Explanations of Terms Used

Char

The character afforded by the representation. Characters are numbered as in the ATLAS. There is often ambiguity as to which character a representation affords. In the case of such ambiguity, we use conjugacy class representatives mentioned in the notes to resolve the ambiguities.

CCL Test

The results of taking traces of various elements in the representation and checking against the character. This is useful for identifying mislabelled representations, particularly in cases where there is more than one representation of a given dimension.

Checker Test

The results of applying the checker (semi-presentation) to the representation in question. Passing this test means that (providing the matrices generate the correct group) the generators given are standard generators. If no check has been carried out, the field is left blank.

Comments

Usually records the way the representation was constructed. If the representation was not found by us, we credit the finder here. The notation perm rep on n points means that the representation was found by using the SPLIT-P package on a permutation representation.

Dim

The size of the matrices of the representation.

File

A link to the file containing the representation in question. All representations are in GAP-readable format, and can be read with the command:

rep := ReadAsFunction(filename)();;
gens := rep.generators;;
x := gens[1];
y := gens[2];

Group

The group for which the representation in question is faithful.

Ind

The Frobenius-Schur indicator of the representation (or of one of the irreducible constituents of the representation). As in the ATLAS, the notations -, o and + indicate -1, 0 and +1 respectively.

Max

The maximum absolute entry of the generators of a representation. (N.B. The maximum entry over the whole group is not recorded.)

Representation genus

A 3-tuple containing the representation's group, dimension and indicator. Note that a single representation genus often applies to several inequivalent (and algebraically non-conjugate) representations.

Ring

A ring over which the representation can be written. In general, this is not the smallest such ring (especially if the ring specified is C. However, rational representations are always marked with Z or Q, and representations writeable over Z(i) or Z(ω) (where ω is a cube root of unity) are always marked thus.

Size (Kb)

The size in kilobytes of the GAP-readable file containing the representation.

Sparsity

The percentage of zero entries in the generators of a representation. (N.B. The average sparsity over the group is not recorded.)

Last updated 8th September 2005