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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 — Function
conjugacy_class_type(t::CharTable, class::Int64)

Return the generic conjugacy class class of t.

source
conjugacy_class_type(t::SimpleCharTable{T}, class::Int64) where T <: NfPoly

Return the generic conjugacy class class of t.

source
Oscar.number_of_conjugacy_classes — Method
number_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
source
Oscar.number_of_conjugacy_classes — Method
number_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 - 1
source
Oscar.number_of_conjugacy_classes — Method
number_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))
1
source
Oscar.number_of_conjugacy_classes — Method
number_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 - 1
source
AbstractAlgebra.order — Method
order(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 + q
source
AbstractAlgebra.order — Method
order(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 + q
source
GenericCharacterTables.centralizer_order — Function
centralizer_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
source
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
source
GenericCharacterTables.parameters — Method
parameters(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)ℤ
source
GenericCharacterTables.parameters — Method
parameters(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))
source
GenericCharacterTables.parameters — Method
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
source
GenericCharacterTables.info — Method
info(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]]
source

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 — Method
norm(class::GenericConjugacyClass)

Return the (generic) norm of class.

Examples

julia> g=generic_character_table("GL2");

julia> norm(conjugacy_class_type(g, 2))
1
source
LinearAlgebra.norm — Method
norm(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)
source
LinearAlgebra.norm — Method
norm(t::Table, class::Int64)

Return the (generic) norm of the class type class.

Examples

julia> g=generic_character_table("GL2");

julia> norm(g,2)
1
source
Oscar.scalar_product — Method
scalar_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)ℤ
source
Oscar.scalar_product — Method
scalar_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)
source
Oscar.scalar_product — Method
scalar_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)ℤ
source
Oscar.class_multiplication_coefficient — Method
class_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)
source
Oscar.class_multiplication_coefficient — Method
class_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)ℤ
source
Oscar.class_multiplication_coefficient — Method
class_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)ℤ
source

Iteration

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)