ROOT DATA FOR LIE THEORY

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ROOT DATA FOR LIE THEORY

Introduction
Definition of a root datum
Positive and simple roots
Classification of root data

Cartan matrices

Creating root data

Operators on root data

Properties of root data

Roots, coroots and weights
Accessing (co)roots
Action of (co)roots
Operations and properties for (co)roots
Weights

Creating new root data from old

Constants associated with crystallographic root data

Related structures

Bibliography

DETAILS

Introduction

Definition of a root datum

Positive and simple roots

Classification of root data

Cartan matrices
CartanMatrix( t ) : MonStgElt -> AlgMatElt
IsCartanMatrix( M ) : AlgMatElt -> BoolElt
CartanName( C ) : AlgMatElt -> List
Example RootDtm_CartanMatrices (H35E1)
DynkinDiagram( t ) : List -> .
IsCartanIrreducible( C ) : AlgMatElt -> BoolElt
IsCrystallographic( C ) : AlgMatElt -> BoolElt
FundamentalGroup( t ) : AlgMatElt -> GrpAb
Example RootDtm_CartanMatrixFunctions (H35E2)

Creating root data
RootDatum( t ) : MonStgElt -> RootDtm
Example RootDtm_CreatingRootData (H35E3)
RootDatum( C ) : AlgMatElt -> RootDtm
RootDatum( A, B ) : AlgMatElt, AlgMatElt -> RootDtm
Example RootDtm_GLRootDatum (H35E4)

Properties of root data
IsIrreducible( RD ) : RootDtm -> BoolElt
IsCrystallographic( RD ) : RootDtm -> BoolElt
IsSimplyLaced( RD ) : RootDtm-> BoolElt
IsSemisimple( RD ) : RootDtm-> BoolElt
IsAdjoint( RD ) : RootDtm-> BoolElt
IsSimplyConnected( RD ) : RootDtm-> BoolElt
Example RootDtm_Properties (H35E8)

Roots, coroots and weights

Accessing (co)roots
NumberOfPositiveRoots( RD ) : RootDtm -> RngIntElt
Roots( RD ) : RootDtm -> {@@}
PositiveRoots( RD ) : RootDtm -> {@@}
Root( RD, r ) : RootDtm, RngIntElt -> {@@}
RootPosition( RD, v ) : RootDtm, . -> {@@}
Example RootDtm_RootsCoroots (H35E9)
HighestRoot( RD ) : RootDtm -> .
HighestShortRoot( RD ) : RootDtm -> .
Example RootDtm_HeighestRoots (H35E10)

Action of (co)roots
SimpleReflectionMatrices( RD ) : RootDtm -> []
ReflectionMatrices( RD ) : RootDtm -> []
ReflectionMatrix( RD, r ) : RootDtm, RngIntElt -> []
SimpleReflectionPermutations( RD ) : RootDtm -> []
ReflectionPermutations( RD ) : RootDtm -> []
ReflectionPermutation( RD, r ) : RootDtm, RngIntElt -> []
Example RootDtm_Action (H35E11)

Weights
WeightLattice( RD ) : RootDtm -> Lat
FundamentalWeights( RD ) : RootDtm -> Mtrx
Example RootDtm_Weights (H35E14)

Creating new root data from old
DirectSum( RD1, RD2 ) : RootDtm, RootDtm -> RootDtm
DirectSumDecomposition( RD ) : RootDtm -> []
Dual( RD ) : RootDtm -> RootDtm
Example RootDtm_DirectSumDual (H35E15)
RootSubdatum( RD, a ) : RootDtm, SetEnum -> RootDtm
RootSubdatum( RD, s ) : RootDtm, SeqEnum -> RootDtm
Example RootDtm_RootSubdata (H35E16)

Bibliography