Conjugacy class types
A class type is a family of conjugacy classes which are indexed by a set of parameters, together with ranges of admissible values for each parameters, and a set of excluded parameter values.
Properties
GenericCharacterTables.conjugacy_class_type — Functionconjugacy_class_type(t::CharTable, class::Int64)Return the generic conjugacy class class of t.
conjugacy_class_type(t::SimpleCharTable{T}, class::Int64) where T <: NfPolyReturn the generic conjugacy class class of t.
GenericCharacterTables.number_of_conjugacy_class_types — Functionnumber_of_conjugacy_class_types(t::Table)Return the number of conjugacy class types of table t.
Examples
julia> g=generic_character_table("GL2");
julia> number_of_conjugacy_class_types(g)
4
Oscar.number_of_conjugacy_classes — Methodnumber_of_conjugacy_classes(t::Table)Return the number of conjugacy classes of table t.
Examples
julia> g=generic_character_table("GL2");
julia> number_of_conjugacy_classes(g)
q^2 - 1
Oscar.number_of_conjugacy_classes — Methodnumber_of_conjugacy_classes(class::GenericConjugacyClass)Return the number of conjugacy classes in class.
Examples
julia> g=generic_character_table("GL2");
julia> number_of_conjugacy_classes(conjugacy_class_type(g, 1))
q - 1Oscar.number_of_conjugacy_classes — Methodnumber_of_conjugacy_classes(class::SimpleGenericConjugacyClass)Return the number of conjugacy classes in class.
Examples
julia> g=green_function_table("GL2");
julia> number_of_conjugacy_classes(conjugacy_class_type(g, 1))
1Oscar.number_of_conjugacy_classes — Methodnumber_of_conjugacy_classes(t::Table, class::Int64)Return the number of conjugacy classes in the class type class of the table t.
Examples
julia> g=generic_character_table("GL2");
julia> number_of_conjugacy_classes(g, 1)
q - 1AbstractAlgebra.order — Methodorder(class::AbstractGenericConjugacyClass)Return the order of the conjugacy classes in class.
Examples
julia> g=generic_character_table("GL2");
julia> order(conjugacy_class_type(g, 3))
q^2 + qAbstractAlgebra.order — Methodorder(t::Table, class::Int64)Return the order of the conjugacy classes in the class type class of t.
Examples
julia> g=generic_character_table("GL2");
julia> order(g, 3)
q^2 + qGenericCharacterTables.centralizer_order — Functioncentralizer_order(class::AbstractGenericConjugacyClass)Return the order of the centralizer of the class type class.
Examples
julia> g=generic_character_table("GL2");
julia> centralizer_order(conjugacy_class_type(g, 1))
q^4 - q^3 - q^2 + q
centralizer_order(t::Table, class::Int64)Return the order of the centralizer of the class type class of the table t.
Examples
julia> g=generic_character_table("GL2");
julia> centralizer_order(g, 1)
q^4 - q^3 - q^2 + q
GenericCharacterTables.parameters — Methodparameters(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)ℤGenericCharacterTables.parameters — Methodparameters(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))
GenericCharacterTables.parameters — Methodparameters(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
GenericCharacterTables.info — Methodinfo(class::AbstractGenericConjugacyClass)Return the infolists of the generic conjugacy class class.
GenericCharacterTables.info — Methodinfo(class::AbstractGenericConjugacyClass)Return the infolists of the generic conjugacy class class.
GenericCharacterTables.info — Methodinfo(t::Table, class::Int64)Return the infolists of the class type class of the table t.
Examples
julia> g=generic_character_table("GL2");
julia> info(g, 1)
2-element Vector{Any}:
Any[1, 0]
Any["A_1", [1, 1]]
Norms and scalar products
GenericCharacterTables allows you to compute norms and scalar products of class types. The results are correct for all possible combinations of parameters except possibly for those where the additionally returned exceptions apply. Those consist of multivariate polynomials with coefficients in a rational function field and are satisfied if the evaluation of this polynomial is an integer.
LinearAlgebra.norm — Methodnorm(class::GenericConjugacyClass)Return the (generic) norm of class.
Examples
julia> g=generic_character_table("GL2");
julia> norm(conjugacy_class_type(g, 2))
1LinearAlgebra.norm — Methodnorm(class::SimpleGenericConjugacyClass)Return the (generic) norm of class.
Examples
julia> g=green_function_table("GL3");
julia> norm(conjugacy_class_type(g, 2))
(5*q^2 + 2*q + 3)//(q^5 - 2*q^4 + q^3)LinearAlgebra.norm — Methodnorm(t::Table, class::Int64)Return the (generic) norm of the class type class.
Examples
julia> g=generic_character_table("GL2");
julia> norm(g,2)
1Oscar.scalar_product — Methodscalar_product(class1::GenericConjugacyClass, class2::GenericConjugacyClass)Return the (generic) scalar product between class1 and class2.
Examples
julia> g=generic_character_table("GL2");
julia> scalar_product(conjugacy_class_type(g, 3), conjugacy_class_type(g, 2))
0
With exceptions:
i1 + j1 - 2*i2 ∈ (q - 1)ℤ
i1 - i2 ∈ (q - 1)ℤ
j1 - i2 ∈ (q - 1)ℤOscar.scalar_product — Methodscalar_product(class1::SimpleGenericConjugacyClass, class2::SimpleGenericConjugacyClass)Return the (generic) scalar product between class1 and class2.
Examples
julia> g=green_function_table("GL3");
julia> scalar_product(conjugacy_class_type(g, 2), conjugacy_class_type(g, 2))
(5*q^2 + 2*q + 3)//(q^5 - 2*q^4 + q^3)Oscar.scalar_product — Methodscalar_product(t::Table, class1::Int64, class2::Int64)Return the (generic) scalar product between the class types class1 and class2.
Examples
julia> g=generic_character_table("GL2");
julia> scalar_product(g,3,2)
0
With exceptions:
i1 + j1 - 2*i2 ∈ (q - 1)ℤ
i1 - i2 ∈ (q - 1)ℤ
j1 - i2 ∈ (q - 1)ℤOscar.class_multiplication_coefficient — Methodclass_multiplication_coefficient(class1::SimpleGenericConjugacyClass, class2::SimpleGenericConjugacyClass, class3::SimpleGenericConjugacyClass)Return the (generic) class multiplication coefficient of class1, class2 and class3.
Examples
julia> g=green_function_table("GL3");
julia> class_multiplication_coefficient(conjugacy_class_type(g, 1), conjugacy_class_type(g, 2), conjugacy_class_type(g, 3))
(q + 3)//(q^5 - 2*q^4 + q^3)Oscar.class_multiplication_coefficient — Methodclass_multiplication_coefficient(class1::GenericConjugacyClass, class2::GenericConjugacyClass, class3::GenericConjugacyClass)Return the (generic) class multiplication coefficient of class1, class2 and class3.
Examples
julia> g=generic_character_table("SL2.0");
julia> class_multiplication_coefficient(conjugacy_class_type(g, 2), conjugacy_class_type(g, 2), conjugacy_class_type(g, 4))
q + 1
With exceptions:
a3 ∈ (q + 1)ℤOscar.class_multiplication_coefficient — Methodclass_multiplication_coefficient(t::Table, class1::Int64, class2::Int64, class3::Int64)Return the (generic) class multiplication coefficient of the class types class1, class2 and class3 of the table t.
Examples
julia> g=generic_character_table("SL2.0");
julia> class_multiplication_coefficient(g,2,2,4)
q + 1
With exceptions:
a3 ∈ (q + 1)ℤIteration
To get a specific conjugacy class type one can also use the indexing features of generic character tables. Moreover the conjugacy class type itself implements the standard iteration and indexing interfaces and can thus be treated similarly to a Vector object.
julia> g=generic_character_table("GL2");
julia> cl=g[:,1]
Generic conjugacy class of GL2
with parameters
i ∈ {1,…, q - 1}
of order 1
with values
1: E(q - 1)^(2*i*k)
2: q*E(q - 1)^(2*i*k)
3: (q + 1)*E(q - 1)^(i*l + i*k)
4: (q - 1)*E(q - 1)^(i*k)
julia> cl[1]
E(q - 1)^(2*i*k)
julia> cl[2:3]
2-element Vector{GenericCharacterTables.GenericCyclo}:
q*E(q - 1)^(2*i*k)
(q + 1)*E(q - 1)^(i*l + i*k)
julia> cl[:]
4-element Vector{GenericCharacterTables.GenericCyclo}:
E(q - 1)^(2*i*k)
q*E(q - 1)^(2*i*k)
(q + 1)*E(q - 1)^(i*l + i*k)
(q - 1)*E(q - 1)^(i*k)
julia> for v in cl
println(v)
end
E(q - 1)^(2*i*k)
q*E(q - 1)^(2*i*k)
(q + 1)*E(q - 1)^(i*l + i*k)
(q - 1)*E(q - 1)^(i*k)