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 available yet.
True if the inner product of 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.
True if the inner product of 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. True if the character x is not a constituent of the character y, otherwise false.
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.
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.
True if the character x is an irreducible character, otherwise false.
True if the character x is a linear character, otherwise false.
True if the character x is faithful, i.e. has trivial kernel, otherwise false.
True if the character x is a real character, i.e. takes real values on all of the classes of G, otherwise false.
In this table T is a character table, and x is any class function.
A character table is an enumerated sequence of characters that behaves
only special with respect to printing; 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].
The value of the i-th irreducible character (from the character table T) on the j-th conjugacy class of G.
Given a character table T (or any sequence of character), return the number of entries.
The value of the class function x on the element g of G.
The value of the class function x on the i-th conjugacy class of G.
Given a class function x on G return its length (which equals the number of conjugacy classes group the group G).
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.
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 above.
Let Q(x) be the subfield of Qm generated by 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 Gal(Q(x)/Q) determined by the integer j. The integer j must be coprime to m.
Let Q(x) be the subfield of Qm generated by Q and the values of the G-character x. This function returns the the sequence of Galois conjugates of x under the action of the Galois group Gal(Q(x)/Q).
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).
The degree of the class function x, i.e. the value of x on the identity element of G.
The inner product of the class functions x and y, where x and y are class functions belonging to the same character ring.
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 of the class function x (which is the inner product with itself).
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) = |{ h in G | h^k = g}|. The value of Schur(x, k) is the coefficient a_x in the expression T_k = sum_( x in Irr(G)) a_x x.
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.
Procedure that, given a class function x and a Boolean value b, stores the information with x that the value of the predicate IsCharacter(x) equals b.
Given a class function x on the subgroup H of the group G, construct the class function obtained by induction of x to G. Note that if x is a character of H, then Induction(x, G) will return a character of G.
Given a class function x on the quotient group Q of the group G and the natural homomorphism f from G onto the group Q, construct the extension of x on G.
Given a class function x on the group G and a subgroup H of G, construct the restriction of x to H (a class function). Note that if x is a character of G, then Restriction(x, H) will return a character of H.
Given a class function x and a partition p of n (2 <= n <= 6), this function returns the symmetrized character with respect to p; the partition must be specified in the form of a sequence of positive integers (adding up to n).
Given group G represented as a permutation group, construct the character of G afforded by the defining permutation representation of G.
Given a group G and some subgroup H of G, construct the character of G afforded by the permutation representation of G given by the action of G on the right cosets of H in G.
Given a sequence or table of irreducible characters T for the group G and a sequence q of k elements of Qm (possibly Q), create the class function q_1 * T_1 + ... + q_k * T_k, where T_i is the i-th irreducible character of G as defined by the table T.
Given a sequence or table of class functions T for G of length l and a class function y on G, attempt to express y as a linear combination of the elements of T.The function returns two values: a sequence q=[q_1, ..., q_l] of cyclotomic field elements and a class function z. The i-th term of q is, for 1 <= i <= l defined to be the inner product of y and T_i, where T_i is the i-th entry of T. The sequence q determines a class function x=q_1.T_1 + ... + q_l.T_l which will equal y if T is a complete table of irreducible characters. The difference z=y - x is the second return value; it is the zero class function if and only if y is a linear combination of the T_i.
> A := AlternatingGroup(GrpPerm, 5); > R := CharacterRing(A);The first character will be the principal character
> T1 := R ! 1; > print T1; ( 1, 1, 1, 1, 1 )Next construct the permutation character
> pc := PermutationCharacter(A); > T2 := pc - T1; > print InnerProduct(pc, T1), InnerProduct(T2, T2); 1 1 > print T2; ( 4, 0, 1, -1, -1 )It follows that pc - T1 is an irreducible character
> B := Stabilizer(A, 5); > r := RootOfUnity(3, CyclotomicField(3)); > S := CharacterRing(B); > lambda := S ! [1, 1, r, r^2 ]; > print IsLinear(lambda); trueThis defines a linear character on a subgroup of index 5 in A
> T3 := Induction(lambda, A); > print InnerProduct(T3, T3); 1 > print T3; ( 5, 1, -1, 0, 0 )Finally we use characters on the cyclic subgroup of order 5:
> K := sub<A | (1,2,3,4,5) >; > Y := CharacterTable(K); > print Y;We subtract what we already know from mu and get a new irreducible. We use decomposition with respect to a sequence.
Character Table of Group K --------------------------
------------------------------- Class | 1 2 3 4 5 Size | 1 1 1 1 1 Order | 1 5 5 5 5 ------------------------------- p = 5 1 1 1 1 1 ------------------------------- X.1 + 1 1 1 1 1 X.2 0 1 Z1 Z1#2 Z1#3 Z1#4 X.3 0 1 Z1#2 Z1#4 Z1 Z1#3 X.4 0 1 Z1#3 Z1 Z1#4 Z1#2 X.5 0 1 Z1#4 Z1#3 Z1#2 Z1
Explanation of Symbols: -----------------------
# denotes algebraic conjugation, that is, # k indicates replacing the root of unity w by w^k
Z1 = -1 - zeta_5 - zeta_5^2 - zeta_5^3
> mu := Induction(Y[2], A);
> _, T4 := Decomposition([T1, T2, T3], mu); > print InnerProduct(T4, T4); 1 > print T4; ( 3, -1, 0, (1 + zeta_5^2 + zeta_5^3), (-zeta_5^2 - zeta_5^3) ) > T5 := GaloisConjugate(T4, 2); > print T5; ( 3, -1, 0, (-zeta_5^2 - zeta_5^3), (1 + zeta_5^2 + zeta_5^3) )Compare this to the standard character table:
> print CharacterTable(A);[Next] [Prev] [_____] [Left] [Up] [Index] [Root]
Character Table of Group A --------------------------
--------------------------- Class | 1 2 3 4 5 Size | 1 15 20 12 12 Order | 1 2 3 5 5 --------------------------- p = 2 1 1 3 5 4 p = 3 1 2 1 5 4 p = 5 1 2 3 1 1 --------------------------- X.1 + 1 1 1 1 1 X.2 + 3 -1 0 Z1 Z1#2 X.3 + 3 -1 0 Z1#2 Z1 X.4 + 4 0 1 -1 -1 X.5 + 5 1 -1 0 0
Explanation of Symbols: -----------------------
# denotes algebraic conjugation, that is, # k indicates replacing the root of unity w by w^k
Z1 =(1 + zeta_5^2 + zeta_5^3)