WeylGroups : Table of Contents

• WeylGroups -- Weyl groups

  • RootSystem -- the class of all root systems

    • rootSystem -- obtain a root system
    • cartanMatrix -- the Cartan matrix of a root system
    • rank(RootSystem) -- the rank of a root system
    • rootSystem(DynkinType) -- the root system of a given type
    • rootSystemA -- a root system of type A
    • rootSystemB -- a root system of type B
    • rootSystemC -- a root system of type C
    • rootSystemD -- a root system of type D
    • rootSystemE -- a root system of type E
    • rootSystem(RootSystem,Parabolic) -- the root system of the Levy subgroup of a parabolic
    • rootSystem(DynkinDiagram) -- the root system corresponding to a Dynkin diagram
  • Weight -- the class of weights

    • weight -- construct a weight in the weight lattice of a root system
    • weight(RootSystem,BasicList) -- construct a weight from a list
    • weight(RootSystem,Vector) -- construct a weight from a vector
    • isPositiveRoot -- check whether a weight is a positive root
    • isRoot -- check whether a weight is a root or whether a root is in a parabolic sub root system
  • Root -- the class of roots (or, more generally, elements of the root lattice)

    • addRoots -- adding roots
    • addRoots(RootSystem,Root,Root) -- the sum of two roots
    • ZZ * Root -- multiplication of a root by an integer
    • halfSumOfRoots -- the half-sum of positive roots
    • simpleRoot -- a simple root
    • norm(RootSystem,Root) -- the squared norm of a root
    • rootCoefficients -- coefficients at the simple roots
    • positiveRoots -- the set of all positive roots
    • numberOfPositiveRoots -- the number of positive roots
  • WeylGroupElement -- the class of elements of Weyl groups

    • reduce -- the product of several reflections
    • reducedDecomposition -- the reduced decomposition of an element of a Weyl group
    • isReduced -- check whether a decomposition is of minimal length
    • coxeterLength -- the length of a reduced decomposition of an element of a Weyl group
    • longestWeylGroupElement -- the longest element of a Weyl group
    • reflection -- the reflection with respect to a root
    • isReflection -- checks whether an element of a Weyl group is a reflection
    • whoseReflection -- the positive root whose reflection is a given element of a Weyl group
    • listWeylGroupElements -- list all elements of a given length in a Weyl group
    • neutralWeylGroupElement -- the neutral element of a Weyl group
    • underBruhat -- obtain Weyl group elements less than an element for the Bruhat order
    • aboveBruhat -- obtain Weyl group elements just greater than an element for the Bruhat order
    • isLtBruhat -- compare two Weyl group elements in the Bruhat order
    • intervalBruhat -- obtaining an interval for the Bruhat order
  • Parabolic -- the class of parabolic subgroups of Weyl groups
  • WeylGroupLeftCoset -- the class of left cosets of Weyl groups by parabolic subgroup
  • WeylGroupRightCoset -- the class of right cosets of Weyl groups by parabolic subgroups
  • WeylGroupDoubleCoset -- the class of double cosets of Weyl groups by pairs of parabolic subgroups
  • DynkinDiagram -- the class of Dynkin diagrams
  • DynkinType -- the class of Dynkin Types
  • HasseDiagram -- the class of Hasse diagrams
  • HasseGraph -- the class of Hasse graphs

    • hasseDiagramToGraph -- turning a hasse diagram into a graph (intended for graphic representation)
    • hasseGraphToPicture -- construct the picture of a Hasse Graph
    • storeHasseGraph -- store a Hasse graph in a file
    • loadHasseGraph -- load a Hasse graph from a file
• - Weight -- the negative of a weight
• aboveBruhat(BasicList) -- The Weyl group elements just under the ones in the list for the Bruhat order
• aboveBruhat(WeylGroupElement) -- Weyl group elements just above a given one for the Bruhat order
• cartanMatrix(RootSystem) -- the Cartan matrix of a root system
• connectedComponents -- get the connected components
• connectedComponents(DynkinDiagram) -- the connected components of a Dynkin diagram
• coxeterLength(WeylGroupElement) -- the length of a reduced decomposition of an element of a Weyl group
• dynkinDiagram -- produce a Dynkin Diagram
• dynkinDiagram(DynkinDiagram,Parabolic) -- the Dynkin diagram of the Levy subgroup of a parabolic
• dynkinDiagram(RootSystem) -- the Dynkin diagram of a root system
• dynkinExponents -- the exponents associated to a type
• dynkinExponents(DynkinType) -- the exponents of the Dynkin type
• dynkinType -- obtaining a Dynkin type
• DynkinType ++ DynkinType -- the disjoint union of Dynkin Types
• DynkinType == DynkinType -- the equality of Dynkin Types
• dynkinType(BasicList) -- constructing a Dynkin type
• dynkinType(DynkinDiagram) -- the Dynkin type of a Dynkin diagram
• dynkinType(RootSystem) -- the Dynkin type of a root system
• endVertices -- the vertices with at most one edge
• endVertices(DynkinDiagram) -- the vertices of a Dynkin diagram with at most one neighbor
• eval -- evaluate the dual of a root at something
• eval(RootSystem,Weight,Root) -- evaluate the dual of a root at a Weight
• eval(RootSystem,Weight,ZZ) -- evaluate the dual of a simple root at a Weight
• eval(RootSystem,ZZ,Root) -- evaluate the dual of a root at a fundamental weight
• eval(RootSystem,ZZ,ZZ) -- evaluate the dual of a simple root at another one
• halfSumOfRoots(RootSystem) -- the half-sum of positive roots
• hasseDiagramToGraph(HasseDiagram) -- turning a hasse diagram into a graph (intended for graphic representation)
• hasseGraphToPicture(HasseGraph) -- Obtain a picture from a Hasse graph
• intervalBruhat(WeylGroupElement,WeylGroupElement) -- elements between two given ones for the Bruhat order on a Weyl group
• intervalBruhat(WeylGroupLeftCoset,WeylGroupLeftCoset) -- elements between two given ones for the Bruhat order on a quotient of a Weyl group
• intervalBruhat(WeylGroupRightCoset,WeylGroupRightCoset) -- elements between two given ones for the Bruhat order on a quotient of a Weyl group
• inverse(WeylGroupElement) -- the inverse to an element of a Weyl group
• isLtBruhat(WeylGroupElement,WeylGroupElement) -- compare two Weyl group elements in the Bruhat order
• isMinimalRepresentative -- check whether an element of a Weyl group is the minimal representative of a coset
• isMinimalRepresentative(Parabolic,WeylGroupElement) -- check whether an element of a Weyl group is the minimal representative of a right coset
• isMinimalRepresentative(Parabolic,WeylGroupElement,Parabolic) -- check whether an element of a Weyl group is the minimal representative of a double coset
• isMinimalRepresentative(WeylGroupElement,Parabolic) -- check whether an element of a Weyl group is the minimal representative of a left coset
• isPositiveRoot(RootSystem,Weight) -- check whether a weight is a positive root
• isReduced(BasicList,WeylGroupElement) -- whether an Weyl group element can be multiplied by some simple reflections with length increasing at each step
• isReduced(RootSystem,BasicList) -- whether a decomposition in simple reflections is reduced
• isReflection(WeylGroupElement) -- checks whether an element of a Weyl group is a reflection
• isRoot(RootSystem,Parabolic,Root) -- check whether a root is in the sub root system of the parabolic
• isRoot(RootSystem,Weight) -- check whether a weight is a positive root
• listWeylGroupElements(RootSystem,ZZ) -- list all elements of a given length in a Weyl group
• loadHasseGraph(String) -- load a Hasse graph from a file
• longestWeylGroupElement(RootSystem) -- the longest element of the Weyl group of a root system
• longestWeylGroupElement(RootSystem,Parabolic) -- the longest element of a parabolic subgroup of the Weyl group of a root system
• minimalRepresentative -- the minimal representative of a coset
• minimalRepresentative(WeylGroupDoubleCoset) -- the minimal representative of a coset
• minimalRepresentative(WeylGroupLeftCoset) -- the minimal representative of a coset
• minimalRepresentative(WeylGroupRightCoset) -- the minimal representative of a coset
• neutralWeylGroupElement(RootSystem) -- the neutral element of the Weyl group of a root system
• numberOfPositiveRoots(DynkinType) -- the number of positive roots
• numberOfPositiveRoots(RootSystem) -- the number of positive roots
• parabolic -- construct a parabolic
• Parabolic % WeylGroupElement -- the right coset defined by an element of Weyl group
• Parabolic % WeylGroupLeftCoset -- the double coset defined by a left coset
• Parabolic == Parabolic -- equality of parabolics
• parabolic(RootSystem,Set) -- construct a parabolic from a set of simple roots
• parabolic(WeylGroupDoubleCoset) -- the parabolic associated to a double coset
• poincareSeries -- a generating series for number of elements in a Weyl group
• poincareSeries(HasseDiagram,RingElement) -- the generating series of a Hasse diagram
• poincareSeries(RootSystem,Parabolic,RingElement) -- the generating series of a quotient of the Weyl group by length
• poincareSeries(RootSystem,RingElement) -- the generating series of the Weyl group by length
• positiveRoots(RootSystem) -- the set of all positive roots
• positiveRoots(RootSystem,Parabolic) -- the set of all positive roots in a parabolic sugroups
• rank(DynkinDiagram) -- the rank of a Dynkin diagram
• reduce(RootSystem,BasicList) -- the product of several reflections with respect to simple roots
• reducedDecomposition(WeylGroupElement) -- the reduced decomposition of an element of a Weyl group
• reflect -- apply the reflection with respect to a root
• reflect(RootSystem,BasicList,Root) -- apply to a root several reflections with respect to simple roots
• reflect(RootSystem,BasicList,Weight) -- apply to a weight several reflections with respect to roots
• reflect(RootSystem,ZZ,Root) -- apply to a root the reflection with respect to a simple root
• reflect(RootSystem,ZZ,Weight) -- apply to a weight the reflection with respect to a root
• reflection(RootSystem,Root) -- the reflection with respect to a root
• rootCoefficients(RootSystem,Root) -- the coefficients at the simple roots
• rootCoefficients(RootSystem,Weight) -- the coefficients at the simple roots
• RootSystem ++ RootSystem -- the direct sum of root systems
• RootSystem == RootSystem -- equality of root systems
• rootSystemF4 -- the root system of type F4
• rootSystemG2 -- the root system of type G2
• scalarProduct -- compute a scalar product (invariant by the Weyl group)
• scalarProduct(RootSystem,Weight,Weight) -- the scalar product of two weights
• scalarProduct(RootSystem,ZZ,Weight) -- the scalar product of a fundamental weight and a weight
• scalarProduct(RootSystem,ZZ,ZZ) -- the scalar product of two fundamental weights
• simpleRoot(RootSystem,ZZ) -- the n-th simple root
• storeHasseGraph(HasseGraph,String) -- store a Hasse graph in a file
• underBruhat(BasicList) -- Weyl group elements just under the ones in the list for the Bruhat order
• underBruhat(WeylGroupElement) -- Weyl group elements just under a given one for the Bruhat order
• Weight + Weight -- the sum of two weights
• Weight - Weight -- the difference of two weights
• WeylGroupDoubleCoset == WeylGroupDoubleCoset -- equality of double cosets
• WeylGroupElement % Parabolic -- the left coset defined by an element of Weyl group
• WeylGroupElement * Root -- apply an element of a Weyl group to a root
• WeylGroupElement * Weight -- apply an element of a Weyl group to a weight
• WeylGroupElement * WeylGroupElement -- the product of two elements of a Weyl group
• WeylGroupElement * WeylGroupLeftCoset -- apply an element of a Weyl group to a left coset
• WeylGroupElement == WeylGroupElement -- equality of elements of Weyl groups
• WeylGroupElement ^ ZZ -- the power of an element of a Weyl group
• WeylGroupLeftCoset == WeylGroupLeftCoset -- equality of left cosets
• WeylGroupRightCoset % Parabolic -- the double coset defined by a right coset
• WeylGroupRightCoset * WeylGroupElement -- apply an element of a Weyl group to a left coset
• WeylGroupRightCoset == WeylGroupRightCoset -- equality of right cosets
• whoseReflection(WeylGroupElement) -- the positive root whose reflection is a given element of a Weyl group
• ZZ * Weight -- the multiple of a weight