIntElt -> ModTupRng
Radical(T) : TenSpcElt -> Tup
Coradical(T) : TenSpcElt -> ModTupRng, Map
Example Multilinear_Radicals (H63E40)
Discriminant(B) : TenSpcElt -> RngMPolElt
Example Multilinear_DiscriminatingOctonions (H63E41)
Pfaffian(B) : TenSpcElt -> RngMPolElt
Example Multilinear_Genus2Pfaff (H63E42)
Exporting Tensors
HeisenbergAlgebra(T) : TenSpcElt -> AlgGen
Example Multilinear_CraftingAlgebras (H63E43)
HeisenbergLieAlgebra(T) : TenSpcElt -> AlgLie
Example Multilinear_CraftingLieAlgberas (H63E44)
HeisenbergGroup(T) : TenSpcElt -> GrpMat
Example Multilinear_CraftingPGroups (H63E45)
Example Multilinear_PGroupsHalfFull (H63E46)
Tensor Spaces
Constructions of Tensor and Cotensor Spaces
Universal Tensor Spaces
KTensorSpace(K, S) : Fld, [RngIntElt] -> TenSpc
RTensorSpace(R, S) : Rng, [RngIntElt] -> TenSpc
Example Multilinear_UniversalKTenSpc (H63E47)
TensorSpace(S) : SeqEnum -> TenSpc, List
Example Multilinear_UniversalTenSpc (H63E48)
TensorSpace(V, p, q) : ModTupFld, RngIntElt, RngIntElt -> TenSpc
Example Multilinear_SignaturedTenSpc (H63E49)
Universal Cotensor Spaces
KCotensorSpace(K, S) : Fld, [RngIntElt] -> TenSpc
CotensorSpace(S) : SeqEnum -> TenSpc
Example Multilinear_UniversalCoTenSpc (H63E50)
Some Standard Constructions
AlternatingSpace(T) : TenSpc -> TenSpc, Map
AntisymmetricSpace(T) : TenSpc -> TenSpc, Map
SymmetricSpace(T) : TenSpc -> TenSpc, Map
Example Multilinear_StandardTenSubspcs (H63E51)
ExteriorCotensorSpace(V, n) : ModTupFld, RngIntElt -> TenSpc
SymmetricCotensorSpace(V, n) : ModTupFld, RngIntElt -> TenSpc
Example Multilinear_StandardCoTenSubspcs (H63E52)
Operations on Tensor Spaces
Membership and Comparison with Tensor Spaces
T in TS : TenSpcElt, TenSpc -> BoolElt
TS ! T : TenSpc, TenSpcElt -> TenSpcElt
TS ! S : TenSpc, SeqEnum -> TenSpcElt
T ! n : TenSpc, RngIntElt -> TenSpcElt
IsCoercible(TS, x) : TenSpc, Any -> BoolElt, TenSpcElt
Example Multilinear_Coercion (H63E53)
S eq T : TenSpc, TenSpc -> BoolElt
S subset T : TenSpc, TenSpc -> BoolElt
IsCoercible(T, S) : TenSpc, Any -> BoolElt
Example Multilinear_TenSpcContainment (H63E54)
Tensor Spaces as Modules
Basis(T) : TenSpc -> SeqEnum
T . i : TenSpc, RngIntElt -> TenSpcElt
NumberOfGenerators(T) : TenSpc -> RngIntElt
Dimension(T) : TenSpc -> RngIntElt
# T : TenSpc -> RngIntElt
Example Multilinear_BasicModule (H63E55)
Random(T) : TenSpc -> TenSpcElt
RandomTensor(R, S) : Rng, [RngIntElt] -> TenSpcElt
Example Multilinear_RandomTensors (H63E56)
RandomAlternatingTensor(R, d, n, c) : Rng, RngIntElt, RngIntElt, RngIntElt -> TenSpcElt
RandomAntisymmetricTensor(R, d, n, c) : Rng, RngIntElt, RngIntElt, RngIntElt -> TenSpcElt
RandomSymmetricTensor(R, d, n, c) : Rng, RngIntElt, RngIntElt, RngIntElt -> TenSpcElt
Example Multilinear_RandomSymTen (H63E57)
Properties of Tensor Spaces
Valence(T) : TenSpc -> RngIntElt
Frame(T) : TenSpc -> List
BaseRing(T) : TenSpc -> Rng
Example Multilinear_TenSpcProperties (H63E58)
TensorCategory(T) : TenSpc -> TenCat
IsCovariant(T) : TenSpc -> BoolElt
ChangeTensorCategory(T, C) : TenSpc, TenCat -> TenSpc
Example Multilinear_TenSpcCategories (H63E59)
IsAlternating(T) : TenSpc -> BoolElt
IsAntisymmetric(T) : TenSpc -> BoolElt
IsSymmetric(T) : TenSpc -> BoolElt
UniversalTensorSpace(T) : TenSpc -> TenSpc
Example Multilinear_UniversalConst (H63E60)
Tensor Categories
Constructing Tensor Categories
TensorCategory(A, P) : [RngIntElt], SetEnum -> TenCat
CotensorCategory(A, P) : [RngIntElt], SetEnum -> TenCat
Example Multilinear_BasicCatConst (H63E61)
HomotopismCategory(v : parameters) : RngIntElt -> TenCat
CohomotopismCategory(v) : RngIntElt -> TenCat
AdjointCategory(v, s, t) : RngIntElt, RngIntElt, RngIntElt -> TenCat
Example Multilinear_TenCatSpecial (H63E62)
Operations on Tensor Categories
C1 eq C2 : TenCat, TenCat -> BoolElt
Valence(C) : TenCat -> RngIntElt
Arrows(C) : TenCat -> SeqEnum
RepeatPartition(C) : TenCat -> SetEnum
IsCovariant(C) : TenCat -> BoolElt
Example Multilinear_TenCatProperties (H63E63)
Categorical Operations
Categorical Operations on Tensors
Subtensor(T, S) : TenSpcElt, List -> TenSpcElt
Subtensor(T, D, C) : TenSpcElt, List, Any -> TenSpcElt
IsSubtensor(T, S) : TenSpcElt, TenSpcElt -> BoolElt
Example Multilinear_Subtensors (H63E64)
LocalIdeal(T, S, I) : TenSpcElt, List, RngIntElt -> TenSpcElt
LocalIdeal(T, D, C, I) : TenSpcElt, List, Any, RngIntElt -> TenSpcElt
LocalIdeal(T, S, I) : TenSpcElt, TenSpcElt, RngIntElt -> TenSpcElt
IsLocalIdeal(T, S, I) : TenSpcElt, TenSpcElt, RngIntElt -> BoolElt
Example Multilinear_LocalIdeals (H63E65)
Ideal(T, S) : TenSpcElt, List -> TenSpcElt
Ideal(T, D, C) : TenSpcElt, List, Any -> TenSpcElt
Ideal(T, S) : TenSpcElt, TenSpcElt -> TenSpcElt
IsIdeal(T, S) : TenSpcElt, TenSpcElt -> BoolElt
Example Multilinear_Ideals (H63E66)
LocalQuotient(T, S, I : parameters) : TenSpcElt, TenSpcElt, RngIntElt -> TenSpcElt, Hmtp
Quotient(T, S : parameters) : TenSpcElt, TenSpcElt -> TenSpcElt, Hmtp
Example Multilinear_Quotients (H63E67)
Categorical Operations on Tensor Spaces
SubConstructor(T, L) : TenSpc, Any -> TenSpc, Map
IsSubtensorSpace(T, S) : TenSpc, TenSpc -> BoolElt
Example Multilinear_SubtensorSpaces (H63E68)
QuoConstructor(T, X) : TenSpc, Any -> TenSpc, Map
Example Multilinear_QuotientTensorSpaces (H63E69)
Homotopisms
Constructions of Homotopisms
Homotopism(T, S, M : parameters) : TenSpcElt, TenSpcElt, List -> Hmtp
Homotopism(M, C) : List, TenCat -> Hmtp
IsHomotopism(T, s, H) : TenSpcElt, TenSpcElt, Hmtp -> BoolElt
Example Multilinear_HomotopismConst (H63E70)
Example Multilinear_MixedHomotopisms (H63E71)
Basic Operations with Homotopisms
H1 * H2 : Hmtp, Hmtp -> Hmtp
H . a : Hmtp, RngIntElt -> Map
Example Multilinear_HomotopismOps (H63E72)
Precompose(T, f, a) : TenSpcElt, Map, RngIntElt -> TenSpcElt
T @ H : TenSpcElt, Hmtp -> TenSpcElt
Basic Properties of Homotopisms
Domain(H) : Hmtp -> TenSpcElt
Codomain(H) : Hmtp -> TenSpcElt
Maps(H) : Hmtp -> List
TensorCategory(H) : Hmtp -> TenCat
ChangeTensorCategory(H, C) : Hmtp, TenCat -> Hmtp
Valence(H) : Hmtp -> RngIntElt
Kernel(H) : Hmtp -> TenSpcElt
Image(H) : Hmtp -> TenSpcElt
Example Multilinear_HomotopismProps (H63E73)
Shuffle(H, g) : Hmtp, GrpPermElt -> Hmtp
Linear Invariants of Tensors
Invariants for Bilinear Tensors
AdjointAlgebra(B) : TenSpcElt -> AlgMat
Example Multilinear_AdjointAlge (H63E74)
LeftNucleus(B : parameters) : TenSpcElt -> AlgMat
MidNucleus(B) : TenSpcElt -> AlgMat
RightNucleus(B) : TenSpcElt -> AlgMat
Example Multilinear_GoingNuclear (H63E75)
Invariants of General Multilinear Maps
Centroid(T) : TenSpcElt -> AlgMat
Example Multilinear_Centroid (H63E76)
DerivationAlgebra(T) : TenSpcElt -> AlgMatLie
Nucleus(T, a, b) : TenSpcElt, RngIntElt, RngIntElt -> AlgMat
Example Multilinear_RestrictDerivation (H63E77)
SelfAdjointAlgebra(t, a, b) : TenSpcElt, RngIntElt, RngIntElt -> ModMatFld
TensorOverCentroid(T) : TenSpcElt -> TenSpcElt, Hmtp
Example Multilinear_CentroidUnipotent (H63E78)
Some Extended Examples
Distinguishing Groups
Example Multilinear_Payne_Grps (H63E79)
Simplifying Automorphism Group Computations
Example Multilinear_ExtOverAdj (H63E80)
Bibliography
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