ROOT DATA

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ROOT DATA

Acknowledgements

Introduction
Reflections
Definition of a Split Root Datum
Simple and Positive Roots
The Coxeter Group
Isogeny

Constructing Root Data

Operators on Root Data

Properties of Root Data

Roots, Coroots and Weights
Accessing Roots and Coroots
Reflections
Operations and Properties for Root and Coroot indices
Weights

Creating New Root Data from Old

Constants Associated with Root Data

Related Structures

Bibliography

DETAILS

Introduction

Reflections

Definition of a Split Root Datum

Simple and Positive Roots

The Coxeter Group

Isogeny

Constructing Root Data
RootDatum( N ) : MonStgElt -> RootDtm
Example RootDtm_CreatingRootData (H80E1)
RootDatum( C ) : AlgMatElt -> RootDtm
RootDatum( D ) : GrphDir -> RootDtm
RootDatum( A, B ) : AlgMatElt, AlgMatElt -> RootDtm
Example RootDtm_G2RootSystem (H80E2)
IrreducibleRootDatum( X, n ) : MonStgElt, RngIntElt -> RootDtm
StandardRootDatum( X, n ) : MonStgElt, RngIntElt -> RootDtm
Example RootDtm_IrreducibleRootDatum (H80E3)

Properties of Root Data
IsFinite( R ) : RootSys -> BoolElt
IsIrreducible( R ) : RootDtm -> BoolElt
IsSemisimple( R ) : RootDtm-> BoolElt
IsCrystallographic( R ) : RootSys -> BoolElt
IsSimplyLaced( R ) : RootDtm-> BoolElt
IsAdjoint( R ) : RootDtm-> BoolElt
IsSimplyConnected( R ) : RootDtm-> BoolElt
Example RootDtm_Properties (H80E9)

Roots, Coroots and Weights

Accessing Roots and Coroots
RootSpace( R ) : RootDtm -> Lat
SimpleRoots( R ) : RootDtm -> Mtrx
NumberOfPositiveRoots( R ) : RootDtm -> RngIntElt
Roots( R ) : RootDtm -> {@@}
PositiveRoots( R ) : RootDtm -> {@@}
Root( R, r ) : RootDtm, RngIntElt -> {@@}
RootPosition( R, v ) : RootDtm, . -> {@@}
Example RootDtm_RootsCoroots (H80E10)
HighestRoot( R ) : RootDtm -> .
HighestShortRoot( R ) : RootDtm -> .
Example RootDtm_HeighestRoots (H80E11)
CoxeterForm( R ) : RootSys -> AlgMatElt

Reflections
SimpleReflectionMatrices( R ) : RootDtm -> []
ReflectionMatrices( R ) : RootDtm -> []
ReflectionMatrix( R, r ) : RootDtm, RngIntElt -> []
SimpleReflectionPermutations( R ) : RootDtm -> []
ReflectionPermutations( R ) : RootDtm -> []
ReflectionPermutation( R, r ) : RootDtm, RngIntElt -> []
ReflectionWords( R ) : RootDtm -> []
ReflectionWord( R, r ) : RootDtm, RngIntElt -> []
Example RootDtm_Action (H80E12)

Weights
WeightLattice( R ) : RootDtm -> Lat
CoweightLattice( R ) : RootDtm -> Lat
FundamentalWeights( R ) : RootDtm -> Mtrx
FundamentalCoweights( R ) : RootDtm -> Mtrx
Example RootDtm_Weights (H80E16)

Creating New Root Data from Old
RootSubdatum( R, a ) : RootDtm, SetEnum -> RootDtm
RootSubdatum( R, s ) : RootDtm, SeqEnum -> RootDtm
Example RootDtm_RootSubdata (H80E17)
R1 + R2 : RootDtm, RootDtm -> RootDtm
DirectSumDecomposition( R ) : RootDtm -> []
Dual( R ) : RootDtm -> RootDtm
Example RootDtm_DirectSumDual (H80E18)

Constants Associated with Root Data
SetExtraspecialSigns( R, s ) : RootDtm, . ->
LieConstant_p( R, r, s ) : RootDtm, RngIntElt, RngIntElt -> RngIntElt
LieConstant_q( R, r, s ) : RootDtm, RngIntElt, RngIntElt -> RngIntElt
CartanInteger( R, r, s) : RootDtm, RngIntElt, RngIntElt -> RngIntElt
LieConstant_N( R, r, s ) : RootDtm, RngIntElt, RngIntElt -> RngIntElt
LieConstant_epsilon( R, r, s ) : RootDtm, RngIntElt, RngIntElt -> RngIntElt
LieConstant_M( R, r, s, i ) : RootDtm, RngIntElt, RngIntElt, RngIntElt -> RngIntElt
LieConstant_C( R, i, j, r, s ) : RootDtm, RngIntElt, RngIntElt, RngIntElt, RngIntElt -> RngIntElt
LieConstant_eta( R, r, s ) : RootDtm, RngIntElt, RngIntElt -> RngIntElt
StructureConstants( R ) : RootDtm -> RngIntElt
Example RootDtm_consts (H80E19)

Bibliography