<!-- %W interfac.xml GAP 4 package CTblLib Thomas Breuer -->
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<Chapter Label="ch:interfac">
<Heading>The User Interface to the &GAP; Character Table Library</Heading>
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<Section Label="sect:accessdata">
<Heading>Accessing Data of the <Package>CTblLib</Package> Package</Heading>
<Subsection Label="subsect:admissiblenames">
<Heading>Admissible Names for Character Tables in <Package>CTblLib</Package>
</Heading>
When you access a character table from the &GAP; Character Table Library,
this table is specified by an admissible name.
<P/>
Admissible names for the <E>ordinary character table</E> <M>tbl</M> of
the group <M>G</M> are
<List>
<Item>
an &ATLAS; like name if <M>tbl</M> is an &ATLAS; table
(see Section <Ref Sect="sec:ATLAS Tables"/>), for example
<C>"M22"</C> for the table of the Mathieu group <M>M_{22}</M>,
<C>"L2(13).2"</C> for <M>L_2(13):2</M>, and
<C>"12_1.U4(3).2_1"</C> for <M>12_1.U_4(3).2_1</M>.
<P/>
(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.)
</Item>
<Item>
the names that were admissible for tables of <M>G</M> in the
&CAS; system
if the &CAS; table library contained a table of
<M>G</M>, for example <C>sl42</C> for the table of the alternating
group <M>A_8</M>.
<P/>
(But note that the ordering of rows and columns of the &GAP; table may be
different from that in &CAS;,
see Section <Ref Sect="sec:CAS Tables"/>.)
</Item>
<Item>
some <Q>relative</Q> names, as follows.
<List>
<Item>
If <M>G</M> is the <M>n</M>-th maximal subgroup (in decreasing group
order) of a group whose library table <M>subtbl</M> is available in &GAP;
and stores the <Ref Func="Maxes"/> value,
and if <C>name</C> is an admissible name for <M>subtbl</M>
then <C>name</C>M<M>n</M> is admissible for <M>tbl</M>.
For example, the name <C>"J3M2"</C> can be used to access the second
maximal subgroup of the sporadic simple Janko group <M>J_3</M>
which has the admissible name <C>"J3"</C>.
</Item>
<Item>
If <M>G</M> is a nontrivial Sylow <M>p</M> normalizer in a sporadic
simple group with admissible name <C>name</C>
–where nontrivial means that <M>G</M> is not isomorphic to a
subgroup of <M>p:(p-1)</M>–
then <C>name</C>N<M>p</M> is an admissible name of <M>tbl</M>.
For example, the name <C>"J4N11"</C> can be used to access the table
of the Sylow <M>11</M> normalizer in the sporadic simple Janko group
<M>J_4</M>.
</Item>
<Item>
In a few cases, the table of the Sylow <M>p</M>-subgroup of <M>G</M>
is accessible via the name <C>name</C>Syl<M>p</M> where <C>name</C>
is an admissible name of the table of <M>G</M>.
For example, <C>"A11Syl2"</C> is an admissible name for the table of
the Sylow <M>2</M>-subgroup of the alternating group <M>A_{11}</M>.
</Item>
<Item>
In a few cases, the table of an element centralizer in <M>G</M> is
accessible via the name <C>name</C>C<M>cl</M>
where <C>name</C> is an admissible name of the table of <M>G</M>.
For example, <C>"M11C2"</C> is an admissible name for the table of an
involution centralizer in the Mathieu group <M>M_{11}</M>.
</Item>
</List>
</Item>
</List>
<P/>
The recommended way to access a <E>Brauer table</E> is via applying the
<K>mod</K> operator to the ordinary table and the desired characteristic
(see <Ref Func="BrauerTable" BookName="ref"/> and
Section <Ref Sect="Operators for Character Tables" BookName="ref"/>),
so it is not necessary to define admissible names of Brauer tables.
<P/>
A <E>generic character table</E>
(see Section <Ref Sect="sec:generictables"/>) is
accessible only by the name given by its
<Ref Attr="Identifier" Label="for character tables" BookName="ref"/>
value.
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<Section Label="sect:tbltom">
<Heading>The Interface to the <Package>TomLib</Package> Package</Heading>
The &GAP; Character Table Library contains ordinary character tables of all
groups for which the <Package>TomLib</Package> package <Cite Key="TomLib"/>
contains the table of marks.
This section describes the mapping between these character tables and their
tables of marks.
<P/>
If the <Package>TomLib</Package> package is not loaded then
<Ref Attr="FusionToTom"/> is the only available function from this section,
but of course it is of little interest in this situation.
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<Section Label="sect:tblgrp">
<Heading>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.
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<Section Label="sec:unipotsec">
<Heading>Unipotent Characters of Finite Groups of Lie Type</Heading>
<Index Subkey="unipotent">character</Index>
<#Include Label="sec:unipot">
<#Include Label="UnipotentCharacter"> <!-- crossref. from Ref. Manual, currently commented out -->
<#Include Label="DeligneLusztigNames">
<#Include Label="DeligneLusztigName">
<#Include Label="KnowsDeligneLusztigNames">
</Section>
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<Section Label="sect:ctbllib-browse">
<Heading><Package>Browse</Package> Applications Provided by
<Package>CTblLib</Package></Heading>
The following functions are available only if the &GAP; package
<Package>Browse</Package> (see <Cite Key="Browse"/>) is loaded.
The function <Ref Func="DisplayCTblLibInfo" Label="for a character table"/>
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.
<List>
<Item>
An overview of the &GAP; Character Table Library,
and details pages about ordinary and modular character tables
(see <Ref Func="BrowseCTblLibInfo"/>),
which allow one to navigate to related pages and to pages showing
for example decomposition matrices
(cf. <Ref Func="BrowseDecompositionMatrix" BookName="browse"/>),
</Item>
<Item>
an alternative display function that shows character tables from the
&ATLAS; of Finite Groups <Cite Key="CCN85"/>
and the &ATLAS; of Brauer Characters <Cite Key="JLPW95"/> in a format
similar to the one used in these books
(see <Ref Func="BrowseAtlasTable"/>,
cf. <Ref Meth="Browse (for character tables)" BookName="browse"/>
for the default display format for character tables),
</Item>
<Item>
an overview of the names of simple groups for which the
&ATLAS; of Finite Groups <Cite Key="CCN85"/>
and the &ATLAS; of Brauer Characters <Cite Key="JLPW95"/>
show the character tables and other information
(see <Ref Func="BrowseAtlasContents"/>,
a variant that doe not rely on <Package>Browse</Package> is
<Ref Func="DisplayAtlasContents"/>),
</Item>
<Item>
a function that shows the &ATLAS; map of the bicyclic extensions
of a simple &ATLAS; group
(see <Ref Func="BrowseAtlasMap"/>,
a variant that does not rely on <Package>Browse</Package> is
<Ref Func="DisplayAtlasMap" Label="for the name of a simple group"/>),
</Item>
<Item>
an overview of the <Q>atomic irrationalities</Q> that occur in
&ATLAS; character tables
(see <Ref Func="BrowseCommonIrrationalities"/>),
</Item>
<Item>
an overview of the lists of improvements to the
&ATLAS; of Finite Groups
(see <Ref Func="BrowseAtlasImprovements"/>).
</Item>
<Item>
an overview of the differences between the character table data since version 1.1.3 of the <Package>CTblLib</Package> package
(see <Ref Func="BrowseCTblLibDifferences"/>),
</Item>
</List>
The functions <Ref Func="BrowseCTblLibInfo"/>,
<Ref Func="BrowseCommonIrrationalities"/>,
<Ref Func="BrowseCTblLibDifferences"/>,
<Ref Func="BrowseAtlasContents"/>,
and <Ref Func="BrowseAtlasImprovements"/>
occur also in the list
of choices shown by <Ref Func="BrowseGapData" BookName="browse"/>.
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<Section Label="sec:duplicates">
<Heading>Duplicates of Library Character Tables</Heading>
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<Section Label="sec:attributes">
<Heading>Attributes for Library Character Tables</Heading>
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 <E>computing</E> them,
except for <Ref Meth="InfoText"/>
and the related property <Ref Prop="IsAtlasCharacterTable"/>.
<P/>
Other such attributes and properties are described in other manual sections
because the context fits better.
These attributes are <Ref Attr="FusionToTom"/>,
<Ref Attr="GroupInfoForCharacterTable"/>,
<Ref Prop="KnowsSomeGroupInfo"/>,
<Ref Prop="IsNontrivialDirectProduct"/>,
<Ref Attr="DeligneLusztigNames"/>,
<Ref Attr="DeligneLusztigName"/>,
<Ref Attr="KnowsDeligneLusztigNames"/>,
<Ref Prop="IsDuplicateTable"/>, and
<Ref Attr="CASInfo"/>.
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