WeylGroups -- Weyl groups

Description

This package provides functions to compute in Weyl groups of root systems. In particular, it can compute intervals for the Bruhat order.

Here is a quick tutorial on how to use it.

RootSystem -- the class of all root systems
Weight -- the class of weights
Root -- the class of roots (or, more generally, elements of the root lattice)
WeylGroupElement -- the class of elements of Weyl groups
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

Authors

Baptiste Calmès
Viktor Petrov

Version

This documentation describes version 0.5.3 of WeylGroups.

Citation

If you have used this package in your research, please cite it as follows:

@misc{WeylGroupsSource,
  title = {{WeylGroups: root systems and Weyl groups. Version~0.5.3}},
  author = {Baptiste Calm{\`e}s and Viktor Petrov},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Types
- DynkinDiagram -- the class of Dynkin diagrams
- DynkinType -- the class of Dynkin Types
- HasseDiagram -- the class of Hasse diagrams
- HasseGraph -- the class of Hasse graphs
- Parabolic -- the class of parabolic subgroups of Weyl groups
- Root -- the class of roots (or, more generally, elements of the root lattice)
- RootSystem -- the class of all root systems
- Weight -- the class of weights
- WeylGroupDoubleCoset -- the class of double cosets of Weyl groups by pairs of parabolic subgroups
- WeylGroupElement -- the class of elements 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
Functions and commands
- aboveBruhat -- obtain Weyl group elements just greater than an element for the Bruhat order
- addRoots -- adding roots
- cartanMatrix -- the Cartan matrix of a root system
- connectedComponents -- get the connected components
- coxeterLength -- the length of a reduced decomposition of an element of a Weyl group
- dynkinDiagram -- produce a Dynkin Diagram
- dynkinExponents -- the exponents associated to a type
- dynkinType -- obtaining a Dynkin type
- endVertices -- the vertices with at most one edge
- eval -- evaluate the dual of a root at something
- halfSumOfRoots -- the half-sum of positive roots
- hasseDiagramToGraph -- turning a hasse diagram into a graph (intended for graphic representation)
- hasseGraphToPicture -- construct the picture of a Hasse Graph
- intervalBruhat -- obtaining an interval for the Bruhat order
- isLtBruhat -- 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
- isPositiveRoot -- check whether a weight is a positive root
- isReduced -- check whether a decomposition is of minimal length
- isReflection -- 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
- listWeylGroupElements -- list all elements of a given length in a Weyl group
- loadHasseGraph -- load a Hasse graph from a file
- longestWeylGroupElement -- the longest element of a Weyl group
- minimalRepresentative -- the minimal representative of a coset
- neutralWeylGroupElement -- the neutral element of a Weyl group
- numberOfPositiveRoots -- the number of positive roots
- parabolic -- construct a parabolic
- poincareSeries -- a generating series for number of elements in a Weyl group
- positiveRoots -- the set of all positive roots
- reduce -- the product of several reflections
- reducedDecomposition -- the reduced decomposition of an element of a Weyl group
- reflect -- apply the reflection with respect to a root
- reflection -- the reflection with respect to a root
- rootCoefficients -- coefficients at the simple roots
- rootSystem -- obtain a root system
- 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
- scalarProduct -- compute a scalar product (invariant by the Weyl group)
- simpleRoot -- a simple root
- storeHasseGraph -- store a Hasse graph in a file
- underBruhat -- obtain Weyl group elements less than an element for the Bruhat order
- weight -- construct a weight in the weight lattice of a root system
- whoseReflection -- the positive root whose reflection is a given element of a Weyl group
Methods
- - 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
- addRoots(RootSystem,Root,Root) -- the sum of two roots
- cartanMatrix(RootSystem) -- the Cartan matrix of a root system
- 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(DynkinDiagram,Parabolic) -- the Dynkin diagram of the Levy subgroup of a parabolic
- dynkinDiagram(RootSystem) -- the Dynkin diagram of a root system
- dynkinExponents(DynkinType) -- the exponents of the 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(DynkinDiagram) -- the vertices of a Dynkin diagram with at most one neighbor
- 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(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(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
- norm(RootSystem,Root) -- the squared norm of a root
- numberOfPositiveRoots(DynkinType) -- the number of positive roots
- numberOfPositiveRoots(RootSystem) -- the number of positive roots
- 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(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
- rank(RootSystem) -- the rank of a root system
- 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(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
- 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
- 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
- weight(RootSystem,BasicList) -- construct a weight from a list
- weight(RootSystem,Vector) -- construct a weight from a vector
- 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 * Root -- multiplication of a root by an integer
- ZZ * Weight -- the multiple of a weight
Other things
- rootSystemF4 -- the root system of type F4
- rootSystemG2 -- the root system of type G2

For the programmer

The object WeylGroups is a package, defined in WeylGroups.m2, with auxiliary files in WeylGroups/.

The source of this document is in WeylGroups.m2:1559:0.