The User Interface to the &GAP; Character Table Library
Accessing Data of the `CTblLib` Package
Admissible Names for Character Tables in `CTblLib`

When you access a character table from the &GAP; Character Table Library, this table is specified by an admissible name.

Admissible names for the ordinary character table tbl of the group G are an &ATLAS; like name if tbl is an &ATLAS; table (see Section ), for example "M22" for the table of the Mathieu group M_{22}, "L2(13).2" for L_2(13):2, and "12_1.U4(3).2_1" for 12_1.U_4(3).2_1.

(The difference to the name printed in the &ATLAS; is that subscripts and superscripts are omitted except if they are used to qualify integer values, and double dots are replaced by a single dot.) the names that were admissible for tables of G in the &CAS; system if the &CAS; table library contained a table of G, for example sl42 for the table of the alternating group A_8.

(But note that the ordering of rows and columns of the &GAP; table may be different from that in &CAS;, see Section .) some relative names, as follows. If G is the n-th maximal subgroup (in decreasing group order) of a group whose library table subtbl is available in &GAP; and stores the value, and if name is an admissible name for subtbl then nameMn is admissible for tbl. For example, the name "J3M2" can be used to access the second maximal subgroup of the sporadic simple Janko group J_3 which has the admissible name "J3". If G is a nontrivial Sylow p normalizer in a sporadic simple group with admissible name name –where nontrivial means that G is not isomorphic to a subgroup of p:(p-1)– then nameNp is an admissible name of tbl. For example, the name "J4N11" can be used to access the table of the Sylow 11 normalizer in the sporadic simple Janko group J_4. In a few cases, the table of the Sylow p-subgroup of G is accessible via the name nameSylp where name is an admissible name of the table of G. For example, "A11Syl2" is an admissible name for the table of the Sylow 2-subgroup of the alternating group A_{11}. In a few cases, the table of an element centralizer in G is accessible via the name nameCcl where name is an admissible name of the table of G. For example, "M11C2" is an admissible name for the table of an involution centralizer in the Mathieu group M_{11}.

The recommended way to access a Brauer table is via applying the mod operator to the ordinary table and the desired characteristic (see and Section ), so it is not necessary to define admissible names of Brauer tables.

A generic character table (see Section ) is accessible only by the name given by its value.

<#Include Label="CharacterTable_for_a_string"> <#Include Label="AllCharacterTableNames"> <#Include Label="OneCharacterTableName"> <#Include Label="NameOfEquivalentLibraryCharacterTable">

The Interface to the `TomLib` Package

The &GAP; Character Table Library contains ordinary character tables of all groups for which the `TomLib` package contains the table of marks. This section describes the mapping between these character tables and their tables of marks.

If the `TomLib` package is not loaded then is the only available function from this section, but of course it is of little interest in this situation.

<#Include Label="TableOfMarks"> <#Include Label="CharacterTable_for_tom"> <#Include Label="FusionCharTableTom"> <#Include Label="FusionToTom"> <#Include Label="NameOfLibraryCharacterTable">

The Interface to &GAP;'s Group Libraries

Sometimes it is useful to extend a character-theoretic computation with computations involving a group that has the character table in question. For many character tables in the &GAP; Character Table Library, corresponding groups can be found in the various group libraries that are distributed with &GAP;. This section describes how one can access the library groups that belong to a given character table.

<#Include Label="GroupInfoForCharacterTable"> <#Include Label="KnowsSomeGroupInfo"> <#Include Label="CharacterTableForGroupInfo"> <#Include Label="GroupForGroupInfo"> <#Include Label="GroupForTom"> <#Include Label="AtlasStabilizer"> <#Include Label="IsNontrivialDirectProduct">

Unipotent Characters of Finite Groups of Lie Type
character
<#Include Label="sec:unipot">

<#Include Label="UnipotentCharacter"> <#Include Label="DeligneLusztigNames"> <#Include Label="DeligneLusztigName"> <#Include Label="KnowsDeligneLusztigNames">

Browse
Applications Provided by `CTblLib`

The following functions are available only if the &GAP; package `Browse` (see ) is loaded. The function shows details about an ordinary or modular character table in a pager, the other functions can be used to show the following information via browse tables.

An overview of the &GAP; Character Table Library (see ),
details tables about ordinary and modular character tables (see ),
ordinary and modular character tables, (cf. ),
decomposition matrices (cf. ),
the atomic irrationalities that occur in &ATLAS; character tables (see ),
an overview of the differences between the character table data from version 1.1.3 and version 1.2 of the `CTblLib` package, (see ).

The functions and are also reachable in the list of choices shown by .

<#Include Label="DisplayCTblLibInfo"> <#Include Label="BrowseCTblLibInfo"> <#Include Label="BrowseCommonIrrationalities"> <#Include Label="BrowseCTblLibDifferences">

Duplicates of Library Character Tables

It can be useful to deal with different instances of the same character table. An example is the situation that a group G, say, contains several classes of isomorphic maximal subgroups that have different class fusions; the attribute of the character table of G then contains several entries that belong to the same group, but the identifiers of the character tables are different.

On the other hand, it can be useful to consider only one of the different instances when one searches for character tables with certain properties, for example using .

duplicate character table
For that, we introduce the following concept. A character table t_1 is said to be a duplicate of another character table t_2 if the attribute is set in t_1, if the first entry of the attribute value is "ConstructPermuted", and if the second entry of the attribute value is the value of t_2. We call t_2 the main table of t_1.

<#Include Label="IsDuplicateTable"> <#Include Label="IdentifierOfMainTable"> <#Include Label="IdentifiersOfDuplicateTables">

Attributes for Library Character Tables

This section describes certain attributes which are set only for certain (not necessarily all) character tables from the &GAP; Character Table Library. The attribute values are part of the database, there are no methods for computing them.

Other such attributes and properties are described in manual sections because the context fits better. These attributes are , , , , , , , , and .

<#Include Label="Maxes"> <#Include Label="ProjectivesInfo"> <#Include Label="ExtensionInfoCharacterTable"> <#Include Label="ConstructionInfoCharacterTable">