# 50 Characters

The functions described in this chapter are used to handle characters (see Chapter Character Tables). For this, in many cases one needs maps (see Chapter Maps and Parametrized Maps).

There are four kinds of functions:

First, those functions which get informations about characters; they compute the scalar product of characters (see ScalarProduct, MatScalarProducts), decomposition matrices (see Decomposition, Subroutines of Decomposition), the kernel of a character (see KernelChar), p-blocks (see PrimeBlocks), Frobenius-Schur indicators (see Indicator), eigenvalues of the corresponding representations (see Eigenvalues), and Molien series of characters (see MolienSeries), and decide if a character might be a permutation character candidate (see Permutation Character Candidates).

Second, those functions which construct characters or virtual characters, that is, differences of characters; these functions compute reduced characters (see Reduced, ReducedOrdinary), tensor products (see Tensored), symmetrisations (see Symmetrisations, SymmetricParts, AntiSymmetricParts, MinusCharacter, OrthogonalComponents, SymplecticComponents), and irreducibles differences of virtual characters (see IrreducibleDifferences). Further, one can compute restricted characters (see Restricted), inflated characters (see Inflated), induced characters (see Induced, InducedCyclic), and permutation character candidates (see Permutation Character Candidates, PermChars).

Third, those functions which use general methods for lattices. These are the LLL algorithm to compute a lattice base consisting of short vectors (see LLLReducedBasis, LLLReducedGramMat, LLL), functions to compute all orthogonal embeddings of a lattice (see OrthogonalEmbeddings), and one for the special case of D_n lattices (see DnLattice). A backtrack search for irreducible characters in the span of proper characters is performed by Extract.

Finally, those functions which find special elements of parametrized characters (see More about Maps and Parametrized Maps); they compute the set of contained virtual characters (see ContainedDecomposables) or characters (see ContainedCharacters), the set of contained vectors which possibly are virtual characters (see ContainedSpecialVectors, ContainedPossibleVirtualCharacters) or characters (see ContainedPossibleCharacters).

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GAP 3.4.4
April 1997