next
|
previous
|
forward
|
backward
| up |
top
|
index
|
toc
|
Macaulay2 web site
WeylGroups
::
Parabolic
Parabolic -- the class of parabolic subgroups of Weyl groups
Description
A parabolic subgroup is represented by the set of indices of generating reflections.
Methods that use an object of class Parabolic :
dynkinDiagram(DynkinDiagram,Parabolic)
-- the Dynkin diagram of the Levy subgroup of a parabolic
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
isRoot(RootSystem,Parabolic,Root)
-- check whether a root is in the sub root system of the parabolic
longestWeylGroupElement(RootSystem,Parabolic)
-- the longest element of a parabolic subgroup of the Weyl group of a root system
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
poincareSeries(RootSystem,Parabolic,RingElement)
-- the generating series of a quotient of the Weyl group by length
positiveRoots(RootSystem,Parabolic)
-- the set of all positive roots in a parabolic sugroups
rootSystem(RootSystem,Parabolic)
-- the root system of the Levy subgroup of a parabolic
WeylGroupElement % Parabolic
-- the left coset defined by an element of Weyl group
WeylGroupRightCoset % Parabolic
-- the double coset defined by a right coset
For the programmer
The object
Parabolic
is
a
type
, with ancestor classes
Set
<
Tally
<
HashTable
<
Thing
.