The user interface is the part of the GAP interface that allows one to display information about the current contents of the database and to access individual data (perhaps by downloading them, see Section 4.2-1). The corresponding functions are described in this chapter. See Section 2.4 for some small examples how to use the functions of the interface.
Data extensions of the AtlasRep package are regarded as another part of the GAP interface, they are described in Chapter 5. Finally, the low level part of the interface is described in Chapter 7.
Note that accessing the data means in particular that it is not the aim of this package to construct representations from known ones. For example, if at least one permutation representation for a group \(G\) is stored but no matrix representation in a positive characteristic \(p\), say, then OneAtlasGeneratingSetInfo (3.5-6) returns fail when it is asked for a description of an available set of matrix generators for \(G\) in characteristic \(p\), although such a representation can be obtained by reduction modulo \(p\) of an integral matrix representation, which in turn can be constructed from any permutation representation.
When you access data via the AtlasRep package, you specify the group in question by an admissible name. Thus it is essential to know these names, which are called the GAP names of the group in the following.
For a group \(G\), say, whose character table is available in GAP's Character Table Library (see [Bre22]), the admissible names of \(G\) are the admissible names of this character table. One such name is the Identifier (Reference: Identifier for character tables) value of the character table, see CTblLib: Admissible Names for Character Tables in CTblLib. This name is usually very similar to the name used in the ATLAS of Finite Groups [CCN+85]. For example, "M22" is a GAP name of the Mathieu group \(M_{22}\), "12_1.U4(3).2_1" is a GAP name of \(12_1.U_4(3).2_1\), the two names "S5" and "A5.2" are GAP names of the symmetric group \(S_5\), and the two names "F3+" and "Fi24'" are GAP names of the simple Fischer group \(Fi_{24}^\prime\).
When a GAP name is required as an input of a package function, this input is case insensitive. For example, both "A5" and "a5" are valid arguments of DisplayAtlasInfo (3.5-1).
Internally, for example as part of filenames (see Section 7.6), the package uses names that may differ from the GAP names; these names are called ATLAS-file names. For example, "A5", "TE62", and "F24" are ATLAS-file names. Of these, only "A5" is also a GAP name, but the other two are not; corresponding GAP names are "2E6(2)" and "Fi24'", respectively.
For the general definition of standard generators of a group, see [Wil96].
Several different standard generators may be defined for a group, the definitions for each group that occurs in the ATLAS of Group Representations can be found at
http://atlas.math.rwth-aachen.de/Atlas/v3.
When one specifies the standardization, the \(i\)-th set of standard generators is denoted by the number \(i\). Note that when more than one set of standard generators is defined for a group, one must be careful to use compatible standardization. For example, the straight line programs, straight line decisions and black box programs in the database refer to a specific standardization of their inputs. That is, a straight line program for computing generators of a certain subgroup of a group \(G\) is defined only for a specific set of standard generators of \(G\), and applying the program to matrix or permutation generators of \(G\) but w. r. t. a different standardization may yield unpredictable results. Therefore the results returned by the functions described in this chapter contain information about the standardizations they refer to.
For each straight line program (see AtlasProgram (3.5-4)) that is used to compute lists of class representatives, it is essential to describe the classes in which these elements lie. Therefore, in these cases the records returned by the function AtlasProgram (3.5-4) contain a component outputs with value a list of class names.
Currently we define these class names only for simple groups and certain extensions of simple groups, see Section 3.4-1. The function AtlasClassNames (3.4-2) can be used to compute the list of class names from the character table in the GAP Library.
For the definition of class names of an almost simple group, we assume that the ordinary character tables of all nontrivial normal subgroups are shown in the ATLAS of Finite Groups [CCN+85].
Each class name is a string consisting of the element order of the class in question followed by a combination of capital letters, digits, and the characters ' and - (starting with a capital letter). For example, 1A, 12A1, and 3B' denote the class that contains the identity element, a class of element order \(12\), and a class of element order \(3\), respectively.
For the table of a simple group, the class names are the same as returned by the two argument version of the GAP function ClassNames (Reference: ClassNames), cf. [CCN+85, Chapter 7, Section 5]: The classes are arranged w. r. t. increasing element order and for each element order w. r. t. decreasing centralizer order, the conjugacy classes that contain elements of order \(n\) are named \(n\)A, \(n\)B, \(n\)C, \(\ldots\); the alphabet used here is potentially infinite, and reads A, B, C, \(\ldots\), Z, A1, B1, \(\ldots\), A2, B2, \(\ldots\).
For example, the classes of the alternating group \(A_5\) have the names 1A, 2A, 3A, 5A, and 5B.
Next we consider the case of an upward extension \(G.A\) of a simple group \(G\) by a cyclic group of order \(A\). The ATLAS defines class names for each element \(g\) of \(G.A\) only w. r. t. the group \(G.a\), say, that is generated by \(G\) and \(g\); namely, there is a power of \(g\) (with the exponent coprime to the order of \(g\)) for which the class has a name of the same form as the class names for simple groups, and the name of the class of \(g\) w. r. t. \(G.a\) is then obtained from this name by appending a suitable number of dashes '. So dashed class names refer exactly to those classes that are not printed in the ATLAS.
For example, those classes of the symmetric group \(S_5\) that do not lie in \(A_5\) have the names 2B, 4A, and 6A. The outer classes of the group \(L_2(8).3\) have the names 3B, 6A, 9D, and 3B', 6A', 9D'. The outer elements of order \(5\) in the group \(Sz(32).5\) lie in the classes with names 5B, 5B', 5B'', and 5B'''.
In the group \(G.A\), the class of \(g\) may fuse with other classes. The name of the class of \(g\) in \(G.A\) is obtained from the names of the involved classes of \(G.a\) by concatenating their names after removing the element order part from all of them except the first one.
For example, the elements of order \(9\) in the group \(L_2(27).6\) are contained in the subgroup \(L_2(27).3\) but not in \(L_2(27)\). In \(L_2(27).3\), they lie in the classes 9A, 9A', 9B, and 9B'; in \(L_2(27).6\), these classes fuse to 9AB and 9A'B'.
Now we define class names for general upward extensions \(G.A\) of a simple group \(G\). Each element \(g\) of such a group lies in an upward extension \(G.a\) by a cyclic group, and the class names w. r. t. \(G.a\) are already defined. The name of the class of \(g\) in \(G.A\) is obtained by concatenating the names of the classes in the orbit of \(G.A\) on the classes of cyclic upward extensions of \(G\), after ordering the names lexicographically and removing the element order part from all of them except the first one. An exception is the situation where dashed and non-dashed class names appear in an orbit; in this case, the dashed names are omitted.
For example, the classes 21A and 21B of the group \(U_3(5).3\) fuse in \(U_3(5).S_3\) to the class 21AB, and the class 2B of \(U_3(5).2\) fuses with the involution classes 2B', 2B'' in the groups \(U_3(5).2^{\prime}\) and \(U_3(5).2^{{\prime\prime}}\) to the class 2B of \(U_3(5).S_3\).
It may happen that some names in the outputs component of a record returned by AtlasProgram (3.5-4) do not uniquely determine the classes of the corresponding elements. For example, the (algebraically conjugate) classes 39A and 39B of the group \(Co_1\) have not been distinguished yet. In such cases, the names used contain a minus sign -, and mean one of the classes in the range described by the name before and the name after the minus sign
; the element order part of the name does not appear after the minus sign. So the name 39A-B for the group \(Co_1\) means 39A or 39B, and the name 20A-B''' for the group \(Sz(32).5\) means one of the classes of element order \(20\) in this group (these classes lie outside the simple group \(Sz\)).
For a downward extension \(m.G.A\) of an almost simple group \(G.A\) by a cyclic group of order \(m\), let \(\pi\) denote the natural epimorphism from \(m.G.A\) onto \(G.A\). Each class name of \(m.G.A\) has the form nX_0, nX_1 etc., where nX is the class name of the image under \(\pi\), and the indices 0, 1 etc. are chosen according to the position of the class in the lifting order rows for \(G\), see [CCN+85, Chapter 7, Section 7, and the example in Section 8]).
For example, if \(m = 6\) then 1A_1 and 1A_5 denote the classes containing the generators of the kernel of \(\pi\), that is, central elements of order \(6\).
‣ AtlasClassNames( tbl ) | ( function ) |
Returns: a list of class names.
Let tbl be the ordinary or modular character table of a group \(G\), say, that is almost simple or a downward extension of an almost simple group and such that tbl is an ATLAS table from the GAP Character Table Library, according to its InfoText (Reference: InfoText) value. Then AtlasClassNames returns the list of class names for \(G\), as defined in Section 3.4-1. The ordering of class names is the same as the ordering of the columns of tbl.
(The function may work also for character tables that are not ATLAS tables, but then clearly the class names returned are somewhat arbitrary.)
gap> AtlasClassNames( CharacterTable( "L3(4).3" ) ); [ "1A", "2A", "3A", "4ABC", "5A", "5B", "7A", "7B", "3B", "3B'", "3C", "3C'", "6B", "6B'", "15A", "15A'", "15B", "15B'", "21A", "21A'", "21B", "21B'" ] gap> AtlasClassNames( CharacterTable( "U3(5).2" ) ); [ "1A", "2A", "3A", "4A", "5A", "5B", "5CD", "6A", "7AB", "8AB", "10A", "2B", "4B", "6D", "8C", "10B", "12B", "20A", "20B" ] gap> AtlasClassNames( CharacterTable( "L2(27).6" ) ); [ "1A", "2A", "3AB", "7ABC", "13ABC", "13DEF", "14ABC", "2B", "4A", "26ABC", "26DEF", "28ABC", "28DEF", "3C", "3C'", "6A", "6A'", "9AB", "9A'B'", "6B", "6B'", "12A", "12A'" ] gap> AtlasClassNames( CharacterTable( "L3(4).3.2_2" ) ); [ "1A", "2A", "3A", "4ABC", "5AB", "7A", "7B", "3B", "3C", "6B", "15A", "15B", "21A", "21B", "2C", "4E", "6E", "8D", "14A", "14B" ] gap> AtlasClassNames( CharacterTable( "3.A6" ) ); [ "1A_0", "1A_1", "1A_2", "2A_0", "2A_1", "2A_2", "3A_0", "3B_0", "4A_0", "4A_1", "4A_2", "5A_0", "5A_1", "5A_2", "5B_0", "5B_1", "5B_2" ] gap> AtlasClassNames( CharacterTable( "2.A5.2" ) ); [ "1A_0", "1A_1", "2A_0", "3A_0", "3A_1", "5AB_0", "5AB_1", "2B_0", "4A_0", "4A_1", "6A_0", "6A_1" ]
‣ AtlasCharacterNames( tbl ) | ( function ) |
Returns: a list of character names.
Let tbl be the ordinary or modular character table of a simple group. AtlasCharacterNames returns a list of strings, the \(i\)-th entry being the name of the \(i\)-th irreducible character of tbl; this name consists of the degree of this character followed by distinguishing lowercase letters.
gap> AtlasCharacterNames( CharacterTable( "A5" ) ); [ "1a", "3a", "3b", "4a", "5a" ]
The examples shown in this section refer to the situation that no extensions have been notified, and to a perhaps outdated table of contents. That is, the current version of the database may contain more information than is shown here.
‣ DisplayAtlasInfo( [listofnames][,] [std][,] ["contents", sources][,] [...] ) | ( function ) |
‣ DisplayAtlasInfo( gapname[, std][, ...] ) | ( function ) |
This function lists the information available via the AtlasRep package, for the given input.
There are essentially three ways of calling this function.
If there is no argument or if the first argument is a list listofnames of strings that are GAP names of groups, DisplayAtlasInfo shows an overview of the known information.
If the first argument is a string gapname that is a GAP name of a group, DisplayAtlasInfo shows an overview of the information that is available for this group.
If the string "contents" is the only argument then the function shows which parts of the database are available; these are at least the "core" part, which means the data from the ATLAS of Group Representations, and the "internal" part, which means the data that are distributed with the AtlasRep package. Other parts can become available by calls to AtlasOfGroupRepresentationsNotifyData (5.1-1). Note that the shown numbers of locally available files depend on what has already been downloaded.
In each case, the information will be printed to the screen or will be fed into a pager, see Section 4.2-11. An interactive alternative to DisplayAtlasInfo is the function BrowseAtlasInfo (Browse: BrowseAtlasInfo), see [BL18].
The following paragraphs describe the structure of the output in the two cases. Examples can be found in Section 3.5-2.
Called without arguments, DisplayAtlasInfo shows a general overview for all groups. If some information is available for the group \(G\), say, then one line is shown for \(G\), with the following columns.
groupthe GAP name of \(G\) (see Section 3.2), if applicable followed by a string (by default a star *) indicating that at least one column refers to data not belonging to the core part of the database (see Section 4.2-12).
#the number of faithful representations stored for \(G\) that satisfy the additional conditions given (see below),
maxesthe number of available straight line programs for computing generators of maximal subgroups of \(G\),
cla + sign if at least one program for computing representatives of conjugacy classes of elements of \(G\) is stored,
cyca + sign if at least one program for computing representatives of classes of maximally cyclic subgroups of \(G\) is stored,
outdescriptions of outer automorphisms of \(G\) for which at least one program is stored,
fnda + sign if at least one program is available for finding standard generators,
chka + sign if at least one program is available for checking whether a set of generators is a set of standard generators, and
prsa + sign if at least one program is available that encodes a presentation.
Called with a list listofnames of strings that are GAP names of some groups, DisplayAtlasInfo prints the overview described above but restricted to the groups in this list.
In addition to or instead of listofnames, the string "contents" and a description \(sources\) of the data may be given about which the overview is formed. See below for admissible values of \(sources\).
Called with a string gapname that is a GAP name of a group, DisplayAtlasInfo prints an overview of the information that is available for this group. One line is printed for each faithful representation, showing the number of this representation (which can be used in calls of AtlasGenerators (3.5-3)), and a string of one of the following forms; in both cases, \(id\) is a (possibly empty) string.
G <= Sym(\(n\)\(id\))denotes a permutation representation of degree \(n\), for example G <= Sym(40a) and G <= Sym(40b) denote two (nonequivalent) representations of degree \(40\).
G <= GL(\(n\)\(id\),\(descr\))denotes a matrix representation of dimension \(n\) over a coefficient ring described by \(descr\), which can be a prime power, ℤ (denoting the ring of integers), a description of an algebraic extension field, ℂ (denoting an unspecified algebraic extension field), or ℤ/\(m\)ℤ for an integer \(m\) (denoting the ring of residues mod \(m\)); for example, G <= GL(2a,4) and G <= GL(2b,4) denote two (nonequivalent) representations of dimension \(2\) over the field with four elements.
Below the representations, the programs available for gapname are listed. In each row of the overview, the entry in the first column is followed by a string (by default a star *) if the row refers to data not belonging to the core part of the database (see Section 4.2-12).
The following optional arguments can be used to restrict the overviews.
must be a positive integer or a list of positive integers; if it is given then only those representations are considered that refer to the std-th set of standard generators or the \(i\)-th set of standard generators, for \(i\) in std (see Section 3.3),
"contents" and \(sources\)for a string or a list of strings \(sources\), restrict the data about which the overview is formed; if \(sources\) is the string "core" then only data from the ATLAS of Group Representations are considered, if \(sources\) is a string that denotes a data extension in the sense of a dirid argument of AtlasOfGroupRepresentationsNotifyData (5.1-1) then only the data that belong to this data extension are considered; also a list of such strings may be given, then the union of these data is considered,
Identifier and \(id\)restrict to representations with id component in the list \(id\) (note that this component is itself a list, entering this list is not admissible), or satisfying the function \(id\),
IsPermGroup and true (or false)restrict to permutation representations (or to representations that are not permutation representations),
NrMovedPoints and \(n\)for a positive integer, a list of positive integers, or a property \(n\), restrict to permutation representations of degree equal to \(n\), or in the list \(n\), or satisfying the function \(n\),
NrMovedPoints and the string "minimal"restrict to faithful permutation representations of minimal degree (if this information is available),
IsTransitive and a boolean valuerestrict to transitive or intransitive permutation representations where this information is available (if the value true or false is given), or to representations for which this information is not available (if the value fail is given),
IsPrimitive and a boolean valuerestrict to primitive or imprimitive permutation representations where this information is available (if the value true or false is given), or to representations for which this information is not available (if the value fail is given),
Transitivity and \(n\)for a nonnegative integer, a list of nonnegative integers, or a property \(n\), restrict to permutation representations for which the information is available that the transitivity is equal to \(n\), or is in the list \(n\), or satisfies the function \(n\); if \(n\) is fail then restrict to all permutation representations for which this information is not available,
RankAction and \(n\)for a nonnegative integer, a list of nonnegative integers, or a property \(n\), restrict to permutation representations for which the information is available that the rank is equal to \(n\), or is in the list \(n\), or satisfies the function \(n\); if \(n\) is fail then restrict to all permutation representations for which this information is not available,
IsMatrixGroup and true (or false)restrict to matrix representations (or to representations that are not matrix representations),
Characteristic and \(p\)for a prime integer, a list of prime integers, or a property \(p\), restrict to matrix representations over fields of characteristic equal to \(p\), or in the list \(p\), or satisfying the function \(p\) (representations over residue class rings that are not fields can be addressed by entering fail as the value of \(p\)),
Dimension and \(n\)for a positive integer, a list of positive integers, or a property \(n\), restrict to matrix representations of dimension equal to \(n\), or in the list \(n\), or satisfying the function \(n\),
Characteristic, \(p\), Dimension,
and the string "minimal"for a prime integer \(p\), restrict to faithful matrix representations over fields of characteristic \(p\) that have minimal dimension (if this information is available),
Ring and \(R\)for a ring or a property \(R\), restrict to matrix representations for which the information is available that the ring spanned by the matrix entries is contained in this ring or satisfies this property (note that the representation might be defined over a proper subring); if \(R\) is fail then restrict to all matrix representations for which this information is not available,
Ring, \(R\), Dimension,
and the string "minimal"for a ring \(R\), restrict to faithful matrix representations over this ring that have minimal dimension (if this information is available),
Character and \(chi\)for a class function or a list of class functions \(chi\), restrict to representations with these characters (note that the underlying characteristic of the class function, see Section Reference: UnderlyingCharacteristic, determines the characteristic of the representation),
Character and \(name\)for a string \(name\), restrict to representations for which the character is known to have this name, according to the information shown by DisplayAtlasInfo; if the characteristic is not specified then it defaults to zero,
Character and \(n\)for a positive integer \(n\), restrict to representations for which the character is known to be the \(n\)-th irreducible character in GAP's library character table of the group in question; if the characteristic is not specified then it defaults to zero,
IsStraightLineProgram and truerestrict to straight line programs, straight line decisions (see Section 6.1), and black box programs (see Section 6.2), and
IsStraightLineProgram and falserestrict to representations.
Note that the above conditions refer only to the information that is available without accessing the representations. For example, if it is not stored in the table of contents whether a permutation representation is primitive then this representation does not match an IsPrimitive condition in DisplayAtlasInfo.
If minimality
information is requested and no available representation matches this condition then either no minimal representation is available or the information about the minimality is missing. See MinimalRepresentationInfo (6.3-1) for checking whether the minimality information is available for the group in question. Note that in the cases where the string "minimal" occurs as an argument, MinimalRepresentationInfo (6.3-1) is called with third argument "lookup"; this is because the stored information was precomputed just for the groups in the ATLAS of Group Representations, so trying to compute non-stored minimality information (using other available databases) will hardly be successful.
The representations are ordered as follows. Permutation representations come first (ordered according to their degrees), followed by matrix representations over finite fields (ordered first according to the field size and second according to the dimension), matrix representations over the integers, and then matrix representations over algebraic extension fields (both kinds ordered according to the dimension), the last representations are matrix representations over residue class rings (ordered first according to the modulus and second according to the dimension).
The maximal subgroups are ordered according to decreasing group order. For an extension \(G.p\) of a simple group \(G\) by an outer automorphism of prime order \(p\), this means that \(G\) is the first maximal subgroup and then come the extensions of the maximal subgroups of \(G\) and the novelties; so the \(n\)-th maximal subgroup of \(G\) and the \(n\)-th maximal subgroup of \(G.p\) are in general not related. (This coincides with the numbering used for the Maxes (CTblLib: Maxes) attribute for character tables.)
Here are some examples how DisplayAtlasInfo (3.5-1) can be called, and how its output can be interpreted.
gap> DisplayAtlasInfo( "contents" ); - AtlasRepAccessRemoteFiles: false - AtlasRepDataDirectory: /home/you/gap/pkg/atlasrep/ ID | address, version, files ---------+------------------------------------------------ core | http://atlas.math.rwth-aachen.de/Atlas/, | version 2019-04-08, | 10586 files locally available. ---------+------------------------------------------------ internal | atlasrep/datapkg, | version 2019-05-06, | 276 files locally available. ---------+------------------------------------------------ mfer | http://www.math.rwth-aachen.de/~mfer/datagens/, | version 2015-10-06, | 34 files locally available. ---------+------------------------------------------------ ctblocks | ctblocks/atlas/, | version 2019-04-08, | 121 files locally available.
Note: The above output does not fit to the rest of the manual examples, since data extensions except internal have been removed at the beginning of Chapter 2.
The output tells us that two data extensions have been notified in addition to the core data from the ATLAS of Group Representations and the (local) internal data distributed with the AtlasRep package. The files of the extension mfer must be downloaded before they can be read (but note that the access to remote files is disabled), and the files of the extension ctblocks are locally available in the ctblocks/atlas subdirectory of the GAP package directory. This table (in particular the numbers of locally available files) depends on your installation of the package and how many files you have already downloaded.
gap> DisplayAtlasInfo( [ "M11", "A5" ] ); group | # | maxes | cl | cyc | out | fnd | chk | prs ------+----+-------+----+-----+-----+-----+-----+---- M11 | 42 | 5 | + | + | | + | + | + A5* | 18 | 3 | + | | | | + | +
The above output means that the database provides \(42\) representations of the Mathieu group \(M_{11}\), straight line programs for computing generators of representatives of all five classes of maximal subgroups, for computing representatives of the conjugacy classes of elements and of generators of maximally cyclic subgroups, contains no straight line program for applying outer automorphisms (well, in fact \(M_{11}\) admits no nontrivial outer automorphism), and contains straight line decisions that check a set of generators or a set of group elements for being a set of standard generators. Analogously, \(18\) representations of the alternating group \(A_5\) are available, straight line programs for computing generators of representatives of all three classes of maximal subgroups, and no straight line programs for computing representatives of the conjugacy classes of elements, of generators of maximally cyclic subgroups, and no for computing images under outer automorphisms; straight line decisions for checking the standardization of generators or group elements are available. The star * in the first column of the row for \(A_5\) means that some of the available data do not belong to the core part of the database (see Section 4.2-12).
gap> DisplayAtlasInfo( [ "M11", "A5" ], NrMovedPoints, 11 ); group | # | maxes | cl | cyc | out | fnd | chk | prs ------+---+-------+----+-----+-----+-----+-----+---- M11 | 1 | 5 | + | + | | + | + | +
The given conditions restrict the overview to permutation representations on \(11\) points. The rows for all those groups are omitted for which no such representation is available, and the numbers of those representations are shown that satisfy the given conditions. In the above example, we see that no representation on \(11\) points is available for \(A_5\), and exactly one such representation is available for \(M_{11}\).
gap> DisplayAtlasInfo( "A5", IsPermGroup, true ); Representations for G = A5: (all refer to std. generators 1) --------------------------- 1: G <= Sym(5) 3-trans., on cosets of A4 (1st max.) 2: G <= Sym(6) 2-trans., on cosets of D10 (2nd max.) 3: G <= Sym(10) rank 3, on cosets of S3 (3rd max.) gap> DisplayAtlasInfo( "A5", NrMovedPoints, [ 4 .. 9 ] ); Representations for G = A5: (all refer to std. generators 1) --------------------------- 1: G <= Sym(5) 3-trans., on cosets of A4 (1st max.) 2: G <= Sym(6) 2-trans., on cosets of D10 (2nd max.)
The first three representations stored for \(A_5\) are (in fact primitive) permutation representations.
gap> DisplayAtlasInfo( "A5", Dimension, [ 1 .. 3 ] ); Representations for G = A5: (all refer to std. generators 1) --------------------------- 8: G <= GL(2a,4) character 2a 9: G <= GL(2b,4) character 2b 10: G <= GL(3,5) character 3a 12: G <= GL(3a,9) character 3a 13: G <= GL(3b,9) character 3b 17: G <= GL(3a,Field([Sqrt(5)])) character 3a 18: G <= GL(3b,Field([Sqrt(5)])) character 3b gap> DisplayAtlasInfo( "A5", Characteristic, 0 ); Representations for G = A5: (all refer to std. generators 1) --------------------------- 14: G <= GL(4,Z) character 4a 15: G <= GL(5,Z) character 5a 16: G <= GL(6,Z) character 3ab 17: G <= GL(3a,Field([Sqrt(5)])) character 3a 18: G <= GL(3b,Field([Sqrt(5)])) character 3b
The representations with number between \(4\) and \(13\) are (in fact irreducible) matrix representations over various finite fields, those with numbers \(14\) to \(16\) are integral matrix representations, and the last two are matrix representations over the field generated by \(\sqrt{{5}}\) over the rational number field.
gap> DisplayAtlasInfo( "A5", Identifier, "a" ); Representations for G = A5: (all refer to std. generators 1) --------------------------- 4: G <= GL(4a,2) character 4a 8: G <= GL(2a,4) character 2a 12: G <= GL(3a,9) character 3a 17: G <= GL(3a,Field([Sqrt(5)])) character 3a
Each of the representations with the numbers \(4, 8, 12\), and \(17\) is labeled with the distinguishing letter a.
gap> DisplayAtlasInfo( "A5", NrMovedPoints, IsPrimeInt ); Representations for G = A5: (all refer to std. generators 1) --------------------------- 1: G <= Sym(5) 3-trans., on cosets of A4 (1st max.) gap> DisplayAtlasInfo( "A5", Characteristic, IsOddInt ); Representations for G = A5: (all refer to std. generators 1) --------------------------- 6: G <= GL(4,3) character 4a 7: G <= GL(6,3) character 3ab 10: G <= GL(3,5) character 3a 11: G <= GL(5,5) character 5a 12: G <= GL(3a,9) character 3a 13: G <= GL(3b,9) character 3b gap> DisplayAtlasInfo( "A5", Dimension, IsPrimeInt ); Representations for G = A5: (all refer to std. generators 1) --------------------------- 8: G <= GL(2a,4) character 2a 9: G <= GL(2b,4) character 2b 10: G <= GL(3,5) character 3a 11: G <= GL(5,5) character 5a 12: G <= GL(3a,9) character 3a 13: G <= GL(3b,9) character 3b 15: G <= GL(5,Z) character 5a 17: G <= GL(3a,Field([Sqrt(5)])) character 3a 18: G <= GL(3b,Field([Sqrt(5)])) character 3b gap> DisplayAtlasInfo( "A5", Ring, IsFinite and IsPrimeField ); Representations for G = A5: (all refer to std. generators 1) --------------------------- 4: G <= GL(4a,2) character 4a 5: G <= GL(4b,2) character 2ab 6: G <= GL(4,3) character 4a 7: G <= GL(6,3) character 3ab 10: G <= GL(3,5) character 3a 11: G <= GL(5,5) character 5a
The above examples show how the output can be restricted using a property (a unary function that returns either true or false) that follows NrMovedPoints (Reference: NrMovedPoints for a permutation), Characteristic (Reference: Characteristic), Dimension (Reference: Dimension), or Ring (Reference: Ring) in the argument list of DisplayAtlasInfo (3.5-1).
gap> DisplayAtlasInfo( "A5", IsStraightLineProgram, true ); Programs for G = A5: (all refer to std. generators 1) -------------------- - class repres.* - presentation - maxes (all 3): 1: A4 2: D10 3: S3 - std. gen. checker: (check) (pres)
Straight line programs are available for computing generators of representatives of the three classes of maximal subgroups of \(A_5\), and a straight line decision for checking whether given generators are in fact standard generators is available as well as a presentation in terms of standard generators, see AtlasProgram (3.5-4). A straight line program for computing conjugacy class representatives is available, and the star * says that this program does not belong to the core part of the database (see Section 4.2-12).
‣ AtlasGenerators( gapname, repnr[, maxnr] ) | ( function ) |
‣ AtlasGenerators( identifier ) | ( function ) |
Returns: a record containing generators for a representation, or fail.
In the first form, gapname must be a string denoting a GAP name (see Section 3.2) of a group, and repnr a positive integer. If at least repnr representations for the group with GAP name gapname are available then AtlasGenerators, when called with gapname and repnr, returns an immutable record describing the repnr-th representation; otherwise fail is returned. If a third argument maxnr, a positive integer, is given then an immutable record describing the restriction of the repnr-th representation to the maxnr-th maximal subgroup is returned.
The result record has at least the following components.
contentsthe identifier of the part of the database to which the generators belong, for example "core" or "internal",
generatorsa list of generators for the group,
groupnamethe GAP name of the group (see Section 3.2),
identifiera GAP object (a list of filenames plus additional information) that uniquely determines the representation, see Section 7.7; the value can be used as identifier argument of AtlasGenerators.
repnamea string that is an initial part of the filenames of the generators.
repnrthe number of the representation in the current session, equal to the argument repnr if this is given.
standardizationthe positive integer denoting the underlying standard generators,
typea string that describes the type of the representation ("perm" for a permutation representation, "matff" for a matrix representation over a finite field, "matint" for a matrix representation over the ring of integers, "matalg" for a matrix representation over an algebraic number field).
Additionally, the following describing components may be available if they are known, and depending on the data type of the representation.
sizethe group order,
idthe distinguishing string as described for DisplayAtlasInfo (3.5-1),
characternamea string that describes the character of the representation,
constituentsa list of positive integers denoting the positions of the irreducible constituents of the character of the representation,
p (for permutation representations)for the number of moved points,
dim (for matrix representations)the dimension of the matrices,
ring (for matrix representations)the ring generated by the matrix entries,
transitivity (for permutation representations)a nonnegative integer, see Transitivity (Reference: Transitivity),
orbits (for intransitive permutation representations)the sorted list of orbit lengths on the set of moved points,
rankAction (for transitive permutation representations)the number of orbits of the point stabilizer on the set of moved points, see RankAction (Reference: RankAction),
stabilizer (for transitive permutation representations)a string that describes the structure of the point stabilizers,
isPrimitive (for transitive permutation representations)true if the point stabilizers are maximal subgroups, and false otherwise,
maxnr (for primitive permutation representations)the number of the class of maximal subgroups that contains the point stabilizers, w. r. t. the Maxes (CTblLib: Maxes) list.
It should be noted that the number repnr refers to the number shown by DisplayAtlasInfo (3.5-1) in the current session; it may be that after the addition of new representations (for example after loading a package that provides some), repnr refers to another representation.
The alternative form of AtlasGenerators, with only argument identifier, can be used to fetch the result record with identifier value equal to identifier. The purpose of this variant is to access the same representation also in different GAP sessions.
gap> gens1:= AtlasGenerators( "A5", 1 ); rec( charactername := "1a+4a", constituents := [ 1, 4 ], contents := "core", generators := [ (1,2)(3,4), (1,3,5) ], groupname := "A5", id := "", identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ], isPrimitive := true, maxnr := 1, p := 5, rankAction := 2, repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4", standardization := 1, transitivity := 3, type := "perm" ) gap> gens8:= AtlasGenerators( "A5", 8 ); rec( charactername := "2a", constituents := [ 2 ], contents := "core", dim := 2, generators := [ [ [ Z(2)^0, 0*Z(2) ], [ Z(2^2), Z(2)^0 ] ], [ [ 0*Z(2), Z(2)^0 ], [ Z(2)^0, Z(2)^0 ] ] ], groupname := "A5", id := "a", identifier := [ "A5", [ "A5G1-f4r2aB0.m1", "A5G1-f4r2aB0.m2" ], 1, 4 ], repname := "A5G1-f4r2aB0", repnr := 8, ring := GF(2^2), size := 60, standardization := 1, type := "matff" ) gap> gens17:= AtlasGenerators( "A5", 17 ); rec( charactername := "3a", constituents := [ 2 ], contents := "core", dim := 3, generators := [ [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ -E(5)-E(5)^4, -E(5)-E(5)^4, 1 ] ], [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ], groupname := "A5", id := "a", identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ], polynomial := [ -1, 1, 1 ], repname := "A5G1-Ar3aB0", repnr := 17, ring := NF(5,[ 1, 4 ]), size := 60, standardization := 1, type := "matalg" )
Each of the above pairs of elements generates a group isomorphic to \(A_5\).
gap> gens1max2:= AtlasGenerators( "A5", 1, 2 ); rec( charactername := "1a+4a", constituents := [ 1, 4 ], contents := "core", generators := [ (1,2)(3,4), (2,3)(4,5) ], groupname := "D10", id := "", identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5, 2 ], isPrimitive := true, maxnr := 1, p := 5, rankAction := 2, repname := "A5G1-p5B0", repnr := 1, size := 10, stabilizer := "A4", standardization := 1, transitivity := 3, type := "perm" ) gap> id:= gens1max2.identifier;; gap> gens1max2 = AtlasGenerators( id ); true gap> max2:= Group( gens1max2.generators );; gap> Size( max2 ); 10 gap> IdGroup( max2 ) = IdGroup( DihedralGroup( 10 ) ); true
The elements stored in gens1max2.generators describe the restriction of the first representation of \(A_5\) to a group in the second class of maximal subgroups of \(A_5\) according to the list in the ATLAS of Finite Groups [CCN+85]; this subgroup is isomorphic to the dihedral group \(D_{10}\).
‣ AtlasProgram( gapname[, std][, "contents", sources][, "version", vers], ... ) | ( function ) |
‣ AtlasProgram( identifier ) | ( function ) |
Returns: a record containing a program, or fail.
In the first form, gapname must be a string denoting a GAP name (see Section 3.2) of a group \(G\), say. If the database contains a straight line program (see Section Reference: Straight Line Programs) or straight line decision (see Section 6.1) or black box program (see Section 6.2) as described by the arguments indicated by ... (see below) then AtlasProgram returns an immutable record containing this program. Otherwise fail is returned.
If the optional argument std is given, only those straight line programs/decisions are considered that take generators from the std-th set of standard generators of \(G\) as input, see Section 3.3.
If the optional arguments "contents" and sources are given then the latter must be either a string or a list of strings, with the same meaning as described for DisplayAtlasInfo (3.5-1).
If the optional arguments "version" and vers are given then the latter must be either a number or a list of numbers, and only those straight line programs/decisions are considered whose version number fits to vers.
The result record has at least the following components.
groupnamethe string gapname,
identifiera GAP object (a list of filenames plus additional information) that uniquely determines the program; the value can be used as identifier argument of AtlasProgram (see below),
programthe required straight line program/decision, or black box program,
standardizationthe positive integer denoting the underlying standard generators of \(G\),
versionthe substring of the filename of the program that denotes the version of the program.
If the program computes generators of the restriction to a maximal subgroup then also the following components are present.
sizethe order of the maximal subgroup,
subgroupnamea string denoting a name of the maximal subgroup.
In the first form, the arguments indicated by ... must be as follows.
"maxes" and) a positive integer \(maxnr\)
the required program computes generators of the \(maxnr\)-th maximal subgroup of the group with GAP name \(gapname\).
In this case, the result record of AtlasProgram also may contain a component size, whose value is the order of the maximal subgroup in question.
"maxes"
and two positive integers \(maxnr\) and \(std2\)the required program computes standard generators of the \(maxnr\)-th maximal subgroup of the group with GAP name \(gapname\), w. r. t. the standardization \(std2\).
A prescribed "version" parameter refers to the straight line program for computing the restriction, not to the program for standardizing the result of the restriction.
The meaning of the component size in the result, if present, is the same as in the previous case.
"maxstd" and three positive integers
\(maxnr\), \(vers\), \(substd\)the required program computes standard generators of the \(maxnr\)-th maximal subgroup of the group with GAP name \(gapname\) w. r. t. standardization \(substd\); in this case, the inputs of the program are not standard generators of the group with GAP name \(gapname\) but the outputs of the straight line program with version \(vers\) for computing generators of its \(maxnr\)-th maximal subgroup.
"kernel" and a string \(factname\)the required program computes generators of the kernel of an epimorphism from \(G\) to a group with GAP name \(factname\).
"classes" or "cyclic"the required program computes representatives of conjugacy classes of elements or representatives of generators of maximally cyclic subgroups of \(G\), respectively.
See [BSWW01] and [SWW00] for the background concerning these straight line programs. In these cases, the result record of AtlasProgram also contains a component outputs, whose value is a list of class names of the outputs, as described in Section 3.4.
"cyc2ccl" (and the string \(vers\))the required program computes representatives of conjugacy classes of elements from representatives of generators of maximally cyclic subgroups of \(G\). Thus the inputs are the outputs of the program of type "cyclic" whose version is \(vers\).
"cyc2ccl", \(vers1\), "version", \(vers2\)the required program computes representatives of conjugacy classes of elements from representatives of generators of maximally cyclic subgroups of \(G\), where the inputs are the outputs of the program of type "cyclic" whose version is \(vers1\) and the required program itself has version \(vers2\).
"automorphism" and \(autname\)the required program computes images of standard generators under the outer automorphism of \(G\) that is given by this string.
Note that a value "2" of \(autname\) means that the square of the automorphism is an inner automorphism of \(G\) (not necessarily the identity mapping) but the automorphism itself is not.
"check"the required result is a straight line decision that takes a list of generators for \(G\) and returns true if these generators are standard generators of \(G\) w. r. t. the standardization std, and false otherwise.
"presentation"the required result is a straight line decision that takes a list of group elements and returns true if these elements are standard generators of \(G\) w. r. t. the standardization std, and false otherwise.
See StraightLineProgramFromStraightLineDecision (6.1-9) for an example how to derive defining relators for \(G\) in terms of the standard generators from such a straight line decision.
"find"the required result is a black box program that takes \(G\) and returns a list of standard generators of \(G\), w. r. t. the standardization std.
"restandardize" and an integer \(std2\)the required result is a straight line program that computes standard generators of \(G\) w. r. t. the \(std2\)-th set of standard generators of \(G\); in this case, the argument std must be given.
"other" and \(descr\)the required program is described by \(descr\).
The second form of AtlasProgram, with only argument the list identifier, can be used to fetch the result record with identifier value equal to identifier.
gap> prog:= AtlasProgram( "A5", 2 ); rec( groupname := "A5", identifier := [ "A5", "A5G1-max2W1", 1 ], program := <straight line program>, size := 10, standardization := 1, subgroupname := "D10", version := "1" ) gap> StringOfResultOfStraightLineProgram( prog.program, [ "a", "b" ] ); "[ a, bbab ]" gap> gens1:= AtlasGenerators( "A5", 1 ); rec( charactername := "1a+4a", constituents := [ 1, 4 ], contents := "core", generators := [ (1,2)(3,4), (1,3,5) ], groupname := "A5", id := "", identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ], isPrimitive := true, maxnr := 1, p := 5, rankAction := 2, repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4", standardization := 1, transitivity := 3, type := "perm" ) gap> maxgens:= ResultOfStraightLineProgram( prog.program, > gens1.generators ); [ (1,2)(3,4), (2,3)(4,5) ] gap> maxgens = gens1max2.generators; true
The above example shows that for restricting representations given by standard generators to a maximal subgroup of \(A_5\), we can also fetch and apply the appropriate straight line program. Such a program (see Reference: Straight Line Programs) takes standard generators of a group –in this example \(A_5\)– as its input, and returns a list of elements in this group –in this example generators of the \(D_{10}\) subgroup we had met above– which are computed essentially by evaluating structured words in terms of the standard generators.
gap> prog:= AtlasProgram( "J1", "cyclic" ); rec( groupname := "J1", identifier := [ "J1", "J1G1-cycW1", 1 ], outputs := [ "6A", "7A", "10B", "11A", "15B", "19A" ], program := <straight line program>, standardization := 1, version := "1" ) gap> gens:= GeneratorsOfGroup( FreeGroup( "x", "y" ) );; gap> ResultOfStraightLineProgram( prog.program, gens ); [ (x*y)^2*((y*x)^2*y^2*x)^2*y^2, x*y, (x*(y*x*y)^2)^2*y, (x*y*x*(y*x*y)^3*x*y^2)^2*x*y*x*(y*x*y)^2*y, x*y*x*(y*x*y)^2*y, (x*y)^2*y ]
The above example shows how to fetch and use straight line programs for computing generators of representatives of maximally cyclic subgroups of a given group.
‣ AtlasProgramInfo( gapname[, std][, "contents", sources][, "version", vers], ... ) | ( function ) |
Returns: a record describing a program, or fail.
AtlasProgramInfo takes the same arguments as AtlasProgram (3.5-4), and returns a similar result. The only difference is that the records returned by AtlasProgramInfo have no components program and outputs. The idea is that one can use AtlasProgramInfo for testing whether the program in question is available at all, but without downloading files. The identifier component of the result of AtlasProgramInfo can then be used to fetch the program with AtlasProgram (3.5-4).
gap> AtlasProgramInfo( "J1", "cyclic" ); rec( groupname := "J1", identifier := [ "J1", "J1G1-cycW1", 1 ], standardization := 1, version := "1" )
‣ OneAtlasGeneratingSetInfo( [gapname][,] [std][,] [...] ) | ( function ) |
Returns: a record describing a representation that satisfies the conditions, or fail.
Let gapname be a string denoting a GAP name (see Section 3.2) of a group \(G\), say. If the database contains at least one representation for \(G\) with the required properties then OneAtlasGeneratingSetInfo returns a record \(r\) whose components are the same as those of the records returned by AtlasGenerators (3.5-3), except that the component generators is not contained, and an additional component givenRing is present if Ring is one of the arguments in the function call.
The information in givenRing can be used later to construct the matrices over the prescribed ring. Note that this ring may be for example a domain constructed with AlgebraicExtension (Reference: AlgebraicExtension) instead of a field of cyclotomics or of a finite field constructed with GF (Reference: GF for field size).
The component identifier of \(r\) can be used as input for AtlasGenerators (3.5-3) in order to fetch the generators. If no representation satisfying the given conditions is available then fail is returned.
If the argument std is given then it must be a positive integer or a list of positive integers, denoting the sets of standard generators w. r. t. which the representation shall be given (see Section 3.3).
The argument gapname can be missing (then all available groups are considered), or a list of group names can be given instead.
Further restrictions can be entered as arguments, with the same meaning as described for DisplayAtlasInfo (3.5-1). The result of OneAtlasGeneratingSetInfo describes the first generating set for \(G\) that matches the restrictions, in the ordering shown by DisplayAtlasInfo (3.5-1).
Note that even in the case that the user preference AtlasRepAccessRemoteFiles has the value true (see Section 4.2-1), OneAtlasGeneratingSetInfo does not attempt to transfer remote data files, just the table of contents is evaluated. So this function (as well as AllAtlasGeneratingSetInfos (3.5-7)) can be used to check for the availability of certain representations, and afterwards one can call AtlasGenerators (3.5-3) for those representations one wants to work with.
In the following example, we try to access information about permutation representations for the alternating group \(A_5\).
gap> info:= OneAtlasGeneratingSetInfo( "A5" ); rec( charactername := "1a+4a", constituents := [ 1, 4 ], contents := "core", groupname := "A5", id := "", identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ], isPrimitive := true, maxnr := 1, p := 5, rankAction := 2, repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4", standardization := 1, transitivity := 3, type := "perm" ) gap> gens:= AtlasGenerators( info.identifier ); rec( charactername := "1a+4a", constituents := [ 1, 4 ], contents := "core", generators := [ (1,2)(3,4), (1,3,5) ], groupname := "A5", id := "", identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ], isPrimitive := true, maxnr := 1, p := 5, rankAction := 2, repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4", standardization := 1, transitivity := 3, type := "perm" ) gap> info = OneAtlasGeneratingSetInfo( "A5", IsPermGroup, true ); true gap> info = OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, "minimal" ); true gap> info = OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, [ 1 .. 10 ] ); true gap> OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, 20 ); fail
Note that a permutation representation of degree \(20\) could be obtained by taking twice the primitive representation on \(10\) points; however, the database does not store this imprimitive representation (cf. Section 3.1).
We continue this example. Next we access matrix representations of \(A_5\).
gap> info:= OneAtlasGeneratingSetInfo( "A5", IsMatrixGroup, true ); rec( charactername := "4a", constituents := [ 4 ], contents := "core", dim := 4, groupname := "A5", id := "a", identifier := [ "A5", [ "A5G1-f2r4aB0.m1", "A5G1-f2r4aB0.m2" ], 1, 2 ], repname := "A5G1-f2r4aB0", repnr := 4, ring := GF(2), size := 60, standardization := 1, type := "matff" ) gap> gens:= AtlasGenerators( info.identifier ); rec( charactername := "4a", constituents := [ 4 ], contents := "core", dim := 4, generators := [ <an immutable 4x4 matrix over GF2>, <an immutable 4x4 matrix over GF2> ], groupname := "A5", id := "a", identifier := [ "A5", [ "A5G1-f2r4aB0.m1", "A5G1-f2r4aB0.m2" ], 1, 2 ], repname := "A5G1-f2r4aB0", repnr := 4, ring := GF(2), size := 60, standardization := 1, type := "matff" ) gap> info = OneAtlasGeneratingSetInfo( "A5", Dimension, 4 ); true gap> info = OneAtlasGeneratingSetInfo( "A5", Characteristic, 2 ); true gap> info2:= OneAtlasGeneratingSetInfo( "A5", Ring, GF(2) );; gap> info.identifier = info2.identifier; true gap> OneAtlasGeneratingSetInfo( "A5", Characteristic, [2,5], Dimension, 2 ); rec( charactername := "2a", constituents := [ 2 ], contents := "core", dim := 2, groupname := "A5", id := "a", identifier := [ "A5", [ "A5G1-f4r2aB0.m1", "A5G1-f4r2aB0.m2" ], 1, 4 ], repname := "A5G1-f4r2aB0", repnr := 8, ring := GF(2^2), size := 60, standardization := 1, type := "matff" ) gap> OneAtlasGeneratingSetInfo( "A5", Characteristic, [2,5], Dimension, 1 ); fail gap> info:= OneAtlasGeneratingSetInfo( "A5", Characteristic, 0, > Dimension, 4 ); rec( charactername := "4a", constituents := [ 4 ], contents := "core", dim := 4, groupname := "A5", id := "", identifier := [ "A5", "A5G1-Zr4B0.g", 1, 4 ], repname := "A5G1-Zr4B0", repnr := 14, ring := Integers, size := 60, standardization := 1, type := "matint" ) gap> gens:= AtlasGenerators( info.identifier ); rec( charactername := "4a", constituents := [ 4 ], contents := "core", dim := 4, generators := [ [ [ 1, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ -1, -1, -1, -1 ] ], [ [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 1, 0, 0, 0 ] ] ], groupname := "A5", id := "", identifier := [ "A5", "A5G1-Zr4B0.g", 1, 4 ], repname := "A5G1-Zr4B0", repnr := 14, ring := Integers, size := 60, standardization := 1, type := "matint" ) gap> info = OneAtlasGeneratingSetInfo( "A5", Ring, Integers ); true gap> info2:= OneAtlasGeneratingSetInfo( "A5", Ring, CF(37) );; gap> info = info2; false gap> Difference( RecNames( info2 ), RecNames( info ) ); [ "givenRing" ] gap> info2.givenRing; CF(37) gap> OneAtlasGeneratingSetInfo( "A5", Ring, Integers mod 77 ); fail gap> info:= OneAtlasGeneratingSetInfo( "A5", Ring, CF(5), Dimension, 3 ); rec( charactername := "3a", constituents := [ 2 ], contents := "core", dim := 3, givenRing := CF(5), groupname := "A5", id := "a", identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ], polynomial := [ -1, 1, 1 ], repname := "A5G1-Ar3aB0", repnr := 17, ring := NF(5,[ 1, 4 ]), size := 60, standardization := 1, type := "matalg" ) gap> gens:= AtlasGenerators( info ); rec( charactername := "3a", constituents := [ 2 ], contents := "core", dim := 3, generators := [ [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ -E(5)-E(5)^4, -E(5)-E(5)^4, 1 ] ], [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ], givenRing := CF(5), groupname := "A5", id := "a", identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ], polynomial := [ -1, 1, 1 ], repname := "A5G1-Ar3aB0", repnr := 17, ring := NF(5,[ 1, 4 ]), size := 60, standardization := 1, type := "matalg" ) gap> gens2:= AtlasGenerators( info.identifier );; gap> Difference( RecNames( gens ), RecNames( gens2 ) ); [ "givenRing" ] gap> OneAtlasGeneratingSetInfo( "A5", Ring, GF(17) ); fail
‣ AllAtlasGeneratingSetInfos( [gapname][,] [std][,] [...] ) | ( function ) |
Returns: the list of all records describing representations that satisfy the conditions.
AllAtlasGeneratingSetInfos is similar to OneAtlasGeneratingSetInfo (3.5-6). The difference is that the list of all records describing the available representations with the given properties is returned instead of just one such component. In particular an empty list is returned if no such representation is available.
gap> AllAtlasGeneratingSetInfos( "A5", IsPermGroup, true ); [ rec( charactername := "1a+4a", constituents := [ 1, 4 ], contents := "core", groupname := "A5", id := "", identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ] , isPrimitive := true, maxnr := 1, p := 5, rankAction := 2, repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4", standardization := 1, transitivity := 3, type := "perm" ), rec( charactername := "1a+5a", constituents := [ 1, 5 ], contents := "core", groupname := "A5", id := "", identifier := [ "A5", [ "A5G1-p6B0.m1", "A5G1-p6B0.m2" ], 1, 6 ] , isPrimitive := true, maxnr := 2, p := 6, rankAction := 2, repname := "A5G1-p6B0", repnr := 2, size := 60, stabilizer := "D10", standardization := 1, transitivity := 2, type := "perm" ), rec( charactername := "1a+4a+5a", constituents := [ 1, 4, 5 ], contents := "core", groupname := "A5", id := "", identifier := [ "A5", [ "A5G1-p10B0.m1", "A5G1-p10B0.m2" ], 1, 10 ], isPrimitive := true, maxnr := 3, p := 10, rankAction := 3, repname := "A5G1-p10B0", repnr := 3, size := 60, stabilizer := "S3", standardization := 1, transitivity := 1, type := "perm" ) ]
Note that a matrix representation in any characteristic can be obtained by reducing a permutation representation or an integral matrix representation; however, the database does not store such a representation (cf. Section 3.1).
‣ AtlasGroup( [gapname][,] [std][,] [...] ) | ( function ) |
‣ AtlasGroup( identifier ) | ( function ) |
Returns: a group that satisfies the conditions, or fail.
AtlasGroup takes the same arguments as OneAtlasGeneratingSetInfo (3.5-6), and returns the group generated by the generators component of the record that is returned by OneAtlasGeneratingSetInfo (3.5-6) with these arguments; if OneAtlasGeneratingSetInfo (3.5-6) returns fail then also AtlasGroup returns fail.
gap> g:= AtlasGroup( "A5" ); Group([ (1,2)(3,4), (1,3,5) ])
Alternatively, it is possible to enter exactly one argument, a record identifier as returned by OneAtlasGeneratingSetInfo (3.5-6) or AllAtlasGeneratingSetInfos (3.5-7), or the identifier component of such a record.
gap> info:= OneAtlasGeneratingSetInfo( "A5" ); rec( charactername := "1a+4a", constituents := [ 1, 4 ], contents := "core", groupname := "A5", id := "", identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ], isPrimitive := true, maxnr := 1, p := 5, rankAction := 2, repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4", standardization := 1, transitivity := 3, type := "perm" ) gap> AtlasGroup( info ); Group([ (1,2)(3,4), (1,3,5) ]) gap> AtlasGroup( info.identifier ); Group([ (1,2)(3,4), (1,3,5) ])
In the groups returned by AtlasGroup, the value of the attribute AtlasRepInfoRecord (3.5-10) is set. This information is used for example by AtlasSubgroup (3.5-9) when this function is called with second argument a group created by AtlasGroup.
‣ AtlasSubgroup( gapname[, std][, ...], maxnr ) | ( function ) |
‣ AtlasSubgroup( identifier, maxnr ) | ( function ) |
‣ AtlasSubgroup( G, maxnr ) | ( function ) |
Returns: a group that satisfies the conditions, or fail.
The arguments of AtlasSubgroup, except the last argument maxnr, are the same as for AtlasGroup (3.5-8). If the database provides a straight line program for restricting representations of the group with name gapname (given w. r. t. the std-th standard generators) to the maxnr-th maximal subgroup and if a representation with the required properties is available, in the sense that calling AtlasGroup (3.5-8) with the same arguments except maxnr yields a group, then AtlasSubgroup returns the restriction of this representation to the maxnr-th maximal subgroup.
In all other cases, fail is returned.
Note that the conditions refer to the group and not to the subgroup. It may happen that in the restriction of a permutation representation to a subgroup, fewer points are moved, or that the restriction of a matrix representation turns out to be defined over a smaller ring. Here is an example.
gap> g:= AtlasSubgroup( "A5", NrMovedPoints, 5, 1 ); Group([ (1,5)(2,3), (1,3,5) ]) gap> NrMovedPoints( g ); 4
Alternatively, it is possible to enter exactly two arguments, the first being a record identifier as returned by OneAtlasGeneratingSetInfo (3.5-6) or AllAtlasGeneratingSetInfos (3.5-7), or the identifier component of such a record, or a group G constructed with AtlasGroup (3.5-8).
gap> info:= OneAtlasGeneratingSetInfo( "A5" ); rec( charactername := "1a+4a", constituents := [ 1, 4 ], contents := "core", groupname := "A5", id := "", identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ], isPrimitive := true, maxnr := 1, p := 5, rankAction := 2, repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4", standardization := 1, transitivity := 3, type := "perm" ) gap> AtlasSubgroup( info, 1 ); Group([ (1,5)(2,3), (1,3,5) ]) gap> AtlasSubgroup( info.identifier, 1 ); Group([ (1,5)(2,3), (1,3,5) ]) gap> AtlasSubgroup( AtlasGroup( "A5" ), 1 ); Group([ (1,5)(2,3), (1,3,5) ])
‣ AtlasRepInfoRecord( G ) | ( attribute ) |
‣ AtlasRepInfoRecord( name ) | ( attribute ) |
Returns: the record stored in the group G when this was constructed with AtlasGroup (3.5-8), or a record with information about the group with name name.
For a group G that has been constructed with AtlasGroup (3.5-8), the value of this attribute is the info record that describes G, in the sense that this record was the first argument of the call to AtlasGroup (3.5-8), or it is the result of the call to OneAtlasGeneratingSetInfo (3.5-6) with the conditions that were listed in the call to AtlasGroup (3.5-8).
gap> AtlasRepInfoRecord( AtlasGroup( "A5" ) ); rec( charactername := "1a+4a", constituents := [ 1, 4 ], contents := "core", groupname := "A5", id := "", identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ], isPrimitive := true, maxnr := 1, p := 5, rankAction := 2, repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4", standardization := 1, transitivity := 3, type := "perm" )
For a string name that is a GAP name of a group \(G\), say, AtlasRepInfoRecord returns a record that contains information about \(G\) which is used by DisplayAtlasInfo (3.5-1). The following components may be bound in the record.
namethe string name,
nrMaxesthe number of conjugacy classes of maximal subgroups of \(G\),
sizethe order of \(G\),
sizesMaxesa list which contains at position \(i\), if bound, the order of a subgroup in the \(i\)-th class of maximal subgroups of \(G\),
slpMaxesa list of length two; the first entry is a list of positions \(i\) such that a straight line program for computing the restriction of representations of \(G\) to a subgroup in the \(i\)-th class of maximal subgroups is available via AtlasRep; the second entry is the corresponding list of standardizations of the generators of \(G\) for which these straight line programs are available,
structureMaxesa list which contains at position \(i\), if bound, a string that describes the structure of the subgroups in the \(i\)-th class of maximal subgroups of \(G\).
gap> AtlasRepInfoRecord( "A5" ); rec( name := "A5", nrMaxes := 3, size := 60, sizesMaxes := [ 12, 10, 6 ], slpMaxes := [ [ 1 .. 3 ], [ [ 1 ], [ 1 ], [ 1 ] ] ], structureMaxes := [ "A4", "D10", "S3" ] ) gap> AtlasRepInfoRecord( "J5" ); rec( )
‣ EvaluatePresentation( G, gapname[, std] ) | ( operation ) |
‣ EvaluatePresentation( gens, gapname[, std] ) | ( operation ) |
Returns: a list of group elements or fail.
The first argument must be either a group G or a list gens of group generators, and gapname must be a string that is a GAP name (see Section 3.2) of a group \(H\), say. The optional argument std, if given, must be a positive integer that denotes a standardization of generators of \(H\), the default is \(1\).
EvaluatePresentation returns fail if no presentation for \(H\) w. r. t. the standardization std is stored in the database, and otherwise returns the list of results of evaluating the relators of a presentation for \(H\) at gens or the GeneratorsOfGroup (Reference: GeneratorsOfGroup) value of G, respectively. (An error is signalled if the number of generators is not equal to the number of inputs of the presentation.)
The result can be used as follows. Let \(N\) be the normal closure of the the result in G. The factor group G\(/N\) is an epimorphic image of \(H\). In particular, if all entries of the result have order \(1\) then G itself is an epimorphic image of \(H\). Moreover, an epimorphism is given by mapping the std-th standard generators of \(H\) to the \(N\)-cosets of the given generators of G.
gap> g:= MathieuGroup( 12 );; gap> gens:= GeneratorsOfGroup( g );; # switch to 2 generators gap> g:= Group( gens[1] * gens[3], gens[2] * gens[3] );; gap> EvaluatePresentation( g, "J0" ); # no pres. for group "J0" fail gap> relimgs:= EvaluatePresentation( g, "M11" );; gap> List( relimgs, Order ); # wrong group [ 3, 1, 5, 4, 10 ] gap> relimgs:= EvaluatePresentation( g, "M12" );; gap> List( relimgs, Order ); # generators are not standard [ 3, 4, 5, 4, 4 ] gap> g:= AtlasGroup( "M12" );; gap> relimgs:= EvaluatePresentation( g, "M12", 1 );; gap> List( relimgs, Order ); # right group, std. generators [ 1, 1, 1, 1, 1 ] gap> g:= AtlasGroup( "2.M12" );; gap> relimgs:= EvaluatePresentation( g, "M12", 1 );; gap> List( relimgs, Order ); # std. generators for extension [ 1, 2, 1, 1, 2 ] gap> Size( NormalClosure( g, SubgroupNC( g, relimgs ) ) ); 2
‣ StandardGeneratorsData( G, gapname[, std] ) | ( operation ) |
‣ StandardGeneratorsData( gens, gapname[, std] ) | ( operation ) |
Returns: a record that describes standard generators of the group in question, or fail, or the string "timeout".
The first argument must be either a group G or a list gens of group generators, and gapname must be a string that is a GAP name (see Section 3.2) of a group \(H\), say. The optional argument std, if given, must be a positive integer that denotes a standardization of generators of \(H\), the default is \(1\).
If the global option projective is given then the group elements must be matrices over a finite field, and the group must be a central extension of the group \(H\) by a normal subgroup that consists of scalar matrices. In this case, all computations will be carried out modulo scalar matrices (in particular, element orders will be computed using ProjectiveOrder (Reference: ProjectiveOrder)), and the returned standard generators will belong to \(H\).
StandardGeneratorsData returns
failif no black box program for computing standard generators of \(H\) w. r. t. the standardization std is stored in the database, or if the black box program returns fail because a runtime error occurred or the program has proved that the given group or generators cannot generate a group isomorphic to \(H\),
"timeout"if the black box program returns "timeout", typically because some elements of a given order were not found among a reasonable number of random elements, or
otherwise.
When the result is not a record then either the group is not isomorphic to \(H\) (modulo scalars if applicable), or we were unlucky with choosing random elements.
When a record is returned and G or the group generated by gens, respectively, is isomorphic to \(H\) (or to a central extension of \(H\) by a group of scalar matrices if the global option projective is given) then the result describes the desired standard generators.
If G or the group generated by gens, respectively, is not isomorphic to \(H\) then it may still happen that StandardGeneratorsData returns a record. For a proof that the returned record describes the desired standard generators, one can use a presentation of \(H\) whose generators correspond to the std-th standard generators, see EvaluatePresentation (3.5-11).
A returned record has the following components.
gapnamethe string gapname,
givengensthe list of group generators from which standard generators were computed, either gens or the GeneratorsOfGroup (Reference: GeneratorsOfGroup) value of G,
stdgensa list of standard generators of the group,
givengenstostdgensa straight line program that takes givengens as inputs, and returns stdgens,
stdthe underlying standardization std.
The first examples show three cases of failure, due to the unavailability of a suitable black box program or to a wrong choice of gapname. (In the search for standard generators of \(M_{11}\) in the group \(M_{12}\), one may or may not find an element whose order does not appear in \(M_{11}\); in the first case, the result is fail, whereas a record is returned in the second case. Both cases occur.)
gap> StandardGeneratorsData( MathieuGroup( 11 ), "J0" ); fail gap> StandardGeneratorsData( MathieuGroup( 11 ), "M12" ); "timeout" gap> repeat > res:= StandardGeneratorsData( MathieuGroup( 12 ), "M11" ); > until res = fail;
The next example shows a computation of standard generators for the Mathieu group \(M_{12}\). Using a presentation of \(M_{12}\) w. r. t. these standard generators, we prove that the given group is isomorphic to \(M_{12}\).
gap> gens:= GeneratorsOfGroup( MathieuGroup( 12 ) );; gap> std:= 1;; gap> res:= StandardGeneratorsData( gens, "M12", std );; gap> Set( RecNames( res ) ); [ "gapname", "givengens", "givengenstostdgens", "std", "stdgens" ] gap> gens = res.givengens; true gap> ResultOfStraightLineProgram( res.givengenstostdgens, gens ) > = res.stdgens; true gap> evl:= EvaluatePresentation( res.stdgens, "M12", std );; gap> ForAll( evl, IsOne ); true
The next example shows the use of the global option projective. We take an irreducible matrix representation of the double cover of the Mathieu group \(M_{12}\) (thus the center is represented by scalar matrices) and compute standard generators of the factor group \(M_{12}\). Using a presentation of \(M_{12}\) w. r. t. these standard generators, we prove that the given group is modulo scalars isomorphic to \(M_{12}\), and we get generators for the kernel.
gap> g:= AtlasGroup( "2.M12", IsMatrixGroup, Characteristic, IsPosInt );; gap> gens:= Permuted( GeneratorsOfGroup( g ), (1,2) );; gap> res:= StandardGeneratorsData( gens, "M12", std : projective );; gap> gens = res.givengens; true gap> ResultOfStraightLineProgram( res.givengenstostdgens, gens ) > = res.stdgens; true gap> evl:= EvaluatePresentation( res.stdgens, "M12", std );; gap> ForAll( evl, IsOne ); false gap> ForAll( evl, x -> IsCentral( g, x ) ); true
The functions BrowseMinimalDegrees (3.6-1), BrowseBibliographySporadicSimple (3.6-2), and BrowseAtlasInfo (Browse: BrowseAtlasInfo) (an alternative to DisplayAtlasInfo (3.5-1)) are available only if the GAP package Browse (see [BL18]) is loaded.
‣ BrowseMinimalDegrees( [gapnames] ) | ( function ) |
Returns: the list of info records for the clicked representations.
If the GAP package Browse (see [BL18]) is loaded then this function is available. It opens a browse table whose rows correspond to the groups for which AtlasRep knows some information about minimal degrees, whose columns correspond to the characteristics that occur, and whose entries are the known minimal degrees.
gap> if IsBound( BrowseMinimalDegrees ) then > down:= NCurses.keys.DOWN;; DOWN:= NCurses.keys.NPAGE;; > right:= NCurses.keys.RIGHT;; END:= NCurses.keys.END;; > enter:= NCurses.keys.ENTER;; nop:= [ 14, 14, 14 ];; > # just scroll in the table > BrowseData.SetReplay( Concatenation( [ DOWN, DOWN, DOWN, > right, right, right ], "sedddrrrddd", nop, nop, "Q" ) ); > BrowseMinimalDegrees();; > # restrict the table to the groups with minimal ordinary degree 6 > BrowseData.SetReplay( Concatenation( "scf6", > [ down, down, right, enter, enter ] , nop, nop, "Q" ) ); > BrowseMinimalDegrees();; > BrowseData.SetReplay( false ); > fi;
If an argument gapnames is given then it must be a list of GAP names of groups. The browse table is then restricted to the rows corresponding to these group names and to the columns that are relevant for these groups. A perhaps interesting example is the subtable with the data concerning sporadic simple groups and their covering groups, which has been published in [Jan05]. This table can be shown as follows.
gap> if IsBound( BrowseMinimalDegrees ) then > # just scroll in the table > BrowseData.SetReplay( Concatenation( [ DOWN, DOWN, DOWN, END ], > "rrrrrrrrrrrrrr", nop, nop, "Q" ) ); > BrowseMinimalDegrees( BibliographySporadicSimple.groupNamesJan05 );; > fi;
The browse table does not contain rows for the groups \(6.M_{22}\), \(12.M_{22}\), \(6.Fi_{22}\). Note that in spite of the title of [Jan05], the entries in Table 1 of this paper are in fact the minimal degrees of faithful irreducible representations, and in the above three cases, these degrees are larger than the minimal degrees of faithful representations. The underlying data of the browse table is about the minimal faithful (but not necessarily irreducible) degrees.
The return value of BrowseMinimalDegrees is the list of OneAtlasGeneratingSetInfo (3.5-6) values for those representations that have been clicked
in visual mode.
The variant without arguments of this function is also available in the menu shown by BrowseGapData (Browse: BrowseGapData).
‣ BrowseBibliographySporadicSimple( ) | ( function ) |
Returns: a record as returned by ParseBibXMLExtString (GAPDoc: ParseBibXMLextString).
If the GAP package Browse (see [BL18]) is loaded then this function is available. It opens a browse table whose rows correspond to the entries of the bibliographies in the ATLAS of Finite Groups [CCN+85] and in the ATLAS of Brauer Characters [JLPW95].
The function is based on BrowseBibliography (Browse: BrowseBibliography), see the documentation of this function for details, e.g., about the return value.
The returned record encodes the bibliography entries corresponding to those rows of the table that are clicked
in visual mode, in the same format as the return value of ParseBibXMLExtString (GAPDoc: ParseBibXMLextString), see the manual of the GAP package GAPDoc [LN18] for details.
BrowseBibliographySporadicSimple can be called also via the menu shown by BrowseGapData (Browse: BrowseGapData).
gap> if IsBound( BrowseBibliographySporadicSimple ) then > enter:= NCurses.keys.ENTER;; nop:= [ 14, 14, 14 ];; > BrowseData.SetReplay( Concatenation( > # choose the application > "/Bibliography of Sporadic Simple Groups", [ enter, enter ], > # search in the title column for the Atlas of Finite Groups > "scr/Atlas of finite groups", [ enter, > # and quit > nop, nop, nop, nop ], "Q" ) ); > BrowseGapData();; > BrowseData.SetReplay( false ); > fi;
The bibliographies contained in the ATLAS of Finite Groups [CCN+85] and in the ATLAS of Brauer Characters [JLPW95] are available online in HTML format, see http://www.math.rwth-aachen.de/~Thomas.Breuer/atlasrep/bibl/index.html.
The source data in BibXMLext format, which are used by BrowseBibliographySporadicSimple, are distributed with the AtlasRep package, in four files with suffix xml in the package's bibl directory. Note that each of the two books contains two bibliographies.
Details about the BibXMLext format, including information how to transform the data into other formats such as BibTeX, can be found in the GAP package GAPDoc (see [LN18]).
generated by GAPDoc2HTML