WeylGroups : Table of Contents
- WeylGroups -- Weyl groups
- - Weight -- the negative of a weight
- aboveBruhat -- obtain Weyl group elements just greater than an element for the Bruhat order
- 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
- addRoots -- adding roots
- addRoots(RootSystem,Root,Root) -- the sum of two roots
- cartanMatrix -- the Cartan matrix of a root system
- cartanMatrix(RootSystem) -- the Cartan matrix of a root system
- connectedComponents -- get the connected components
- connectedComponents(DynkinDiagram) -- the connected components of a Dynkin diagram
- coxeterLength -- the length of a reduced decomposition of an element of a Weyl group
- coxeterLength(WeylGroupElement) -- the length of a reduced decomposition of an element of a Weyl group
- DynkinDiagram -- the class of Dynkin diagrams
- 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 -- the class of Dynkin Types
- 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 -- the half-sum of positive roots
- halfSumOfRoots(RootSystem) -- the half-sum of positive roots
- HasseDiagram -- the class of Hasse diagrams
- hasseDiagramToGraph -- turning a hasse diagram into a graph (intended for graphic representation)
- hasseDiagramToGraph(HasseDiagram) -- turning a hasse diagram into a graph (intended for graphic representation)
- HasseGraph -- the class of Hasse graphs
- hasseGraphToPicture -- construct the picture of a Hasse Graph
- hasseGraphToPicture(HasseGraph) -- Obtain a picture from a Hasse graph
- intervalBruhat -- obtaining an interval for the Bruhat order
- 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 -- compare two Weyl group elements in the Bruhat order
- 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 -- check whether a weight is a positive root
- isPositiveRoot(RootSystem,Weight) -- check whether a weight is a positive root
- isReduced -- check whether a decomposition is of minimal length
- 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 -- checks whether an element of a Weyl group is a reflection
- isReflection(WeylGroupElement) -- checks whether an element of a Weyl group is a reflection
- isRoot -- check whether a weight is a root or whether a root is in a parabolic sub root system
- 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 -- list all elements of a given length in a Weyl group
- listWeylGroupElements(RootSystem,ZZ) -- list all elements of a given length in a Weyl group
- loadHasseGraph -- load a Hasse graph from a file
- loadHasseGraph(String) -- load a Hasse graph from a file
- longestWeylGroupElement -- the longest element of a Weyl group
- 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 -- the neutral element of a Weyl group
- neutralWeylGroupElement(RootSystem) -- the neutral element of the Weyl group of a root system
- norm(RootSystem,Root) -- the squared norm of a root
- numberOfPositiveRoots -- the number of positive roots
- numberOfPositiveRoots(DynkinType) -- the number of positive roots
- numberOfPositiveRoots(RootSystem) -- the number of positive roots
- Parabolic -- the class of parabolic subgroups of Weyl groups
- 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 -- the set of all positive roots
- 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
- rank(RootSystem) -- the rank of a root system
- reduce -- the product of several reflections
- reduce(RootSystem,BasicList) -- the product of several reflections with respect to simple roots
- reducedDecomposition -- the reduced decomposition of an element of a Weyl group
- 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 -- the reflection with respect to a root
- reflection(RootSystem,Root) -- the reflection with respect to a root
- Root -- the class of roots (or, more generally, elements of the root lattice)
- rootCoefficients -- coefficients at the simple roots
- rootCoefficients(RootSystem,Root) -- the coefficients at the simple roots
- rootCoefficients(RootSystem,Weight) -- the coefficients at the simple roots
- RootSystem -- the class of all root systems
- rootSystem -- obtain a root system
- RootSystem ++ RootSystem -- the direct sum of root systems
- RootSystem == RootSystem -- equality of root systems
- rootSystem(DynkinDiagram) -- the root system corresponding to a Dynkin diagram
- rootSystem(DynkinType) -- the root system of a given type
- rootSystem(RootSystem,Parabolic) -- the root system of the Levy subgroup of a parabolic
- 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
- 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 -- a simple root
- simpleRoot(RootSystem,ZZ) -- the n-th simple root
- storeHasseGraph -- store a Hasse graph in a file
- storeHasseGraph(HasseGraph,String) -- store a Hasse graph in a file
- underBruhat -- obtain Weyl group elements less than an element for the Bruhat order
- 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 -- the class of weights
- weight -- construct a weight in the weight lattice of a root system
- Weight + Weight -- the sum of two weights
- Weight - Weight -- the difference of two weights
- weight(RootSystem,BasicList) -- construct a weight from a list
- weight(RootSystem,Vector) -- construct a weight from a vector
- WeylGroupDoubleCoset -- the class of double cosets of Weyl groups by pairs of parabolic subgroups
- WeylGroupDoubleCoset == WeylGroupDoubleCoset -- equality of double cosets
- WeylGroupElement -- the class of elements of Weyl groups
- 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 -- the class of left cosets of Weyl groups by parabolic subgroup
- WeylGroupLeftCoset == WeylGroupLeftCoset -- equality of left cosets
- WeylGroupRightCoset -- the class of right cosets of Weyl groups by parabolic subgroups
- 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 -- the positive root whose reflection is a given element of a Weyl group
- whoseReflection(WeylGroupElement) -- the positive root whose reflection is a given element of a Weyl group
- ZZ * Root -- multiplication of a root by an integer