The current elements of the machinery for Lie theory comprise:
Coxeter matrices, Coxeter graphs, Cartan matrices, Dynkin digraphs, and Cartan names.
Root systems
Root data
Coxeter groups
Coxeter groups as permutation groups
Reflection groups
Groups of Lie type
Finite-dimensional Lie algebras (See section on Algebras)
We support all the standard descriptions for Coxeter systems and reflection groups: Coxeter matrices, Coxeter graphs, Cartan matrices, Dynkin digraphs, and Cartan names.
Creation of these descriptions.
Conversion between them.
Isomorphism and Cartan equivalence of Coxeter systems.
Testing for finite, affine, hyperbolic, and compact hyperbolic Coxeter systems.
Constructing all finite, affine and hyperbolic systems.
Computing the size and number of roots of (finite) Coxeter systems.
Printing Dynkin diagrams for finite systems.
Predicates: IsIrreducible, IsCrystallographic, IsSimplyLaced.
A datatype for finite root systems has been implemented by Scott H. Murray and Don Taylor.
Any finite root system can be constructed by giving its simple roots and coroots, including nonsemisimple systems (where the dimension of the vector space is larger than the rank). Special function exist to construct semisimple systems from a Coxeter matrix, Coxeter graph, Cartan matrix, Dynkin digraph, or Cartan name. Standard root systems can be constructed—these are systems whose pairing is the Coxeter form, which is how root systems are frequently given in the literature.
equality, isomorphism and Cartan equivalence.
Operations: Cartan name, Coxeter diagram, Dynkin diagram, Coxeter matrix, Coxeter graph, Cartan matrix, Dynkin digraph, base field, rank, dimension, Coxeter group order.
Properties: IsIrreducible, IsSemisimple, IsSimplyLaced.
The (co)roots are stored in an indexed set, with positive roots first. The can be described and manipulated via their index, or as vectors with respect to either the standard basis or the basis of simple (co)roots.
Root space and coroot space.
construction of the complete set of roots or coroots.
conversion between indices and vectors.
Highest long or short root.
Reflection actions of the (co)roots: given as matrices, permutations, or words in the simple reflections.
Basic arithmetic with (co)root indices: Sum, Negative, IsPositive, IsNegative, heights, norms.
Coxeter form and dual Coxeter form.
A datatype for finite root data has been implemented by Scott H. Murray and Don Taylor. This is designed primarily for use with Lie algebras and groups of Lie type.
[add non-reduced and extended root data; Sergei]
Any split (untwisted) root datum can be constructed by giving its simple roots and coroots. Special function exist to construct semisimple systems from a Cartan matrix, Dynkin digraph, or Cartan name. By default the adjoint datum is returned, by the isogeny type can be specified by optional parameters. Standard classical root data can be constructed—these are systems whose pairing is the Coxeter form, which is how root data are frequently given in the literature. For technical reasons it is impossible to define standard root data for the exceptional types.
equality, isomorphism, Cartan equivalence, and isogeny.
Operations: Cartan name, Coxeter diagram, Dynkin diagram, Coxeter matrix, Coxeter graph, Cartan matrix, Dynkin digraph, base field, rank, dimension, Coxeter group order, group of Lie type order.
Fundamental and (co)isogeny groups.
Properties: IsIrreducible, IsSemisimple, IsCrystallographic, IsSimplyLaced, IsAdjoint, IsSimplyConnected.
The (co)roots are stored in an indexed set, with positive roots first. The can be described and manipulated via their index, or as vectors with respect to the standard basis or the basis of simple (co)roots or the basis of fundamental weights.
root space and coroot space.
construction of the complete set of roots or coroots.
conversion between indices and vectors.
Highest long or short root.
Reflection actions of the (co)roots: given as matrices, permutations, or words in the simple reflections.
Basic arithmetic with (co)root indices: Sum, Negative, IsPositive, IsNegative, heights, norms.
Left and right strings through one root in the direction of another.
Coxeter form and dual Coxeter form.
(co)weight lattice and fundamental (co)weights
Coxeter groups are implemented as a subclass of finitely presented groups so that they inherit all the operations for finitely presented groups as well as having many specialized functions. The main difference is that every word is automatically converted into normal form using an algorithm designed and implemented by Bob Howlett. This module was implemented by Bob Howlett, Scott Murray, and Don Taylor.
A Coxeter group can be constructed from a Cartan matrix, Dynkin digraph, Cartan name, root system, or root datum. Note that a Coxeter group does not have a unique reflection representation, and so does not have a unique Cartan matrix.
isomorphism as Coxeter groups.
Operations: Cartan name, Coxeter diagram, Coxeter matrix, Coxeter graph, rank.
Properties: IsFinite, IsAffine, IsHyperbolic, IsCompactHyperbolic, IsIrreducible, IsSimplyLaced.
Arithmetic of words: identity, multiplications, inversion, powers.
Degrees of the basic invariant polynomials.
Coxeter element and Coxeter number.
Braid group and pure braid group
Conversion to and from permutation and reflection representations.
Finite Coxeter groups are implemented as a subclass of permutation groups so that they inherit all the operations for permutation groups as well as having many specialized functions.
A permutation Coxeter group can be constructed from a Cartan matrix, Dynkin digraph, Cartan name, root system, or root datum.
In addition to the standard functions for groups, almost all of the functions for root systems and root data also work on permutation Coxeter groups.
A reflection subgroup can be represented two ways: as a permutation group on the roots of the larger groups, or as a permutation group on its own roots. We use an efficient algorithm due to Don Taylor to compute transversals of reflection subgroups.
We can also compute the "standard" permutation action of a Coxeter group, which is usually the smallest degree permutation action. For example, the standard action of the group of type An, gives the symmetric group on n + 1 points.
construction and identification of a reflection group over an arbitrary ring, given the simple roots, coroots and orders
construction of real reflection groups from a Cartan matrix, Dynkin digraph, Cartan name, root system, or root datum
construction of all finite complex reflection groups
Most of the functions available for Coxeter groups are also available for real reflection groups.
We have implemented code for computing in split (untwisted) groups of Lie type with the Steinberg presentation. These groups can be defined over any Magma field. Elements can be normalised using the Bruhat decomposition: a flag for each group determines whether this is done automatically or manually by the user.
A group of Lie type can be created from a field and a Cartan name, Weyl group, root datum, Cartan matrix or Dynkin digraph.
Most of the operations and properties for root data also apply to groups of Lie type.
equality, algebraic isomorphism, isogeny.
algebraic group generators; abstract group generators for certain fields
element arithmetic and normalisation
Bruhat decomposition and multiplicative Jordan decomposition.