3 Irreducible Objects

3.3 Attributes

3.3-1 UnderlyingCharacter

3.3-2 UnderlyingGroup

3.3-3 UnderlyingCharacterTable

3.3-4 UnderlyingIrreducibleCharacters

3.3-5 UnderlyingCharacterNumber

3.3-6 Dimension

3.3-7 Dual

3.3-8 UnderlyingCharacter

3.3-9 UnderlyingDegree

3.3-10 UnderlyingGroup

3.3-11 UnderlyingCharacterTable

3.3-12 UnderlyingIrreducibleCharacters

3.3-13 UnderlyingCharacterNumber

3.3-14 Dimension

3.3-15 Dual

3.3-1 UnderlyingCharacter

3.3-2 UnderlyingGroup

3.3-3 UnderlyingCharacterTable

3.3-4 UnderlyingIrreducibleCharacters

3.3-5 UnderlyingCharacterNumber

3.3-6 Dimension

3.3-7 Dual

3.3-8 UnderlyingCharacter

3.3-9 UnderlyingDegree

3.3-10 UnderlyingGroup

3.3-11 UnderlyingCharacterTable

3.3-12 UnderlyingIrreducibleCharacters

3.3-13 UnderlyingCharacterNumber

3.3-14 Dimension

3.3-15 Dual

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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