Tables
This section describes how to instantiate generic character tables and tables of Green functions, as well as various functions for accessing properties of these tables.
Loading tables
Before doing anything you need to load a table first. GenericCharacterTables comes with a variety of precomputed tables.
GenericCharacterTables.generic_character_table — Functiongeneric_character_table(x::String)Return the generic character table with name x.
Omitting x will return the names of all importable character tables.
Examples
julia> g = generic_character_table("GL2")
Generic character table GL2
of order q^4 - q^3 - q^2 + q
with 4 irreducible character types
with 4 class types
with parameters (i, j, l, k)GenericCharacterTables also provides some tables of Green functions. They can be loaded similarly.
GenericCharacterTables.green_function_table — Functiongreen_function_table(x::String)Return the greenfunction table with name x.
Omitting x will return the names of all importable greenfunctions.
Examples
julia> g = green_function_table("GL2")
Generic character table GL2
of order q^4 - q^3 - q^2 + q
with 2 irreducible character types
with 2 class types
without parametersProperties
GenericCharacterTables.info — Methodinfo(t::Table)Return the metadata of t in LaTeX format. This usually includes the time the table was first computed.
GenericCharacterTables.number_of_characters — Methodnumber_of_characters(t::Table)Return the number of irreducible characters of table t.
Examples
julia> g=generic_character_table("GL2");
julia> number_of_characters(g)
q^2 - 1
GenericCharacterTables.number_of_parameters — Functionnumber_of_parameters(t::CharTable)Return the number of class and character parameters of the table t.
Examples
julia> g=generic_character_table("GL2");
julia> number_of_parameters(g)
4
GenericCharacterTables.parameters — Functionparameters(class::GenericConjugacyClass)Return the parameters of the conjugacy class type class. This includes the parameter names, ranges and exceptions.
Examples
julia> g=generic_character_table("GL2");
julia> parameters(conjugacy_class_type(g, 3))
i ∈ {1,…, q - 1}, j ∈ {1,…, q - 1} except i - j ∈ (q - 1)ℤparameters(class::SimpleGenericConjugacyClass)Return the parameters of the conjugacy class type class. This includes the parameter names, ranges and exceptions.
Examples
julia> g=generic_character_table("uniGL2");
julia> parameters(conjugacy_class_type(g, 1))
parameters(t::Table, class::Int64)Return the parameters of the class type class of the table t. This includes the parameter names, ranges and exceptions.
Examples
julia> g=generic_character_table("GL2");
julia> parameters(g, 3)
i ∈ {1,…, q - 1}, j ∈ {1,…, q - 1} except i - j ∈ (q - 1)ℤ
parameters(t::CharTable)Return all parameters the table t depends on.
Examples
julia> g=generic_character_table("GL2");
julia> parameters(g)
(q, (i, j, l, k))
parameters(char::AbstractGenericCharacter)Return the parameters of the character type char. This includes the parameter names, ranges and exceptions.
Examples
julia> g=generic_character_table("GL2");
julia> parameters(g[3])
k ∈ {1,…, q - 1}, l ∈ {1,…, q - 1} except -l + k ∈ (q - 1)ℤAbstractAlgebra.order — Methodorder(t::Table)Return the order of the table t.
Examples
julia> g=generic_character_table("GL2");
julia> order(g)
q^4 - q^3 - q^2 + qIteration
Tables implement Julia's iteration interface to iterate over the irreducible character types stored in the table. For a table T,
length(T)returns the number of character types in the table,T[i]returns the $i$th character type.
For example we can use this to compute the order of the underlying group type. (Of course this can also be checked via order, which retrieves a precomputed value.)
julia> g=generic_character_table("GL2");
julia> sum(number_of_characters(c)*degree(c)^2 for c in g)
q^4 - q^3 - q^2 + q
julia> order(g)
q^4 - q^3 - q^2 + q