For a semisimple category C of the form \bigoplus_{i \in I} k\mathrm{-vec} to become a rigid symmetric closed monoidal skeletal category, the set I has to be equipped with extra strucutre. To become a skeletal category, we need:
a total ordering on I.
To become a monoidal category, we need:
a function \texttt{IsYieldingIdentities}, deciding whether an object yields the identity whenever it is part of an associator triple or a braiding pair,
functions \texttt{Multiplicity} and \texttt{*}, defining the tensor product on objects,
a function \texttt{AssociatorFromData}, defining the tensor product on morphisms.
To become a symmetric monoidal category, we need:
a function \texttt{ExteriorPower}.
To become a rigid symmetric monoidal category, we need:
a function \texttt{Dual}, defining duals on objects.
In the following, two families of such sets I are introduced:
\texttt{GIrreducibleObject}: For a group G, the set I consists of the irreducible characters of G. We call the elements in I the G-irreducible objects.
\texttt{GZGradedIrreducibleObject}: For a group G, the set I consists of the irreducible characters of G together with a degree n \in \mathbb{Z}. We call the elements in I the G-\mathbb{Z}-irreducible objects.
‣ GIrreducibleObject( c ) | ( attribute ) |
Returns: a G-irreducible object
The argument is a character c of a group. The output is its associated G-irreducible object.
‣ GZGradedIrreducibleObject( n, c ) | ( operation ) |
Returns: a G-\mathbb{Z}-irreducible object
The argument is an integer n and a character c of a group. The output is their associated G-\mathbb{Z}-irreducible object.
‣ UnderlyingCharacter( i ) | ( attribute ) |
Returns: an irreducible character
The argument is a G-irreducible object i. The output is its underlying character.
‣ UnderlyingGroup( i ) | ( attribute ) |
Returns: a group
The argument is a G-irreducible object i. The output is its underlying group.
‣ UnderlyingCharacterTable( i ) | ( attribute ) |
Returns: a character table
The argument is a G-irreducible object i. The output is the character table of its underlying group.
‣ UnderlyingIrreducibleCharacters( i ) | ( attribute ) |
Returns: a list
The argument is a G-irreducible object i. The output is a list consisting of the irreducible characters of its underlying group.
‣ UnderlyingCharacterNumber( i ) | ( attribute ) |
Returns: an integer
The argument is a G-irreducible object i. The output is the integer n such that the n-th entry of the list of the underlying irreducible characters is the underlying irreducible character of i.
‣ Dimension( i ) | ( attribute ) |
Returns: an integer
The argument is a G-irreducible object i. The output is the dimension of its underlying irreducible character.
‣ Dual( i ) | ( attribute ) |
Returns: a G-irreducible object
The argument is a G-irreducible object i. The output is the G-irreducible object associated to the dual character of c, where c is the associated character of i.
‣ UnderlyingCharacter( i ) | ( attribute ) |
Returns: an irreducible character
The argument is a G-\mathbb{Z}-irreducible object i. The output is its underlying character.
‣ UnderlyingDegree( i ) | ( attribute ) |
Returns: an integer
The argument is a G-\mathbb{Z}-irreducible object i. The output is its underlying degree.
‣ UnderlyingGroup( i ) | ( attribute ) |
Returns: a group
The argument is a G-\mathbb{Z}-irreducible object i. The output is its underlying group.
‣ UnderlyingCharacterTable( i ) | ( attribute ) |
Returns: a character table
The argument is a G-\mathbb{Z}-irreducible object i. The output is the character table of its underlying group.
‣ UnderlyingIrreducibleCharacters( i ) | ( attribute ) |
Returns: a list
The argument is a G-\mathbb{Z}-irreducible object i. The output is a list consisting of the irreducible characters of its underlying group.
‣ UnderlyingCharacterNumber( i ) | ( attribute ) |
Returns: an integer
The argument is a G-\mathbb{Z}-irreducible object i. The output is the integer n such that the n-th entry of the list of the underlying irreducible characters is the underlying irreducible character of i.
‣ Dimension( i ) | ( attribute ) |
Returns: an integer
The argument is a G-\mathbb{Z}-irreducible object i. The output is the dimension of its underlying irreducible character.
‣ Dual( i ) | ( attribute ) |
Returns: a G-\mathbb{Z}-irreducible object
The argument is a G-\mathbb{Z}-irreducible object i. The output is the G-\mathbb{Z}-irreducible object associated to the degree -n and the dual character of c, where n is the underlying degree and c is the underlying character of i.
‣ IsYieldingIdentities( i ) | ( property ) |
Returns: a boolean
The argument is a G-irreducible object i. The output is true if the underlying character of i is the trivial one, false otherwise.
‣ IsYieldingIdentities( i ) | ( property ) |
Returns: a boolean
The argument is a G-\mathbb{Z}-irreducible object i. The output is true if the underlying character of i is the trivial one, false otherwise.
‣ Multiplicity( i, j, k ) | ( operation ) |
Returns: an integer
The arguments are 3 G-irreducible objects i,j,k. Let their underlying characters be denoted by a,b,c, respectively. Then the output is the number \langle a, b\cdot c \rangle, i.e., the multiplicity of a in the product of characters b \cdot c.
3.5-2 \*‣ \*( i, j ) | ( operation ) |
Returns: a list
The arguments are 2 G-irreducible objects i,j with underlying irreducible characters a,b, respectively. The output is a list L = [ [ n_1, k_1 ], \dots, [ n_l, k_l ] ] consisting of positive integers n_c and G-irreducible objects k_c representing the character decomposition into irreducibles of the product a\cdot b.
‣ AssociatorFromData( i, j, k, A, F, L ) | ( operation ) |
Returns: a list
The arguments are
three G-irreducible objects i,j,k,
a list A containing the associator on all irreducibles as strings, e.g., the list constructed by the methods provided in this package,
a homalg field F,
a list L = [ [ n_1, h_1 ], \dots, [ n_l, h_l ] ] consisting of positive integers n_c and G-irreducible objects h_c representing the character decomposition into irreducibles of the product of i,j,k.
The output is the list [ [ \alpha_{h_1}, h_1 ], \dots, [ \alpha_{h_l}, h_l ] ], where \alpha_{h_c} is the F-vector space homomorphism representing the h_c-th component of the associator of i,j,k.
‣ ExteriorPower( i, j ) | ( operation ) |
Returns: a list
The arguments are two G-irreducible objects i, j. The output is the empty list if i is not equal to j. Otherwise, the output is a list L = [ [ n_1, k_1 ], \dots, [ n_1, k_l ] ] consisting of positive integers n_j and G-irreducible objects k_j, corresponding to the decomposition of the second exterior power character \wedge^2 c into irreducibles. Here, c is the associated character of i.
‣ Multiplicity( i, j, k ) | ( operation ) |
Returns: an integer
The arguments are 3 G-\mathbb{Z}-irreducible objects i,j,k. Let their underlying characters be denoted by a,b,c, respectively, and their underlying degrees by n_i, n_j, n_k, respectively. The output is 0 if n_i is not equal to n_j + n_k. Otherwise, the output is the number \langle a, b\cdot c \rangle, i.e., the multiplicity of a in the product of characters b \cdot c.
Let their underlying characters be denoted by a,b, respectively, and their underlying degrees by n_i, n_j, respectively. if n_i = n_j and the underlying character number of j
3.5-6 \*‣ \*( i, j ) | ( operation ) |
Returns: a list
The arguments are 2 G-\mathbb{Z}-irreducible objects i,j with underlying irreducible characters a,b, respectively. The output is a list L = [ [ n_1, k_1 ], \dots, [ n_l, k_l ] ] consisting of positive integers n_c and G-irreducible objects k_c representing the character decomposition into irreducibles of the product a\cdot b. The underlying degrees of k_c are given by the sum of the underlying degrees of i and j.
‣ AssociatorFromData( i, j, k, A, F, L ) | ( operation ) |
Returns: a list
The arguments are
three G-\mathbb{Z}-irreducible objects i,j,k,
a list A containing the associator on all irreducibles (of G-irreducible objects) as strings, e.g., the list constructed by the methods provided in this package,
a homalg field F,
a list L = [ [ n_1, h_1 ], \dots, [ n_l, h_l ] ] consisting of positive integers n_c and G-\mathbb{Z}-irreducible objects h_c representing the character decomposition into irreducibles of the product of i,j,k.
The output is the list [ [ \alpha_{h_1}, h_1 ], \dots, [ \alpha_{h_l}, h_l ] ], where \alpha_{h_c} is the F-vector space homomorphism representing the h_c-th component of the associator of i,j,k.
‣ ExteriorPower( i, j ) | ( operation ) |
Returns: a list
The arguments are two G-\mathbb{Z}-irreducible objects i, j. The output is the empty list if the underlying characters of i and j are unequal. Otherwise, the output is a list L = [ [ n_1, k_1 ], \dots, [ n_1, k_l ] ] consisting of positive integers n_j and G-\mathbb{Z}-irreducible objects k_a, corresponding to the decomposition of the second exterior power character \wedge^2 c into irreducibles. Here, c is the associated character of i and j. The underlying degree of k_a is the sum of the underlying degrees of i and j.
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