Element Operations

Arithmetic

In the list of arithmetic operations below \(x\) and \(y\) denote class functions in the same ring, and \(a\) denotes a scalar, which is any element coercible into a cyclotomic field. Also, \(j\) denotes an integer.

+ y: AlgChtrElt -> AlgChtrElt

- y: AlgChtrElt -> AlgChtrElt

x + y: AlgChtrElt, AlgChtrElt -> AlgChtrElt

x - y: AlgChtrElt, AlgChtrElt -> AlgChtrElt

x * y: AlgChtrElt, AlgChtrElt -> AlgChtrElt

a * x: FldCycElt, AlgChtrElt -> AlgChtrElt

x ^ j: AlgChtrElt, RngIntElt -> AlgChtrElt

Predicates and Booleans

The following Boolean-valued functions are available. Note that with the exception of in, notin, IsReal, and IsFaithful, these functions use the table of irreducible characters, which will be created if it is not yet available.

x in y: AlgChtrElt, AlgChtrElt -> BoolElt: Returns true if the inner product of class functions \(x\) and \(y\) is non-zero, otherwise false. If \(x\) is irreducible and \(y\) is a character, this tests whether or not \(x\) is a constituent of \(y\).

x notin y: AlgChtrElt, AlgChtrElt -> BoolElt: Returns true if the inner product of class functions \(x\) and \(y\) is zero, otherwise false. If \(x\) is irreducible and \(y\) is a character, this tests whether or not \(x\) is not a constituent of \(y\). Returns true if the character \(x\) is not a constituent of the character \(y\), otherwise false.

a in F: FldFunElt, FldFun -> BoolElt
a notin F: FldFunElt, FldFun -> BoolElt
x eq y: AlgChtrElt, AlgChtrElt -> BoolElt
x ne y: AlgChtrElt, AlgChtrElt -> BoolElt
IsCharacter(x): AlgChtrElt -> BoolElt: Returns true if the class function \(x\) is a character, otherwise false. A class function is a character if and only if all inner products with the irreducible characters are non-negative integers.

IsGeneralizedCharacter(x): AlgChtrElt -> BoolElt: Returns true if the class function \(x\) is a generalized character, otherwise false. A class function is a generalized character if and only if all inner products with the irreducible characters are integers.

IsIrreducible(x): AlgChtrElt -> BoolElt: Returns true if the character \(x\) is an irreducible character, otherwise false.

IsLinear(x): AlgChtrElt -> BoolElt: Returns true if the character \(x\) is a linear character, otherwise false.

IsFaithful(x): AlgChtrElt -> BoolElt: Returns true if the character \(x\) is faithful, i.e. has trivial kernel, otherwise false.

IsReal(x): AlgChtrElt -> BoolElt: Returns true if the character \(x\) is a real character, i.e. takes real values on all of the classes of \(G\), otherwise false.

IsOne(x): AlgChtrElt -> BoolElt
IsMinusOne(x): AlgChtrElt -> BoolElt
IsZero(x): AlgChtrElt -> BoolElt
IsPermutationCharacter(c): AlgChtrElt -> BoolElt: Test if a character \(c\) of \(G\) is a virtual permutation character, that is \({\mathbb{C}}[X]-{\mathbb{C}}[Y]\) for some \(G\)-sets \(X\) and \(Y\). (The implementation relies on BurnsideCokernel, and does not compute the subgroup lattice.)

Example: Testing for Permutation Characters (ex-cd3c30)

In a symmetric group \(S_n\) every character is a virtual permutation character, but not in the quaternion group \(Q_8\).

> C:=CharacterTable(Group("S5"));
> [IsPermutationCharacter(c): c in C];
[ true, true, true, true, true, true, true ]
> C:=CharacterTable(Group("Q8"));
> [IsPermutationCharacter(c): c in C];
[ true, true, true, true, false ]

Run in calculator (opens in a new tab)

Accessing Class Functions

In this subsection \(T\) is a character table, and \(x\) is any class function. A character table is an enumerated sequence of characters that has a special print function attached. In particular, its entries can be accessed with the ordinary sequence indexing operations.

T[i]: TabChtr, RngIntElt -> AlgChtrElt: Given the table \(T\) of ordinary characters of \(G\), return the \(i\)-th character of \(G\), where \(i\) is an integer in the range \([1...k]\).

T[i][j]: TabChtr, RngIntElt, RngIntElt -> FldCycElt: The value of the \(i\)-th irreducible character (from the character table \(T\)) on the \(j\)-th conjugacy class of \(G\).

# T: SeqEnum -> RngIntElt: Given a character table \(T\) (or any sequence of characters), return the number of entries.

x(g): AlgChtrElt, GrpElt -> FldCycElt
g @ x: GrpElt, AlgChtrElt -> FldCycElt: The value of the class function \(x\) on the element \(g\) of \(G\).

x[i]: AlgChtrElt, RngIntElt -> FldCycElt: The value of the class function \(x\) on the \(i\)-th conjugacy class of \(G\).

# x: AlgChtrElt -> RngIntElt: Given a class function \(x\) on \(G\) return its length (which equals the number of conjugacy classes of the group \(G\)).

Conjugation of Class Functions

x ^ g: AlgChtrElt, GrpElt -> AlgChtrElt: Given a class function \(x\) on a normal subgroup \(N\) of the group \(G\), and an element \(g\) of \(G\), construct the conjugate class function \(x^g\) of \(x\) which is defined as follows: \(x^g(n) = x( g^{-1}ng)\), for all \(n\) in \(N\).

x ^ H: AlgChtrElt, Grp -> { AlgChtrElt }: Given a class function \(x\) on a normal subgroup \(N\) of the group \(G\), and a subgroup \(H\) of \(G\), construct the sequence of conjugates of \(x\) under the action of the subgroup \(H\). The action of an element of \(H\) on \(x\) is that defined for \(x^g\).

GaloisConjugate(x, j): AlgChtrElt, RngIntElt -> AlgChtrElt: Let \({\mathbb{Q}}(x)\) be the subfield of \({\mathbb{Q}(\zeta_m)}\) generated by \({\mathbb{Q}}\) and the values of the \(G\)-character \(x\). This function returns the Galois conjugate \(x^j\) of \(x\) under the action of the element of the Galois group \({\operatorname{Gal}}({\mathbb{Q}}(x)/{\mathbb{Q}})\) determined by the integer \(j\). The integer \(j\) must be coprime to \(m\).

GaloisOrbit(x): AlgChtrElt -> { AlgChtrElt }: Let \({\mathbb{Q}}(x)\) be the subfield of \({\mathbb{Q}(\zeta_m)}\) generated by \({\mathbb{Q}}\) and the values of the \(G\)-character \(x\). This function returns the sequence of Galois conjugates of \(x\) under the action of the Galois group \({\operatorname{Gal}}({\mathbb{Q}}(x)/{\mathbb{Q}})\).

IsConjugate(x, y): AlgChtrElt, AlgChtrElt -> BoolElt, RngIntElt: Determine if \(x\) and \(y\) are galois conjugate characters. Return false if not. If so, return true, and \(j\) such that GaloisConjugate(x, j) is \(y\).

ClassPowerCharacter(x, j): AlgChtrElt, RngIntElt -> AlgChtrElt: Given a class function \(x\) on the group \(G\) and a positive integer \(j\), construct the class function \(x^j\) which is defined as follows: \(x^j(g) = x(g^j)\).

Functions Returning a Scalar

Degree(x): AlgChtrElt -> RngIntElt: The degree of the class function \(x\), i.e. the value of \(x\) on the identity element of \(G\).

InnerProduct(x, y): AlgChtrElt, AlgChtrElt -> FldCycElt: The inner product of the class functions \(x\) and \(y\), where \(x\) and \(y\) are class functions belonging to the same character ring.

Order(x): AlgChtrElt -> RngIntElt: Given a linear character of the group \(G\), determine the order of \(x\) as an element of the group of linear characters of \(G\).

Norm(x): AlgChtrElt -> FldCycElt: Norm of the class function \(x\) (which is the inner product with itself).

Schur(x, k): AlgChtrElt, RngIntElt -> FldCycElt

Indicator(x): AlgChtrElt -> FldCycElt

Given class function \(x\) and a positive integer \(k\), return the generalised Frobenius–Schur indicator which is defined as follows: Suppose \(g\) is some element of \(G\), and set \(T_k(g) = \vert\{ h\in G | h^k = g\}\vert\). The value of Schur(x, k) is the coefficient \(a_x\) in the expression \(T_k = \sum_{ x\in{\operatorname{Irr}}(G)} a_x x\).

The call Indicator(x) is equivalent to Schur(x,2).

StructureConstant(G, i, j, k): Grp, RngIntElt, RngIntElt, RngIntElt -> RngIntElt: The structure constant \(a_{i,j,k}\) for the centre of the group algebra of the group \(G\). If \(K_i\) is the formal sum of the elements of the \(i\)-th conjugacy class, \(a_{i,j,k}\) is defined by the equation \(K_i * K_j = \sum_k a_{i,j,k} * K_k\).

Element Operations#

Arithmetic#

Predicates and Booleans#

Accessing Class Functions#

Conjugation of Class Functions#

Functions Returning a Scalar#