RootSystem -- the class of all root systems

Methods that use an object of class RootSystem :

• addRoots(RootSystem,Root,Root) -- the sum of two roots
• cartanMatrix(RootSystem) -- the Cartan matrix of a root system
• dynkinDiagram(RootSystem) -- the Dynkin diagram of a root system
• dynkinType(RootSystem) -- the Dynkin type of a root system
• 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
• isPositiveRoot(RootSystem,Weight) -- check whether a weight is a positive root
• isReduced(RootSystem,BasicList) -- whether a decomposition in simple reflections is reduced
• 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
• 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
• neutralWeylGroupElement(RootSystem) -- the neutral element of the Weyl group of a root system
• norm(RootSystem,Root) -- the squared norm of a root
• numberOfPositiveRoots(RootSystem) -- the number of positive roots
• parabolic(RootSystem,Set) -- construct a parabolic from a set of simple roots
• 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(RootSystem) -- the rank of a root system
• reduce(RootSystem,BasicList) -- the product of several reflections with respect to simple roots
• 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(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
• weight(RootSystem,BasicList) -- construct a weight from a list
• weight(RootSystem,Vector) -- construct a weight from a vector

Fixed objects of class RootSystem :

• rootSystemF4 -- the root system of type F4
• rootSystemG2 -- the root system of type G2

For the programmer

The object RootSystem is a type, with ancestor classes MutableHashTable < HashTable < Thing.