Unexported Functions

CharTable <: Table

The type for generic character tables. This is used to model generic character tables containing generic cyclotomic entries.

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)

(t::CharTable)(c::GenericCharacter)

Return c as a generic character of t. This will only work if t is a version of the parent table of c with a more restricted congruence.

GenericCharacter <: AbstractGenericCharacter

The type for generic characters. These are the generic characters used in CharTable.

Examples

julia> g=generic_character_table("GL2");

julia> g[1]
Generic character of GL2
  with parameters
    k ∈ {1,…, q - 1}
  of degree 1
  with values
    exp(2π𝑖((2*i*k)//(q - 1)))
    exp(2π𝑖((2*i*k)//(q - 1)))
    exp(2π𝑖((i*k + j*k)//(q - 1)))
    exp(2π𝑖((i*k)//(q - 1)))

GenericConjugacyClass <: AbstractGenericConjugacyClass

The type for generic conjugacy classes. These are the generic conjugacy classes used in CharTable.

GenericCyclo <: RingElem

The type for generic cyclotomic numbers.

Examples

julia> R = universal_polynomial_ring(QQ; cached=false);

julia> q = gen(R, "q");

julia> S = generic_cyclotomic_ring(R);

julia> S(q; exponent=1//(q-1))
q*exp(2π𝑖(1//(q - 1)))

GenericCycloFrac

The type for fractions of generic cyclotomic numbers.

Examples

julia> R = universal_polynomial_ring(QQ; cached=false);

julia> q = gen(R, "q");

julia> S = generic_cyclotomic_ring(R);

julia> a = S(q; exponent=1//(q-1))
q*exp(2π𝑖(1//(q - 1)))

julia> b = S(q^2; exponent=1//(q^2-1))
q^2*exp(2π𝑖(1//(q^2 - 1)))

julia> a//b
q*exp(2π𝑖(1//(q - 1)))//(q^2*exp(2π𝑖(1//(q^2 - 1))))

GenericCycloRing <: Ring

The ring of generic cyclotomic numbers.

Examples

julia> R = universal_polynomial_ring(QQ; cached=false);

julia> q = gen(R, "q");

julia> S = generic_cyclotomic_ring(R)
Generic cyclotomic ring
  over Rational field
  dependent on q

Parameter

A paramter of a generic character or class type unique up to a polynomial modulus. They are used in Parameters.

ParameterExceptions

A collection of parameter exceptions used in GenericCycloFrac.

ParameterSubstitution

A substitution of paramters used in Parameters. They are generated by for example specclassparam!.

Parameters

Parameters of generic characters and class types. This is used in GenericCharacter and CharTable and is only of internal use.

SimpleCharTable{T} <: Table

The type for simple generic character tables. This is used to model generic character tables containing polynomial entries. The type parameter T is the type of the table entries.

Examples

julia> g=generic_character_table("uniGL2")
Generic character table uniGL2
  of order q^4 - q^3 - q^2 + q
  with 2 irreducible character types
  with 4 class types
  without parameters

SimpleGenericCharacter <: AbstractGenericCharacter

The type for simple generic characters. These are the generic characters used in SimpleCharTable.

Examples

julia> g=generic_character_table("uniGL2")
Generic character table uniGL2
  of order q^4 - q^3 - q^2 + q
  with 2 irreducible character types
  with 4 class types
  without parameters

julia> g[1]
Generic character of uniGL2
  of degree q
  with values
    q
    0
    1
    -1

SimpleGenericConjugacyClass <: AbstractGenericConjugacyClass

The type for simple generic conjugacy classes. These are the generic conjugacy classes used in SimpleCharTable.

normal_form(f::ZZUPoly, m::Int64)

Return a normal form of f modulo m, such that normal_form(f,m) is equal to normal_form(g,m) if and only if f and g are congruent modulo m.

Examples

julia> R=universal_polynomial_ring(ZZ);

julia> x=gen(R, :x);

julia> normal_form(4*x^2,6)
x^2 + 3*x

julia> normal_form(4*x^2-(x^2+3*x),6)
0

julia> normal_form(4*x^9+x^7,12)
x^3 + 4*x

julia> normal_form(4*x^9+x^7-(x^3+4*x),12)
0

iszero(x::GenericCycloFrac; ignore_exceptions::Bool=false)

Return if x is zero. If ignore_exceptions is true then the exceptions of x will not be considered.

show(io::IO, c::AbstractGenericCharacter)

Display a summary of the generic character c.

Examples

julia> g=generic_character_table("GL2");

julia> g[3]
Generic character of GL2
  with parameters
    k ∈ {1,…, q - 1}, l ∈ {1,…, q - 1} except -l + k ∈ (q - 1)ℤ
  of degree q + 1
  with values
    (q + 1)*exp(2π𝑖((i*l + i*k)//(q - 1)))
    exp(2π𝑖((i*l + i*k)//(q - 1)))
    exp(2π𝑖((i*l + j*k)//(q - 1))) + exp(2π𝑖((i*k + j*l)//(q - 1)))
    0

julia> [g[3]]
1-element Vector{GenericCharacterTables.GenericCharacter}:
 Generic character of GL2

show(io::IO, c::AbstractGenericConjugacyClass)

Display a summary of the generic character c.

Examples

julia> g=generic_character_table("GL2");

julia> conjugacy_class_type(g, 3)
Generic conjugacy class of GL2
  with parameters 
    i ∈ {1,…, q - 1}, j ∈ {1,…, q - 1} except i - j ∈ (q - 1)ℤ
  of order q^2 + q
  with values
    exp(2π𝑖((i*k + j*k)//(q - 1)))
    exp(2π𝑖((i*k + j*k)//(q - 1)))
    exp(2π𝑖((i*l + j*k)//(q - 1))) + exp(2π𝑖((i*k + j*l)//(q - 1)))
    0

julia> [conjugacy_class_type(g, 3)]
1-element Vector{GenericCharacterTables.GenericConjugacyClass}:
 Generic conjugacy class of GL2

show(io::IO, t::Table)

Display a summary of the generic character table t.

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)

julia> [g]
1-element Vector{GenericCharacterTables.CharTable}:
 Generic character table GL2

add_exception!(a::ParameterExceptions, exception::UPolyFrac)

Include exception into a. This also removes all now redundant exceptions from a.

is_integer(x::UPolyFrac)

Return if x represents an integer.

is_restriction(x::ParameterExceptions)

Return if x actually restricts something.

merge(x::ParameterExceptions, y::ParameterExceptions)

Return a new collection of parameter exceptions composed of x and y where all redundant exceptions are omitted.

nesum(a::GenericCyclo, var::Int64, lower::Int64, upper::Union{Int64,UPoly})

Return the sum of a, from var=lower to upper as GenericCycloFrac using the closed formular for geometric sums. If this is not possible an exception will be thrown.

Examples

julia> R = universal_polynomial_ring(QQ; cached=false);

julia> q = gen(R, "q");

julia> S = generic_cyclotomic_ring(R);

julia> i, = gens(R, ["i"]);

julia> a = S(Dict(1//(q-1)*i => R(1)))
exp(2π𝑖(i//(q - 1)))

julia> GenericCharacterTables.nesum(a, i, 1, q-1)
0
With exceptions:
  1 ∈ (q - 1)ℤ

shift_char_parameters(t::CharTable, a::Union{Parameters,GenericCyclo,GenericCycloFrac}, steps::Int64)

Replace all character parameters of t in a by their counterparts suffixed with steps.

This is done by shifting them steps*number_of_parameters(t) steps further in t.argumentring.

shift_class_parameters(t::CharTable, a::Union{Parameters,GenericCyclo,GenericCycloFrac}, steps::Int64)

Replace all class parameters of t in a by their counterparts suffixed with steps.

This is done by shifting them steps*number_of_parameters(t) steps further in t.argumentring.

shrink(a::GenericCycloFrac{<:NfPoly})

Remove exceptions from a that follow from the others. And try to simplify the representation of a.