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, vec, 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 matrix category vec of 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, vec, 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 matrix category vec of 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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