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### 3 Irreducible Objects

#### 3.1 Introduction

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.

#### 3.2 Constructors

##### 3.2-1 GIrreducibleObject
 ‣ 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.

#### 3.3 Attributes

##### 3.3-1 UnderlyingCharacter
 ‣ UnderlyingCharacter( i ) ( attribute )

Returns: an irreducible character

The argument is a G-irreducible object i. The output is its underlying character.

##### 3.3-2 UnderlyingGroup
 ‣ UnderlyingGroup( i ) ( attribute )

Returns: a group

The argument is a G-irreducible object i. The output is its underlying group.

##### 3.3-3 UnderlyingCharacterTable
 ‣ 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.

##### 3.3-4 UnderlyingIrreducibleCharacters
 ‣ 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.

##### 3.3-5 UnderlyingCharacterNumber
 ‣ 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.

##### 3.3-6 Dimension
 ‣ Dimension( i ) ( attribute )

Returns: an integer

The argument is a G-irreducible object i. The output is the dimension of its underlying irreducible character.

##### 3.3-7 Dual
 ‣ 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.

##### 3.3-8 UnderlyingCharacter
 ‣ UnderlyingCharacter( i ) ( attribute )

Returns: an irreducible character

The argument is a G-\mathbb{Z}-irreducible object i. The output is its underlying character.

##### 3.3-9 UnderlyingDegree
 ‣ UnderlyingDegree( i ) ( attribute )

Returns: an integer

The argument is a G-\mathbb{Z}-irreducible object i. The output is its underlying degree.

##### 3.3-10 UnderlyingGroup
 ‣ UnderlyingGroup( i ) ( attribute )

Returns: a group

The argument is a G-\mathbb{Z}-irreducible object i. The output is its underlying group.

##### 3.3-11 UnderlyingCharacterTable
 ‣ 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.

##### 3.3-12 UnderlyingIrreducibleCharacters
 ‣ 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.

##### 3.3-13 UnderlyingCharacterNumber
 ‣ 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.

##### 3.3-14 Dimension
 ‣ 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.

##### 3.3-15 Dual
 ‣ 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.

#### 3.4 Properties

##### 3.4-1 IsYieldingIdentities
 ‣ 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.

##### 3.4-2 IsYieldingIdentities
 ‣ 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.

#### 3.5 Operations

##### 3.5-1 Multiplicity
 ‣ 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.

##### 3.5-3 AssociatorFromData
 ‣ 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.

##### 3.5-4 ExteriorPower
 ‣ 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.

##### 3.5-5 Multiplicity
 ‣ 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.

##### 3.5-7 AssociatorFromData
 ‣ 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.

##### 3.5-8 ExteriorPower
 ‣ 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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