This chapter describes GAP objects and functions that are provided by the AtlasRep package but that might be of general interest.
The new objects are straight line decisions (see Section 6.1) and black box programs (see Section 6.2).
The new functions are concerned with representations of minimal degree, see Section 6.3, and a JSON interface, see Section 6.4.
Straight line decisions are similar to straight line programs (see Section Reference: Straight Line Programs) but return true or false. A straight line decision checks whether its inputs have some property. An important example is to check whether a given list of group generators is in fact a list of standard generators (cf. Section3.3) for this group.
A straight line decision in GAP is represented by an object in the filter IsStraightLineDecision (6.1-1) that stores a list of "lines" each of which has one of the following three forms.
a nonempty dense list l of integers,
a pair [ l, i ] where l is a list of form 1. and i is a positive integer,
a list [ "Order", i, n ] where i and n are positive integers.
The first two forms have the same meaning as for straight line programs (see Section Reference: Straight Line Programs), the last form means a check whether the element stored at the i-th label has the order n.
For the meaning of the list of lines, see ResultOfStraightLineDecision (6.1-6).
Straight line decisions can be constructed using StraightLineDecision (6.1-5), defining attributes for straight line decisions are NrInputsOfStraightLineDecision (6.1-3) and LinesOfStraightLineDecision (6.1-2), an operation for straight line decisions is ResultOfStraightLineDecision (6.1-6).
Special methods applicable to straight line decisions are installed for the operations Display (Reference: Display), IsInternallyConsistent (Reference: IsInternallyConsistent), PrintObj (Reference: PrintObj), and ViewObj (Reference: ViewObj).
For a straight line decision prog, the default Display (Reference: Display) method prints the interpretation of prog as a sequence of assignments of associative words and of order checks; a record with components gensnames (with value a list of strings) and listname (a string) may be entered as second argument of Display (Reference: Display), in this case these names are used, the default for gensnames is [ g1, g2, ... ], the default for listname is r.
‣ IsStraightLineDecision( obj ) | ( category ) |
Each straight line decision in GAP lies in the filter IsStraightLineDecision.
‣ LinesOfStraightLineDecision( prog ) | ( operation ) |
Returns: the list of lines that define the straight line decision.
This defining attribute for the straight line decision prog (see IsStraightLineDecision (6.1-1)) corresponds to LinesOfStraightLineProgram (Reference: LinesOfStraightLineProgram) for straight line programs.
gap> dec:= StraightLineDecision( [ [ [ 1, 1, 2, 1 ], 3 ], > [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] ); <straight line decision> gap> LinesOfStraightLineDecision( dec ); [ [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ]
‣ NrInputsOfStraightLineDecision( prog ) | ( operation ) |
Returns: the number of inputs required for the straight line decision.
This defining attribute corresponds to NrInputsOfStraightLineProgram (Reference: NrInputsOfStraightLineProgram).
gap> NrInputsOfStraightLineDecision( dec ); 2
‣ ScanStraightLineDecision( string ) | ( function ) |
Returns: a record containing the straight line decision, or fail.
Let string be a string that encodes a straight line decision in the sense that it consists of the lines listed for ScanStraightLineProgram (7.4-1), except that oup lines are not allowed, and instead lines of the following form may occur.
chor a bmeans that it is checked whether the order of the element at label a is b.
ScanStraightLineDecision returns a record containing as the value of its component program the corresponding GAP straight line decision (see IsStraightLineDecision (6.1-1)) if the input string satisfies the syntax rules stated above, and returns fail otherwise. In the latter case, information about the first corrupted line of the program is printed if the info level of InfoCMeatAxe (7.1-2) is at least 1.
gap> str:= "inp 2\nchor 1 2\nchor 2 3\nmu 1 2 3\nchor 3 5";; gap> prg:= ScanStraightLineDecision( str ); rec( program := <straight line decision> ) gap> prg:= prg.program;; gap> Display( prg ); # input: r:= [ g1, g2 ]; # program: if Order( r[1] ) <> 2 then return false; fi; if Order( r[2] ) <> 3 then return false; fi; r[3]:= r[1]*r[2]; if Order( r[3] ) <> 5 then return false; fi; # return value: true
‣ StraightLineDecision( lines[, nrgens] ) | ( function ) |
‣ StraightLineDecisionNC( lines[, nrgens] ) | ( function ) |
Returns: the straight line decision given by the list of lines.
Let lines be a list of lists that defines a unique straight line decision (see IsStraightLineDecision (6.1-1)); in this case StraightLineDecision returns this program, otherwise an error is signalled. The optional argument nrgens specifies the number of input generators of the program; if a list of integers (a line of form 1. in the definition above) occurs in lines then this number is not determined by lines and therefore must be specified by the argument nrgens; if not then StraightLineDecision returns fail.
StraightLineDecisionNC does the same as StraightLineDecision, except that the internal consistency of the program is not checked.
‣ ResultOfStraightLineDecision( prog, gens[, orderfunc] ) | ( operation ) |
Returns: true if all checks succeed, otherwise false.
ResultOfStraightLineDecision evaluates the straight line decision (see IsStraightLineDecision (6.1-1)) prog at the group elements in the list gens.
The function for computing the order of a group element can be given as the optional argument orderfunc. For example, this may be a function that gives up at a certain limit if one has to be aware of extremely huge orders in failure cases.
The result of a straight line decision with lines p_1, p_2, ..., p_k when applied to gens is defined as follows.
First a list r of intermediate values is initialized with a shallow copy of gens.
For i ≤ k, before the i-th step, let r be of length n. If p_i is the external representation of an associative word in the first n generators then the image of this word under the homomorphism that is given by mapping r to these first n generators is added to r. If p_i is a pair [ l, j ], for a list l, then the same element is computed, but instead of being added to r, it replaces the j-th entry of r. If p_i is a triple ["Order", i, n ] then it is checked whether the order of r[i] is n; if not then false is returned immediately.
If all k lines have been processed and no order check has failed then true is returned.
Here are some examples.
gap> dec:= StraightLineDecision( [ ], 1 ); <straight line decision> gap> ResultOfStraightLineDecision( dec, [ () ] ); true
The above straight line decision dec returns true –for any input of the right length.
gap> dec:= StraightLineDecision( [ [ [ 1, 1, 2, 1 ], 3 ], > [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] ); <straight line decision> gap> LinesOfStraightLineDecision( dec ); [ [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] gap> ResultOfStraightLineDecision( dec, [ (), () ] ); false gap> ResultOfStraightLineDecision( dec, [ (1,2)(3,4), (1,4,5) ] ); true
The above straight line decision admits two inputs; it tests whether the orders of the inputs are 2 and 3, and the order of their product is 5.
We can associate a finitely presented group F / R to each straight line decision dec, say, as follows. The free generators of the free group F are in bijection with the inputs, and the defining relators generating R as a normal subgroup of F are given by those words w^k for which dec contains a check whether the order of w equals k.
So if dec returns true for the input list [ g_1, g_2, ..., g_n ] then mapping the free generators of F to the inputs defines an epimorphism Φ from F to the group G, say, that is generated by these inputs, such that R is contained in the kernel of Φ.
(Note that "satisfying dec" is a stronger property than "satisfying a presentation". For example, ⟨ x ∣ x^2 = x^3 = 1 ⟩ is a presentation for the trivial group, but the straight line decision that checks whether the order of x is both 2 and 3 clearly always returns false.)
AtlasRep supports the following two kinds of straight line decisions.
A presentation is a straight line decision dec that is defined for a set of standard generators of a group G and that returns true if and only if the list of inputs is in fact a sequence of such standard generators for G. In other words, the relators derived from the order checks in the way described above are defining relators for G, and moreover these relators are words in terms of standard generators. (In particular the kernel of the map Φ equals R whenever dec returns true.)
A semi-presentation is a straight line decision dec that is defined for a set of standard generators of a group G and that returns true for a list of inputs that is known to generate a group isomorphic with G if and only if these inputs form in fact a sequence of standard generators for G. In other words, the relators derived from the order checks in the way described above are not necessarily defining relators for G, but if we assume that the g_i generate G then they are standard generators. (In particular, F / R may be a larger group than G but in this case Φ maps the free generators of F to standard generators of G.)
More about semi-presentations can be found in [NW05].
Available presentations and semi-presentations are listed by DisplayAtlasInfo (3.5-1), they can be accessed via AtlasProgram (3.5-4). (Clearly each presentation is also a semi-presentation. So a semi-presentation for some standard generators of a group is regarded as available whenever a presentation for these standard generators and this group is available.)
Note that different groups can have the same semi-presentation. We illustrate this with an example that is mentioned in [NW05]. The groups L_2(7) ≅ L_3(2) and L_2(8) are generated by elements of the orders 2 and 3 such that their product has order 7, and no further conditions are necessary to define standard generators.
gap> check:= AtlasProgram( "L2(8)", "check" ); rec( groupname := "L2(8)", identifier := [ "L2(8)", "L28G1-check1", 1, 1 ], program := <straight line decision>, standardization := 1, version := "1" ) gap> gens:= AtlasGenerators( "L2(8)", 1 ); rec( charactername := "1a+8a", constituents := [ 1, 6 ], contents := "core", generators := [ (1,2)(3,4)(6,7)(8,9), (1,3,2)(4,5,6)(7,8,9) ], groupname := "L2(8)", id := "", identifier := [ "L2(8)", [ "L28G1-p9B0.m1", "L28G1-p9B0.m2" ], 1, 9 ], isPrimitive := true, maxnr := 1, p := 9, rankAction := 2, repname := "L28G1-p9B0", repnr := 1, size := 504, stabilizer := "2^3:7", standardization := 1, transitivity := 3, type := "perm" ) gap> ResultOfStraightLineDecision( check.program, gens.generators ); true gap> gens:= AtlasGenerators( "L3(2)", 1 ); rec( contents := "core", generators := [ (2,4)(3,5), (1,2,3)(5,6,7) ], groupname := "L3(2)", id := "a", identifier := [ "L3(2)", [ "L27G1-p7aB0.m1", "L27G1-p7aB0.m2" ], 1, 7 ], isPrimitive := true, maxnr := 1, p := 7, rankAction := 2, repname := "L27G1-p7aB0", repnr := 1, size := 168, stabilizer := "S4", standardization := 1, transitivity := 2, type := "perm" ) gap> ResultOfStraightLineDecision( check.program, gens.generators ); true
‣ AsStraightLineDecision( bbox ) | ( attribute ) |
Returns: an equivalent straight line decision for the given black box program, or fail.
For a black box program (see IsBBoxProgram (6.2-1)) bbox, AsStraightLineDecision returns a straight line decision (see IsStraightLineDecision (6.1-1)) with the same output as bbox, in the sense of AsBBoxProgram (6.2-5), if such a straight line decision exists, and fail otherwise.
gap> lines:= [ [ "Order", 1, 2 ], [ "Order", 2, 3 ], > [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 3, 5 ] ];; gap> dec:= StraightLineDecision( lines, 2 ); <straight line decision> gap> bboxdec:= AsBBoxProgram( dec ); <black box program> gap> asdec:= AsStraightLineDecision( bboxdec ); <straight line decision> gap> LinesOfStraightLineDecision( asdec ); [ [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 3, 5 ] ]
‣ StraightLineProgramFromStraightLineDecision( dec ) | ( operation ) |
Returns: the straight line program associated to the given straight line decision.
For a straight line decision dec (see IsStraightLineDecision (6.1-1), StraightLineProgramFromStraightLineDecision returns the straight line program (see IsStraightLineProgram (Reference: IsStraightLineProgram) obtained by replacing each line of type 3. (i.e, each order check) by an assignment of the power in question to a new slot, and by declaring the list of these elements as the return value.
This means that the return value describes exactly the defining relators of the presentation that is associated to the straight line decision, see 6.1-7.
For example, one can use the return value for printing the relators with StringOfResultOfStraightLineProgram (Reference: StringOfResultOfStraightLineProgram), or for explicitly constructing the relators as words in terms of free generators, by applying ResultOfStraightLineProgram (Reference: ResultOfStraightLineProgram) to the program and to these generators.
gap> dec:= StraightLineDecision( [ [ [ 1, 1, 2, 1 ], 3 ], > [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] ); <straight line decision> gap> prog:= StraightLineProgramFromStraightLineDecision( dec ); <straight line program> gap> Display( prog ); # input: r:= [ g1, g2 ]; # program: r[3]:= r[1]*r[2]; r[4]:= r[1]^2; r[5]:= r[2]^3; r[6]:= r[3]^5; # return values: [ r[4], r[5], r[6] ] gap> StringOfResultOfStraightLineProgram( prog, [ "a", "b" ] ); "[ a^2, b^3, (ab)^5 ]" gap> gens:= GeneratorsOfGroup( FreeGroup( "a", "b" ) ); [ a, b ] gap> ResultOfStraightLineProgram( prog, gens ); [ a^2, b^3, (a*b)^5 ]
Black box programs formalize the idea that one takes some group elements, forms arithmetic expressions in terms of them, tests properties of these expressions, executes conditional statements (including jumps inside the program) depending on the results of these tests, and eventually returns some result.
A specification of the language can be found in [Nic06], see also
http://atlas.math.rwth-aachen.de/Atlas/info/blackbox.html.
The inputs of a black box program may be explicit group elements, and the program may also ask for random elements from a given group. The program steps form products, inverses, conjugates, commutators, etc. of known elements, tests concern essentially the orders of elements, and the result is a list of group elements or true or false or fail.
Examples that can be modeled by black box programs are
which require a fixed number of input elements and form arithmetic expressions of elements but do not use random elements, tests, conditional statements and jumps; the return value is always a list of elements; these programs are described in Section Reference: Straight Line Programs.
which differ from straight line programs only in the sense that also order tests are admissible, and that the return value is true if all these tests are satisfied, and false as soon as the first such test fails; they are described in Section 6.1.
which take a group and a function to generate a random element in this group but no explicit input elements, admit all control structures, and return either a list of standard generators or fail; see ResultOfBBoxProgram (6.2-4) for examples.
In the case of general black box programs, currently GAP provides only the possibility to read an existing program via ScanBBoxProgram (6.2-2), and to run the program using RunBBoxProgram (6.2-3). It is not our aim to write such programs in GAP.
The special case of the "find" scripts mentioned above is also admissible as an argument of ResultOfBBoxProgram (6.2-4), which returns either the set of found generators or fail.
Contrary to the general situation, more support is provided for straight line programs and straight line decisions in GAP, see Section Reference: Straight Line Programs for functions that manipulate them (compose, restrict etc.).
The functions AsStraightLineProgram (6.2-6) and AsStraightLineDecision (6.1-8) can be used to transform a general black box program object into a straight line program or a straight line decision if this is possible.
Conversely, one can create an equivalent general black box program from a straight line program or from a straight line decision with AsBBoxProgram (6.2-5).
Computing a straight line program related to a given straight line decision is supported in the sense of StraightLineProgramFromStraightLineDecision (6.1-9).
Note that none of these three kinds of objects is a special case of another: Running a black box program with RunBBoxProgram (6.2-3) yields a record, running a straight line program with ResultOfStraightLineProgram (Reference: ResultOfStraightLineProgram) yields a list of elements, and running a straight line decision with ResultOfStraightLineDecision (6.1-6) yields true or false.
‣ IsBBoxProgram( obj ) | ( category ) |
Each black box program in GAP lies in the filter IsBBoxProgram.
‣ ScanBBoxProgram( string ) | ( function ) |
Returns: a record containing the black box program encoded by the input string, or fail.
For a string string that describes a black box program, e.g., the return value of StringFile (GAPDoc: StringFile), ScanBBoxProgram computes this black box program. If this is successful then the return value is a record containing as the value of its component program the corresponding GAP object that represents the program, otherwise fail is returned.
As the first example, we construct a black box program that tries to find standard generators for the alternating group A_5; these standard generators are any pair of elements of the orders 2 and 3, respectively, such that their product has order 5.
gap> findstr:= "\ > set V 0\n\ > lbl START1\n\ > rand 1\n\ > ord 1 A\n\ > incr V\n\ > if V gt 100 then timeout\n\ > if A notin 1 2 3 5 then fail\n\ > if A noteq 2 then jmp START1\n\ > lbl START2\n\ > rand 2\n\ > ord 2 B\n\ > incr V\n\ > if V gt 100 then timeout\n\ > if B notin 1 2 3 5 then fail\n\ > if B noteq 3 then jmp START2\n\ > # The elements 1 and 2 have the orders 2 and 3, respectively.\n\ > set X 0\n\ > lbl CONJ\n\ > incr X\n\ > if X gt 100 then timeout\n\ > rand 3\n\ > cjr 2 3\n\ > mu 1 2 4 # ab\n\ > ord 4 C\n\ > if C notin 2 3 5 then fail\n\ > if C noteq 5 then jmp CONJ\n\ > oup 2 1 2";; gap> find:= ScanBBoxProgram( findstr ); rec( program := <black box program> )
The second example is a black box program that checks whether its two inputs are standard generators for A_5.
gap> checkstr:= "\ > chor 1 2\n\ > chor 2 3\n\ > mu 1 2 3\n\ > chor 3 5";; gap> check:= ScanBBoxProgram( checkstr ); rec( program := <black box program> )
‣ RunBBoxProgram( prog, G, input, options ) | ( function ) |
Returns: a record describing the result and the statistics of running the black box program prog, or fail, or the string "timeout".
For a black box program prog, a group G, a list input of group elements, and a record options, RunBBoxProgram applies prog to input, where G is used only to compute random elements.
The return value is fail if a syntax error or an explicit fail statement is reached at runtime, and the string "timeout" if a timeout statement is reached. (The latter might mean that the random choices were unlucky.) Otherwise a record with the following components is returned.
gensa list of group elements, bound if an oup statement was reached,
resulttrue if a true statement was reached, false if either a false statement or a failed order check was reached,
The other components serve as statistical information about the numbers of the various operations (multiply, invert, power, order, random, conjugate, conjugateinplace, commutator), and the runtime in milliseconds (timetaken).
The following components of options are supported.
randomfunctionthe function called with argument G in order to compute a random element of G (default PseudoRandom (Reference: PseudoRandom))
orderfunctionthe function for computing element orders (default Order (Reference: Order)),
quietif true then ignore echo statements (default false),
verboseif true then print information about the line that is currently processed, and about order checks (default false),
allowbreaksif true then call Error (Reference: Error) when a break statement is reached, otherwise ignore break statements (default true).
As an example, we run the black box programs constructed in the example for ScanBBoxProgram (6.2-2).
gap> g:= AlternatingGroup( 5 );; gap> res:= RunBBoxProgram( find.program, g, [], rec() );; gap> IsBound( res.gens ); IsBound( res.result ); true false gap> List( res.gens, Order ); [ 2, 3 ] gap> Order( Product( res.gens ) ); 5 gap> res:= RunBBoxProgram( check.program, "dummy", res.gens, rec() );; gap> IsBound( res.gens ); IsBound( res.result ); false true gap> res.result; true gap> othergens:= GeneratorsOfGroup( g );; gap> res:= RunBBoxProgram( check.program, "dummy", othergens, rec() );; gap> res.result; false
‣ ResultOfBBoxProgram( prog, G[, options] ) | ( function ) |
Returns: a list of group elements or true, false, fail, or the string "timeout".
This function calls RunBBoxProgram (6.2-3) with the black box program prog and second argument either a group or a list of group elements; if options is not given then the default options of RunBBoxProgram (6.2-3) are assumed. The return value is fail if this call yields fail, otherwise the gens component of the result, if bound, or the result component if not.
Note that a group G is used as the second argument in the call of RunBBoxProgram (6.2-3) (the source for random elements), whereas a list G is used as the third argument (the inputs).
As an example, we run the black box programs constructed in the example for ScanBBoxProgram (6.2-2).
gap> g:= AlternatingGroup( 5 );; gap> res:= ResultOfBBoxProgram( find.program, g );; gap> List( res, Order ); [ 2, 3 ] gap> Order( Product( res ) ); 5 gap> res:= ResultOfBBoxProgram( check.program, res ); true gap> othergens:= GeneratorsOfGroup( g );; gap> res:= ResultOfBBoxProgram( check.program, othergens ); false
‣ AsBBoxProgram( slp ) | ( attribute ) |
Returns: an equivalent black box program for the given straight line program or straight line decision.
Let slp be a straight line program (see IsStraightLineProgram (Reference: IsStraightLineProgram)) or a straight line decision (see IsStraightLineDecision (6.1-1)). Then AsBBoxProgram returns a black box program bbox (see IsBBoxProgram (6.2-1)) with the "same" output as slp, in the sense that ResultOfBBoxProgram (6.2-4) yields the same result for bbox as ResultOfStraightLineProgram (Reference: ResultOfStraightLineProgram) or ResultOfStraightLineDecision (6.1-6), respectively, for slp.
gap> f:= FreeGroup( "x", "y" );; gens:= GeneratorsOfGroup( f );; gap> slp:= StraightLineProgram( [ [1,2,2,3], [3,-1] ], 2 ); <straight line program> gap> ResultOfStraightLineProgram( slp, gens ); y^-3*x^-2 gap> bboxslp:= AsBBoxProgram( slp ); <black box program> gap> ResultOfBBoxProgram( bboxslp, gens ); [ y^-3*x^-2 ] gap> lines:= [ [ "Order", 1, 2 ], [ "Order", 2, 3 ], > [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 3, 5 ] ];; gap> dec:= StraightLineDecision( lines, 2 ); <straight line decision> gap> ResultOfStraightLineDecision( dec, [ (1,2)(3,4), (1,3,5) ] ); true gap> ResultOfStraightLineDecision( dec, [ (1,2)(3,4), (1,3,4) ] ); false gap> bboxdec:= AsBBoxProgram( dec ); <black box program> gap> ResultOfBBoxProgram( bboxdec, [ (1,2)(3,4), (1,3,5) ] ); true gap> ResultOfBBoxProgram( bboxdec, [ (1,2)(3,4), (1,3,4) ] ); false
‣ AsStraightLineProgram( bbox ) | ( attribute ) |
Returns: an equivalent straight line program for the given black box program, or fail.
For a black box program (see AsBBoxProgram (6.2-5)) bbox, AsStraightLineProgram returns a straight line program (see IsStraightLineProgram (Reference: IsStraightLineProgram)) with the same output as bbox if such a straight line program exists, and fail otherwise.
gap> Display( AsStraightLineProgram( bboxslp ) ); # input: r:= [ g1, g2 ]; # program: r[3]:= r[1]^2; r[4]:= r[2]^3; r[5]:= r[3]*r[4]; r[3]:= r[5]^-1; # return values: [ r[3] ] gap> AsStraightLineProgram( bboxdec ); fail
This section deals with minimal degrees of permutation and matrix representations. We do not provide an algorithm that computes these degrees for an arbitrary group, we only provide some tools for evaluating known databases, mainly concerning "bicyclic extensions" (see [CCN+85, Section 6.5]) of simple groups, in order to derive the minimal degrees, see Section 6.3-4.
In the AtlasRep package, this information can be used for prescribing "minimality conditions" in DisplayAtlasInfo (3.5-1), OneAtlasGeneratingSetInfo (3.5-6), and AllAtlasGeneratingSetInfos (3.5-7). An overview of the stored minimal degrees can be shown with BrowseMinimalDegrees (3.6-1).
‣ MinimalRepresentationInfo( grpname, conditions ) | ( function ) |
Returns: a record with the components value and source, or fail
Let grpname be the GAP name of a group G, say. If the information described by conditions about minimal representations of this group can be computed or is stored then MinimalRepresentationInfo returns a record with the components value and source, otherwise fail is returned.
The following values for conditions are supported.
If conditions is NrMovedPoints (Reference: NrMovedPoints for a permutation) then value, if known, is the degree of a minimal faithful (not necessarily transitive) permutation representation for G.
If conditions consists of Characteristic (Reference: Characteristic) and a prime integer p then value, if known, is the dimension of a minimal faithful (not necessarily irreducible) matrix representation in characteristic p for G.
If conditions consists of Size (Reference: Size) and a prime power q then value, if known, is the dimension of a minimal faithful (not necessarily irreducible) matrix representation over the field of size q for G.
In all cases, the value of the component source is a list of strings that describe sources of the information, which can be the ordinary or modular character table of G (see [CCN+85], [JLPW95], [HL89]), the table of marks of G, or [Jan05]. For an overview of minimal degrees of faithful matrix representations for sporadic simple groups and their covering groups, see also
http://www.math.rwth-aachen.de/~MOC/mindeg/.
Note that MinimalRepresentationInfo cannot provide any information about minimal representations over prescribed fields in characteristic zero.
Information about groups that occur in the AtlasRep package is precomputed in MinimalRepresentationInfoData (6.3-2), so the packages CTblLib and TomLib are not needed when MinimalRepresentationInfo is called for these groups. (The only case that is not covered by this list is that one asks for the minimal degree of matrix representations over a prescribed field in characteristic coprime to the group order.)
One of the following strings can be given as an additional last argument.
"cache"means that the function tries to compute (and then store) values that are not stored in MinimalRepresentationInfoData (6.3-2), but stored values are preferred; this is also the default.
"lookup"means that stored values are returned but the function does not attempt to compute values that are not stored in MinimalRepresentationInfoData (6.3-2).
"recompute"means that the function always tries to compute the desired value, and checks the result against stored values.
gap> MinimalRepresentationInfo( "A5", NrMovedPoints ); rec( source := [ "computed (alternating group)", "computed (char. table)", "computed (subgroup tables)", "computed (subgroup tables, known repres.)", "computed (table of marks)" ], value := 5 ) gap> MinimalRepresentationInfo( "A5", Characteristic, 2 ); rec( source := [ "computed (char. table)" ], value := 2 ) gap> MinimalRepresentationInfo( "A5", Size, 2 ); rec( source := [ "computed (char. table)" ], value := 4 )
‣ MinimalRepresentationInfoData | ( global variable ) |
This is a record whose components are GAP names of groups for which information about minimal permutation and matrix representations were known in advance or have been computed in the current GAP session. The value for the group G, say, is a record with the following components.
NrMovedPointsa record with the components value (the degree of a smallest faithful permutation representation of G) and source (a string describing the source of this information).
Characteristica record whose components are at most 0 and strings corresponding to prime integers, each bound to a record with the components value (the degree of a smallest faithful matrix representation of G in this characteristic) and source (a string describing the source of this information).
CharacteristicAndSizea record whose components are strings corresponding to prime integers p, each bound to a record with the components sizes (a list of powers q of p), dimensions (the corresponding list of minimal dimensions of faithful matrix representations of G over a field of size q), sources (the corresponding list of strings describing the source of this information), and complete (a record with the components val (true if the minimal dimension over any finite field in characteristic p can be derived from the values in the record, and false otherwise) and source (a string describing the source of this information)).
The values are set by SetMinimalRepresentationInfo (6.3-3).
‣ SetMinimalRepresentationInfo( grpname, op, value, source ) | ( function ) |
Returns: true if the values were successfully set, false if stored values contradict the given ones.
This function sets an entry in MinimalRepresentationInfoData (6.3-2) for the group G, say, with GAP name grpname.
Supported values for op are
"NrMovedPoints" (see NrMovedPoints (Reference: NrMovedPoints for a permutation)), which means that value is the degree of minimal faithful (not necessarily transitive) permutation representations of G,
a list of length two with first entry "Characteristic" (see Characteristic (Reference: Characteristic)) and second entry char either zero or a prime integer, which means that value is the dimension of minimal faithful (not necessarily irreducible) matrix representations of G in characteristic char,
a list of length two with first entry "Size" (see Size (Reference: Size)) and second entry a prime power q, which means that value is the dimension of minimal faithful (not necessarily irreducible) matrix representations of G over the field with q elements, and
a list of length three with first entry "Characteristic" (see Characteristic (Reference: Characteristic)), second entry a prime integer p, and third entry the string "complete", which means that the information stored for characteristic p is complete in the sense that for any given power q of p, the minimal faithful degree over the field with q elements equals that for the largest stored field size of which q is a power.
In each case, source is a string describing the source of the data; computed values are detected from the prefix "comp" of source.
If the intended value is already stored and differs from value then an error message is printed.
gap> SetMinimalRepresentationInfo( "A5", "NrMovedPoints", 5, > "computed (alternating group)" ); true gap> SetMinimalRepresentationInfo( "A5", [ "Characteristic", 0 ], 3, > "computed (char. table)" ); true gap> SetMinimalRepresentationInfo( "A5", [ "Characteristic", 2 ], 2, > "computed (char. table)" ); true gap> SetMinimalRepresentationInfo( "A5", [ "Size", 2 ], 4, > "computed (char. table)" ); true gap> SetMinimalRepresentationInfo( "A5", [ "Size", 4 ], 2, > "computed (char. table)" ); true gap> SetMinimalRepresentationInfo( "A5", [ "Characteristic", 3 ], 3, > "computed (char. table)" ); true
The information about the minimal degree of a faithful matrix representation of G in a given characteristic or over a given field in positive characteristic is derived from the relevant (ordinary or modular) character table of G, except in a few cases where this table itself is not known but enough information about the degrees is available in [HL89] and [Jan05].
The following criteria are used for deriving the minimal degree of a faithful permutation representation of G from the information in the GAP libraries of character tables and of tables of marks.
If the name of G has the form "An" or "An.2" (denoting alternating and symmetric groups, respectively) then the minimal degree is n, except if n is smaller than 3 or 2, respectively.
If the name of G has the form "L2(q)" (denoting projective special linear groups in dimension two) then the minimal degree is q + 1, except if q ∈ { 2, 3, 5, 7, 9, 11 }, see [Hup67, Satz II.8.28].
If the largest maximal subgroup of G is core-free then the index of this subgroup is the minimal degree. (This is used when the two character tables in question and the class fusion are available in GAP's Character Table Library [Bre22]; this happens for many character tables of simple groups.)
If G has a unique minimal normal subgroup then each minimal faithful permutation representation is transitive. (Note that the core of each point stabilizer is either trivial or contains the unique minimal normal subgroup.)
In this case, the minimal degree can be computed directly from the information in the table of marks of G if this is available in GAP's Library of Tables of Marks [NMP18].
Suppose that the largest maximal subgroup of G is not core-free but simple and normal in G, and that the other maximal subgroups of G are core-free. In this case, we take the minimum of the indices of the core-free maximal subgroups and of the product of index and minimal degree of the normal maximal subgroup. (This suffices since no core-free subgroup of the whole group can contain a nontrivial normal subgroup of a normal maximal subgroup.)
Let N be the unique minimal normal subgroup of G, and assume that G/N is simple and has minimal degree n, say. If there is a subgroup U of index n ⋅ |N| in G that intersects N trivially then the minimal degree of G is n ⋅ |N|. (This is used for the case that N is central in G and N × U occurs as a subgroup of G.)
If we know a subgroup of G whose minimal degree is n, say, and if we know either (a class fusion from) a core-free subgroup of index n in G or a faithful permutation representation of degree n for G then n is the minimal degree for G. (This happens often for tables of almost simple groups.)
We define a mapping between certain GAP objects and JSON (JavaScript Object Notation) texts (see [JSO14]), as follows.
The three GAP values true, false, and fail correspond to the JSON texts true, false, and null, respectively.
GAP strings correspond to JSON strings; special characters in a GAP string (control characters ASCII 0 to 31, backslash and double quote) are mapped as defined in JSON's specification, and other ASCII characters are kept as they are; if a GAP string contains non-ASCII characters, it is assumed that it is UTF-8 encoded, and one may choose either to keep non-ASCII characters as they are, or to create an ASCII only JSON string, using JSON's syntax for Unicode code points ("\uXXXX"); in the other direction, JSON strings are assumed to be UTF-8 encoded, and are mapped to UTF-8 encoded GAP strings, by keeping the non-ASCII characters and converting substrings of the form \uXXXX accordingly.
GAP integers (in the sense of IsInt (Reference: IsInt)) are mapped to JSON numbers that consist of digits and optionally a leading sign character -; in the other direction, JSON numbers of this form and also JSON numbers that involve no decimal dots and have no negative exponent (for example "2e3") are mapped to GAP integers.
GAP rationals (in the sense of IsRat (Reference: IsRat)) which are not integers are represented by JSON floating point numbers; the JSON representation (and hence the precision) is given by first applying Float (Reference: Float) and then String (Reference: String).
GAP floats (in the sense of Chapter Reference: Floats in the GAP Reference Manual) are mapped to JSON floating point numbers; the JSON representation (and hence the precision) is given by applying String (Reference: String); in the other direction, JSON numbers that involve a decimal dot or a negative exponent are mapped to GAP floats.
(Nested and not self-referential) dense GAP lists of objects correspond to JSON arrays such that the list entries correspond to each other. (Note that JSON does not support non-dense arrays.)
(Nested and not self-referential) GAP records correspond to JSON objects such that both labels (which are strings in GAP and JSON) and values correspond to each other.
The GAP functions AGR.JsonText (6.4-2) and AGR.GapObjectOfJsonText (6.4-3) can be used to create a JSON text from a suitable GAP object and the GAP object that corresponds to a given JSON text, respectively.
Note that the composition of the two functions is in general not the identity mapping, because AGR.JsonText (6.4-2) accepts non-integer rationals, whereas AGR.GapObjectOfJsonText (6.4-3) does not create such objects.
Note also that the results of AGR.JsonText (6.4-2) do not contain information about dependencies between common subobjects. This is another reason why applying first AGR.JsonText (6.4-2) and then AGR.GapObjectOfJsonText (6.4-3) may yield a GAP object with different behaviour.
Applying AGR.JsonText (6.4-2) to a self-referential object such as [ ~ ] will raise a "recursion depth trap" error.
The aim of this JSON interface is to read and write certain data files with GAP such that these files become easily accessible independent of GAP. The function AGR.JsonText (6.4-2) is intended just as a prototype, variants of this function are very likely to appear in other contexts, for example in order to force certain line formatting or ordering of record components.
It is not the aim of the JSON interface to provide self-contained descriptions of arbitrary GAP objects, in order to read them into a GAP session. Note that those GAP objects for which a JSON equivalent exists (and many more) can be easily written to files as they are, and GAP can read them efficiently. On the other hand, more complicated GAP objects can be written and read via the so-called pickling, for which a framework is provided by the GAP package IO [Neu14].
Here are a few situations which are handled well by pickling but which cannot be addressed with a JSON interface.
Pickling and unpickling take care of common subobjects of the given GAP object. The following example shows that the applying first AGR.JsonText (6.4-2) and then AGR.GapObjectOfJsonText (6.4-3) may yield an object which behaves differently.
gap> l:= [ [ 1 ] ];; l[2]:= l[1];; l; [ [ 1 ], [ 1 ] ] gap> new:= AGR.GapObjectOfJsonText( AGR.JsonText( l ) ).value; [ [ 1 ], [ 1 ] ] gap> Add( l[1], 2 ); l; [ [ 1, 2 ], [ 1, 2 ] ] gap> Add( new[1], 2 ); new; [ [ 1, 2 ], [ 1 ] ]
GAP admits self-referential objects, for example as follows.
gap> l:= [];; l[1]:= l;;
Pickling and unpickling take care of self-referential objects, but AGR.JsonText (6.4-2) does not support the conversion of such objects.
‣ AGR.JsonText( obj[, mode] ) | ( function ) |
Returns: a new mutable string that describes obj as a JSON text, or fail.
If obj is a GAP object for which a corresponding JSON text exists, according to the mapping described above, then such a JSON text is returned. Otherwise, fail is returned.
If the optional argument mode is given and has the value "ASCII" then the result in an ASCII string, otherwise the encoding of strings that are involved in obj is kept.
gap> AGR.JsonText( [] ); "[]" gap> AGR.JsonText( "" ); "\"\"" gap> AGR.JsonText( "abc\ndef\cghi" ); "\"abc\\ndef\\u0003ghi\"" gap> AGR.JsonText( rec() ); "{}" gap> AGR.JsonText( [ , 2 ] ); fail gap> str:= [ '\303', '\266' ];; # umlaut o gap> json:= AGR.JsonText( str );; List( json, IntChar ); [ 34, 195, 182, 34 ] gap> AGR.JsonText( str, "ASCII" ); "\"\\u00F6\""
‣ AGR.GapObjectOfJsonText( string ) | ( function ) |
Returns: a new mutable record whose value component, if bound, contains a mutable GAP object that represents the JSON text string.
If string is a string that represents a JSON text then the result is a record with the components value (the corresponding GAP object in the sense of the above interface) and status (value true). Otherwise, the result is a record with the components status (value false) and errpos (the position in string where the string turns out to be not valid JSON).
gap> AGR.GapObjectOfJsonText( "{ \"a\": 1 }" ); rec( status := true, value := rec( a := 1 ) ) gap> AGR.GapObjectOfJsonText( "{ \"a\": x }" ); rec( errpos := 8, status := false )
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