[Casinfo 0.1.5.lzh | Added By 15
Sharif Garmon
The full list of endpoints provided to your CAS deployment is posted here. Note that you do not need to do anything extra special to get these endpoints added to your deployment; these are all available by default and just need to be turned on and secured for access.
I hope this review was of some help to you and I am sure that both this post as well as the functionality it attempts to explain can be improved in any number of ways. Please feel free to engage and contribute as best as you can.
There are three different kinds of character tables in the GAP library, namely ordinary character tables, Brauer tables, and generic character tables. Note that the Brauer table and the corresponding ordinary table of a group determine the decomposition matrix of the group (and the decomposition matrices of its blocks). These decomposition matrices can be computed from the ordinary and modular irreducibles with GAP, see Section Reference: Operations Concerning Blocks for details. A collection of PDF files of the known decomposition matrices of Atlas tables in the GAP Character Table Library can also be found at
As for the source, there are first of all two big sources, namely the Atlas of Finite Groups (see Section 4.3) and the CAS library of character tables (see [NPP84]). Many Atlas tables are contained in the CAS library, and difficulties may arise because the succession of characters and classes in CAS tables and Atlas tables are in general different, so see Section 4.4 for the relations between these two variants of character tables of the same group. A subset of the CAS tables is the set of tables of Sylow normalizers of sporadic simple groups as published in [Ost86] this may be viewed as another source of character tables. The library also contains the character tables of factor groups of space groups (computed by W. Hanrath, see [Han88]) that are part of [HP89], in the form of two microfiches; these tables are given in CAS format (see Section 4.4) on the microfiches, but they had not been part of the "official" CAS library.
The InfoText (Reference: InfoText) value usually contains more detailed information, for example that the table in question is the character table of a maximal subgroup of an almost simple group. If the table was contained in the CAS library then additional information may be available via the CASInfo (4.4-1) value.
all Atlas tables (see Section 4.3), i. e., the tables of the simple groups which are contained in the Atlas of Finite Groups, and the tables of cyclic and bicyclic extensions of these groups,
the tables of many Sylow p-normalizers of sporadic simple groups; this includes the tables printed in [Ost86] except J_4N2, Co_1N2, Fi_22N2, but also other tables are available; more generally, several tables of normalizers of other radical p-subgroups are available, such as normalizers of defect groups of p-blocks,
Note that class fusions stored on library tables are not guaranteed to be compatible for any two subgroups of a group and their intersection, and they are not guaranteed to be consistent w. r. t. the composition of maps.
The library contains all tables of the Atlas of Brauer Tables ([JLPW95]), and many other Brauer tables of bicyclic extensions of simple groups which are known yet. The Brauer tables in the library contain the information
Generic character tables provide a means for writing down the character tables of all groups in a (usually infinite) series of similar groups, e. g., cyclic groups, or symmetric groups, or the general linear groups GL(2,q) where q ranges over certain prime powers.
For example, both the conjugacy classes and the irreducible characters of the symmetric group S_n are in bijection with the partitions of n. Thus for given n it makes sense to ask for the character corresponding to a particular partition, or just for its character value at another partition.
Currently the only operations for generic tables supported by GAP are the specialisation of the parameter q in order to compute the whole character table of G_q, and local evaluation (see ClassParameters (Reference: ClassParameters) for an example). GAP does not support the computation of, e. g., generic scalar products.
While the numbers of conjugacy classes for the members of a series of groups are usually not bounded, there is always a fixed finite number of types (equivalence classes) of conjugacy classes; very often the equivalence relation is isomorphism of the centralizers of the representatives.
In GAP, the parametrisations of classes and characters for tables computed from generic tables is stored using the attributes ClassParameters (Reference: ClassParameters) and CharacterParameters (Reference: CharacterParameters).
the table of the Frobenius extension of the nontrivial cyclic group of odd order p by the nontrivial cyclic group of order q where q divides p_i-1 for all prime divisors p_i of p; if p is a prime power then q determines the group uniquely and thus the first version can be used, otherwise the action of the residue class of k modulo p is taken for forming orbits of length q each on the nonidentity elements of the group of order p,
In addition to the above calls that really use generic tables, the following calls to CharacterTable (Reference: CharacterTable) are to some extent "generic" constructions. But note that no local evaluation is possible in these cases, as no generic table object exists in GAP that can be asked for local information.
Here are examples for the "local evaluation" of generic character tables, first a character value of the cyclic group shown above, then a character value and a representative order of a symmetric group.
list of functions, one for each class type t, with arguments q and p_t, returning a representative of the class with parameter [t,p_t] (note that this element need not actually lie in the group in question, for example it may be a diagonal matrix but the characteristic polynomial in the group s irreducible),
In the specialized table, the ClassParameters (Reference: ClassParameters) and CharacterParameters (Reference: CharacterParameters) values are the lists of parameters [t,p_t] of classes and characters, respectively.
If the matrix component is present then its value implements a method to compute the complete table of small members G_q more efficiently than via local evaluation; this method will be called when the generic table is used to compute the whole character table for a given q (see CharacterTableSpecialized (4.2-2)).
The GAP character table library contains all character tables of bicyclic extensions of simple groups that are included in the Atlas of Finite Groups ([CCN+85], from now on called Atlas), and the Brauer tables contained in the Atlas of Brauer Characters ([JLPW95]).
For displaying Atlas tables with the row labels used in the Atlas, or for displaying decomposition matrices, see LaTeXStringDecompositionMatrix (Reference: LaTeXStringDecompositionMatrix) and AtlasLabelsOfIrreducibles (4.3-6).
In addition to the information given in Chapters 6 to 8 of the Atlas which tell you how to read the printed tables, there are some rules relating these to the corresponding GAP tables.
For the GAP Character Table Library not the printed versions of the Atlas of Finite Groups and the Atlas of Brauer Characters are relevant but the revised versions given by the currently three lists of improvements that are maintained by Simon Norton. The first such list is contained in [BN95], and is printed in the Appendix of [JLPW95]; it contains the improvements that had been known until the "Atlas of Brauer Characters" was published. The second list contains the improvements to the Atlas of Finite Groups that were found since the publication of [JLPW95]. It can be found in the internet, an HTML version at
Also some tables are regarded as Atlas tables that are not printed in the Atlas but available in Atlas format, according to the lists of improvements mentioned above. Currently these are the tables related to L_2(49), L_2(81), L_6(2), O_8^-(3), O_8^+(3), S_10(2), and ^2E_6(2).3.
If G (or G.a) has a nontrivial Schur multiplier then the attribute ProjectivesInfo (3.7-2) of the GAP table object of G (or G.a) is set; the chars component of the record in question is the list of values lists of those faithful projective irreducibles that are printed in the Atlas (so-called proxy character), and the map component lists the positions of columns in the covering for which the column is printed in the Atlas (a so-called proxy class, this preimage is denoted by g_0 in Chapter 7, Section 14 of the Atlas).
As described in Chapter 6, Section 7 and in Chapter 7, Section 18 of the Atlas, there exist two (often nonisomorphic) groups of structure 2.G.2 for a simple group G, which are isoclinic. The table in the GAP Character Table Library is the one printed in the Atlas, the table of the isoclinic variant can be constructed using CharacterTableIsoclinic (Reference: CharacterTableIsoclinic).
For a > 2, any proxy class and its algebraic conjugates that are not printed in the Atlas are consecutive in the table of G.a; if more than two classes of G.a have the same proxy class (the only case that actually occurs is for a = 5) then the ordering of non-printed classes is the natural one of corresponding Galois conjugacy operators *k (see [CCN+85, Chapter 7, Section 19]).
Each character can be regarded as a faithful character of a factor group m.G, where m divides M. Characters with the same kernel are consecutive as in the Atlas, the ordering of characters with different kernels is given by the order of precedence 1, 2, 4, 3, 6, 12 for the different values of m.
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