<html><head><title>[IRREDSOL] 2 Accessing the data library</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP001.htm">Previous</a>] [<a href ="CHAP003.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <h1>2 Accessing the data library</h1><p> <P> <H3>Sections</H3> <oL> <li> <A HREF="CHAP002.htm#SECT001">Design of the group library</a> <li> <A HREF="CHAP002.htm#SECT002">Low level access functions</a> <li> <A HREF="CHAP002.htm#SECT003">Finding matrix groups with given properties</a> <li> <A HREF="CHAP002.htm#SECT004">Loading and unloading group data manually</a> </ol><p> <p> This chapter describes the design of the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> group library (see Section <a href="CHAP002.htm#SECT001">Design of the group library</a>) and the various ways of accessing groups in the data library. It is possible to construct individual groups in the group library (see Section <a href="CHAP002.htm#SECT002">Low level access functions</a>), or to search for groups with certain properties (see Section <a href="CHAP002.htm#SECT003">Finding matrix groups with given properties</a>). Finally, there are functions for loading and unloading group data manually (see Section <a href="CHAP002.htm#SECT004">Loading and unloading group data manually</a>). <p> <p> <h2><a name="SECT001">2.1 Design of the group library</a></h2> <p><p> To avoid redundancy, the package <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> does not actually store lists of irreducible subgroups of <var>GL(<var>n</var>, <var>q</var>)</var> but only has lists <var>cal A<sub>n,q</sub></var> of subgroups of <var>GL(<var>n</var>, <var>q</var>)</var> such that <p> <dl compact> <dt><var>bullet</var><dd>each group in <var>calA<sub>n,q</sub></var> is absolutely irreducible and solvable <p> <dt><var>bullet</var><dd><var>calA<sub>n,q</sub></var> contains a conjugate of each absolutely irreducible solvable subgroup of <var>GL(n, q)</var> <p> <dt><var>bullet</var><dd>no two groups in <var>calA<sub>n,q</sub></var> are conjugate in <var>GL(n, q)</var> <p> <dt><var>bullet</var><dd>each group in <var>calA<sub>n,q</sub></var> has trace field <var><font face="helvetica,arial">F</font><sub>q</sub></var>. <p> </dl> (For <var>n = 1</var>, such lists are not actually stored but are computed when required.) <p> We will briefly say that <var>calA<sub>n,q</sub></var> contains, up to conjugacy, all absolutely irreducible solvable subgroups of <var>GL(n,q)</var> with trace field <var><font face="helvetica,arial">F</font><sub>q</sub></var>. Here, the <em>trace field</em>of a subgroup <var>G</var> of <var>GL(n, q)</var> is the field generated by the traces of the elements of <var>G</var>. By a theorem of Brauer, an irreducible subgroup of <var>GL(n, q)</var> with trace field <var><font face="helvetica,arial">F</font><sub>q_0</sub></var> has a conjugate lying in <var>GL(n, q<sub>0</sub>)</var>. See also <code>TraceField</code> (<a href="CHAP004.htm#SSEC004.1">TraceField</a>) and <code>ConjugatingMatTraceField</code> (<a href="CHAP004.htm#SSEC004.2">ConjugatingMatTraceField</a>). <p> Note that by the Deuring-Noether theorem, two subgroups of <var>GL(n, q<sub>0</sub>)</var> are conjugate in <var>GL(n, q<sub>0</sub>)</var> if, and only if, they are conjugate in <var>GL(n, q)</var>. Therefore, we obtain, up to conjugacy, all absolutely irreducible solvable subgroups of <var>GL(n, q)</var> by forming the union of the <var>calA<sub>n, q_0</sub></var>, where <var><font face="helvetica,arial">F</font><sub>q_0</sub></var> runs over all subfields of <var><font face="helvetica,arial">F</font><sub>q</sub></var>. <p> The lists <var>calA<sub>n, q</sub></var> are also sufficient to reconstruct lists of all irreducible solvable subgroups of <var>GL(n, q)</var>. Let <var>d</var> be a divisor of <var>n</var>, and let <var>G</var> be an absolutely irreducible subgroup of <var>GL(n/d, q<sup>d</sup>)</var>. By regarding the underlying <var><font face="helvetica,arial">F</font><sub>q^d</sub></var>-vector space of <var>G</var> as an <var><font face="helvetica,arial">F</font><sub>q</sub></var>-vector space, we obtain an irreducible subgroup <var>G^*</var> of <var>GL(n, q)</var> with splitting field <var><font face="helvetica,arial">F</font><sub>q^d</sub></var>. Up to conjugacy, all irreducible subgroups of <var>GL(n, q)</var> arise in that way. If two subgroups <var>G<sub>1</sub></var> and <var>G<sub>2</sub></var> of <var>GL(n, q)</var> are constructed from subgroups <var>G<sub>1</sub>^*</var> and <var>G<sub>2</sub>^*</var>of <var>GL(n/d, q<sup>d</sup>)</var>, then <var>G<sub>1</sub></var> and <var>G<sub>2</sub></var> are conjugate if, and only if, there exists a Galois automophism <var>sigma</var> of <var><font face="helvetica,arial">F</font><sub>q^d</sub>/<font face="helvetica,arial">F</font><sub>q</sub></var> such that <var>(G<sub>1</sub>^*)^sigma</var> and <var>G<sub>2</sub>^*</var> are conjugate in <var>GL(n/d, q<sup>d</sup>)</var>. See, e. g., <a href="biblio.htm#DH"><cite>DH</cite></a>, Theorem B 5.15. The latter information has been precomputed and is also part of the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> library. <p> Note that all of the arguments above apply to nonsolvable groups as well. <p> <p> <h2><a name="SECT002">2.2 Low level access functions</a></h2> <p><p> The access functions described in this section allow to check for the availability of data and to construct irreducible groups in the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> group library. <p> <a name = "SSEC002.1"></a> <li><code>IsAvailableIrreducibleSolvableGroupData(</code><var>n</var><code>, </code><var>q</var><code>) F</code> <p> This function tests whether the irreducible solvable subgroups of <var>GL(n,q)</var> with trace field <var><font face="helvetica,arial">F</font><sub>q</sub></var> are part of the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> library. <p> <a name = "SSEC002.2"></a> <li><code>IndicesIrreducibleSolvableMatrixGroups(</code><var>n</var><code>, </code><var>q</var><code>, </code><var>d</var><code>) F</code> <p> Let <var>n</var> and <var>d</var> be positive integers and <var>q</var> a prime power. This function returns a set of integers parametrising the groups in the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> library which are subgroups of <var>GL(n,q)</var> with trace field <var><font face="helvetica,arial">F</font><sub>q</sub></var> and splitting field <var><font face="helvetica,arial">F</font><sub>q^d</sub></var>. This set is empty unless <var>d</var> divides <var>n</var>. An error is raised if the relevant data is not available, cf. <a href="CHAP002.htm#SSEC002.1">IsAvailableIrreducibleSolvableGroupData</a>. <p> <pre> gap> IndicesIrreducibleSolvableMatrixGroups (6, 2, 2); [ 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12 ] </pre> <p> <a name = "SSEC002.3"></a> <li><code>IrreducibleSolvableMatrixGroup(</code><var>n</var><code>, </code><var>q</var><code>, </code><var>d</var><code>, </code><var>k</var><code>) F</code> <p> Let <var>n</var>, <var>d</var> and <var>k</var> be positive integers and <var>q</var> a prime power. This function returns the irreducible solvable subgroup of <var>GL(n,q)</var> with trace field <var><font face="helvetica,arial">F</font><sub>q</sub></var>, splitting field <var><font face="helvetica,arial">F</font><sub>q^d</sub></var>, and index <var>k</var>. An error is raised if the relevant data is not available, or if <var>k</var> is not in <code>IndicesIrreducibleSolvableMatrixGroups</code>(<var>n</var>, <var>q</var>, <var>d</var>); cf. <a href="CHAP002.htm#SSEC002.1">IsAvailableIrreducibleSolvableGroupData</a> and <a href="CHAP002.htm#SSEC002.2">IndicesIrreducibleSolvableMatrixGroups</a>. The groups returned have the attributes and properties described in Chapter <a href="CHAP004.htm">Additional functionality for matrix groups</a> set to their appropriate values. <p> <a name = "SSEC002.4"></a> <li><code>IsAvailableAbsolutelyIrreducibleSolvableGroupData(</code><var>n</var><code>, </code><var>q</var><code>) F</code> <p> This function tests whether the absolutely irreducible solvable subgroups of <var>GL(n,q)</var> with trace field <var><font face="helvetica,arial">F</font><sub>q</sub></var> are in the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> library. <p> <a name = "SSEC002.5"></a> <li><code>IndicesMaximalAbsolutelyIrreducibleSolvableMatrixGroups(</code><var>n</var><code>, </code><var>q</var><code>) F</code> <p> Let <var>n</var> be a positive integer and <var>q</var> a prime power. This function returns a set of integers parametrising those subgroups of <var>GL(n, q)</var> in the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> library that are maximal among the absolutely irreducible solvable subgroups with trace field <var><font face="helvetica,arial">F</font><sub>q</sub> </var> and that are maximal with respect to being solvable. An error is raised if the relevant data is not available (see <a href="CHAP002.htm#SSEC002.4">IsAvailableAbsolutelyIrreducibleSolvableGroupData</a> for information how to check this first). An integer <var>k</var> in the list return corresponds to <code>IrreducibleSolvableMatrixGroup(</code><var>n</var><code>, </code><var>q</var><code>, 1, </code><var>k</var><code>)</code>, which is the same group as <code>AbsolutelyIrreducibleSolvableMatrixGroup(</code><var>n</var><code>, </code><var>q</var><code>, </code><var>k</var><code>)</code> <p> <pre> gap> inds := IndicesMaximalAbsolutelyIrreducibleSolvableMatrixGroups (2,3); [ 2 ] gap> max := IrreducibleSolvableMatrixGroup (2,3,1,2); Group([ [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0 ] ], [ [ Z(3)^0, Z(3) ], [ 0*Z(3), Z(3)^0 ] ], [ [ Z(3), Z(3)^0 ], [ Z(3)^0, Z(3)^0 ] ], [ [ 0*Z(3), Z(3)^0 ], [ Z(3), 0*Z(3) ] ], [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ] ]) gap> max = GL(2,3); # it is the whole GL true </pre> <p> <a name = "SSEC002.6"></a> <li><code>IndicesAbsolutelyIrreducibleSolvableMatrixGroups(</code><var>n</var><code>, </code><var>q</var><code>) F</code> <a name = "SSEC002.6"></a> <li><code>AbsolutelyIrreducibleSolvableMatrixGroup(</code><var>n</var><code>, </code><var>q</var><code>, </code><var>k</var><code>) F</code> <p> These functions are not available any longer. Please use <code>IndicesIrreducibleSolvableMatrixGroups(</code><var>n</var><code>, </code><var>q</var><code>, 1)</code> and <code>IrreducibleSolvableMatrixGroup(</code><var>n</var><code>, </code><var>q</var><code>, 1, </code><var>k</var><code>)</code> instead. <p> <p> <h2><a name="SECT003">2.3 Finding matrix groups with given properties</a></h2> <p><p> This section describes three functions (<code>AllIrreducibleSolvableMatrixGroups</code>, <code>OneIrreducibleSolvableMatrixGroup</code>, <code>IteratorIrreducibleSolvableMatrixGroups</code>) which allow to find matrix groups with prescribed properties. Using these functions can be more efficient than to construct each group in the library using the functions in Section <a href="CHAP002.htm#SECT002">Low level access functions</a> because they can access additional information about a group in the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> library before actually constructing the group. See the discussion following the description of <code>AllIrreducibleSolvableMatrixGroups</code> for details. <p> <a name = "SSEC003.1"></a> <li><code>AllIrreducibleSolvableMatrixGroups(</code><var>func_1</var><code>, </code><var>arg_1</var><code>, </code><var>func_2</var><code>, </code><var>arg_2</var><code>, ...) F</code> <p> This function returns a list of all irreducible solvable matrix groups <var>G</var> in the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> library for which the return value of <var><var>func<sub>i</sub></var>(G)</var> lies in <var>arg_i</var>. The arguments <var>func_1</var>, <var>func_2</var>, ..., must be <font face="Gill Sans,Helvetica,Arial">GAP</font> functions which take a matrix group as their only argument and return a value, and <var>arg_1</var>, <var>arg_2</var>, ..., must be lists. If <var>arg_i</var> is not a list, <var>arg_i</var> is replaced by the list <code>[</code><var>arg_i</var><code>]</code>. The functions <code>DegreeOfMatrixGroup</code> and <code>FieldOfMatrixGroup</code> (or their equivalents, see below) must be among the <var>func_i</var>. <code>TraceField</code>). Note that all groups in the data library have the property that <var><code>TraceField(</code><var>G</var><code>)</code> = <code>FieldOfMatrixGroup</code>(<var>G</var>)</var>; see Section <a href="CHAP002.htm#SECT001">Design of the group library</a> for details. <p> Note that there is also a function <code>IteratorIrreducibleSolvableMatrixGroups</code> (see <a href="CHAP002.htm#SSEC003.3">IteratorIrreducibleSolvableMatrixGroups</a>) which allows to run through the list produced by <code>AllIrreducibleSolvableMatrixGroups</code> without having to store all of the groups simultaneously. <p> The following functions <var>func_i</var> are handled particularly efficiently, because the return values of these functions can be read off the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> library without actually constructing the relevant matrix group. For the definitions of these functions, see Chapter <a href="CHAP004.htm">Additional functionality for matrix groups</a>. <p> <dl compact> <p> <dt><var>bullet</var><dd><code>DegreeOfMatrixGroup</code> (or <code>Degree</code>, <code>Dimension</code>, <code>DimensionOfMatrixGroup</code>), <dt><var>bullet</var><dd><code>CharacteristicOfField</code> (or <code>Characteristic</code>) <dt><var>bullet</var><dd><code>FieldOfMatrixGroup</code> (or <code>Field</code> or <code>TraceField</code>) <dt><var>bullet</var><dd><code>Order</code> (or <code>Size</code>) <dt><var>bullet</var><dd><code>IsMaximalAbsolutelyIrreducibleSolvableMatrixGroup</code> <dt><var>bullet</var><dd><code>IsAbsolutelyIrreducibleMatrixGroup</code> (or <code>IsAbsolutelyIrreducible</code>) <dt><var>bullet</var><dd><code>MinimalBlockDimensionOfMatrixGroup</code> (or <code>MinimalBlockDimension</code>) <dt><var>bullet</var><dd><code>IsPrimitiveMatrixGroup</code> (or <code>IsPrimitive</code>, <code>IsLinearlyPrimitive</code>) <p> </dl> The groups <var>G</var> passed to the <var>func_i</var> and the groups returned have the attributes and properties described in Chapter <a href="CHAP004.htm">Additional functionality for matrix groups</a> set to their appropriate values. Note that you may speed up computations in <var>G</var> by using an isomorphic copy of <var>G</var>, which can be obtained via <code>RepresentationIsomorphism</code> (see <a href="CHAP004.htm#SSEC001.6">RepresentationIsomorphism</a>). <p> <pre> # get just those groups with trace field GF(9) gap> l := AllIrreducibleSolvableMatrixGroups (Degree, 1, Field, GF(9));; gap> List (l, Order); [ 4, 8 ] # get all irreducible subgroups gap> l := AllIrreducibleSolvableMatrixGroups (Degree, 1, Field, Subfields (GF(9)));; gap> List (l, Order); [ 1, 2, 4, 8 ] # get only maximal absolutely irreducible ones gap> l := AllIrreducibleSolvableMatrixGroups (Degree, 4, Field, GF(3), > IsMaximalAbsolutelyIrreducibleSolvableMatrixGroup, true);; gap> SortedList (List (l, Order)); [ 320, 640, 2304, 4608 ] # get only absolutely irreducible groups gap> l := AllIrreducibleSolvableMatrixGroups (Degree, 4, Field, GF(3), > IsAbsolutelyIrreducibleMatrixGroup, true);; gap> Collected (List (l, Order)); [ [ 20, 1 ], [ 32, 7 ], [ 40, 2 ], [ 64, 10 ], [ 80, 2 ], [ 96, 6 ], [ 128, 9 ], [ 160, 3 ], [ 192, 9 ], [ 256, 6 ], [ 288, 1 ], [ 320, 2 ], [ 384, 4 ], [ 512, 1 ], [ 576, 3 ], [ 640, 1 ], [ 768, 1 ], [ 1152, 4 ], [ 2304, 3 ], [ 4608, 1 ] ] </pre> <p> <a name = "SSEC003.2"></a> <li><code>OneIrreducibleSolvableMatrixGroup(</code><var>func_1</var><code>, </code><var>arg_1</var><code>, </code><var>func_2</var><code>, </code><var>arg_2</var><code>, ...) F</code> <p> This function returns a matrix group <var>G</var> from the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> library such that <var><var>func<sub>i</sub></var>(G)</var> lies in <var>arg_i</var>, or <code>fail</code> if no such group exists. The arguments <var>func_1</var>, <var>func_2</var>, ..., must be <font face="Gill Sans,Helvetica,Arial">GAP</font> functions taking one argument and returning a value, and <var>arg_1</var>, <var>arg_2</var>, ..., must be lists. If <var>arg_i</var> is not a list, <var>arg_i</var> is replaced by the list <code>[</code><var>arg_i</var><code>]</code>. The functions <code>DegreeOfMatrixGroup</code> and <code>FieldOfMatrixGroup</code> (or their equivalents, see below) must be among the <var>func_i</var>. Note that all groups in the data library have the property that <var><code>TraceField(</code><var>G</var><code>)</code> = <code>FieldOfMatrixGroup</code>(<var>G</var>)</var>; see Section <a href="CHAP002.htm#SECT001">Design of the group library</a> for details. The groups passed to the <var>func_i</var> and the groups returned have the attributes and properties described in Chapter <a href="CHAP004.htm">Additional functionality for matrix groups</a> set to their appropriate values. <p> To use this function efficiently, please see the comments in <a href="CHAP002.htm#SSEC003.1">AllIrreducibleSolvableMatrixGroups</a>. <p> <a name = "SSEC003.3"></a> <li><code>IteratorIrreducibleSolvableMatrixGroups(</code><var>func_1</var><code>, </code><var>arg_1</var><code>, </code><var>func_2</var><code>, </code><var>arg_2</var><code>, ...) F</code> <p> This function returns an iterator which runs through the list of all matrix groups <var>G</var> in the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> library such that <var><var>func<sub>i</sub></var>(G)</var> lies in <var>arg_i</var>. The arguments <var>func_1</var>, <var>func_2</var>, ..., must be <font face="Gill Sans,Helvetica,Arial">GAP</font> functions taking one argument and returning a value, and <var>arg_1</var>, <var>arg_2</var>, ..., must be lists. If <var>arg_i</var> is not a list, <var>arg_i</var> is replaced by the list <code>[</code><var>arg_i</var><code>]</code>. The functions <code>DegreeOfMatrixGroup</code> and <code>FieldOfMatrixGroup</code> (or their equivalents, see below) must be among the <var>func_i</var>. <p> Using <p> <code>IteratorIrreducibleSolvableMatrixGroups</code>(<var>func_1</var>, <var>arg_1</var>, <var>func_2</var>, <var>arg_2</var>, ...)) <p> is functionally equivalent to <p> <code>Iterator</code>(<code>AllIrreducibleSolvableMatrixGroups</code>(<var>func_1</var>, <var>arg_1</var>, <var>func_2</var>, <var>arg_2</var>, ...)) <p> (see Section <a href="../../../doc/htm/ref/CHAP028.htm#SECT007">Iterators</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font> reference manual for details) but, unlike <code>AllIrreducibleSolvableMatrixGroups</code>, does not store all relevant matrix groups at the same time. This may save a considerable amount of memory. <p> To use this function efficiently, please see the comments in <a href="CHAP002.htm#SSEC003.1">AllIrreducibleSolvableMatrixGroups</a>. <p> <p> <h2><a name="SECT004">2.4 Loading and unloading group data manually</a></h2> <p><p> The data required by the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> library is loaded into <font face="Gill Sans,Helvetica,Arial">GAP</font>'s workspace automatically whenever required, but is never unloaded automatically. The functions described in this and the following section describe how to load and unload this data manually. They are only relevant if timing or conservation of memory is an issue. <a name = "I0"></a> <p> <a name = "SSEC004.1"></a> <li><code>LoadAbsolutelyIrreducibleSolvableGroupData(</code><var>n</var><code>, </code><var>q</var><code>) F</code> <p> This function loads the data for <var>GL(<var>n</var>, <var>q</var>)</var> into the <font face="Gill Sans,Helvetica,Arial">GAP</font> workspace and does some pre-processing. If the data is already loaded, the function does nothing. This function is called automatically when you access the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> library, so most users will not need this function. <p> <a name = "SSEC004.2"></a> <li><code>LoadedAbsolutelyIrreducibleSolvableGroupData() F</code> <p> This function returns a list. Each entry consists of an integer <var>n</var> and a set <var>l</var>. The set <var>l</var> contains all prime powers <var>q</var> such that the group data for <var>GL(<var>n</var>, <var>q</var>)</var> is currently in memory. <p> <a name = "SSEC004.3"></a> <li><code>UnloadAbsolutelydIrreducibleSolvableGroupData([</code><var>n</var><code> [, </code><var>q</var><code>]]) F</code> <p> This function can be used to delete data for <var>GL(<var>n</var>, <var>q</var>)</var> from the <font face="Gill Sans,Helvetica,Arial">GAP</font> workspace. If no argument is given, all data will be deleted. If only <var>n</var> is given, all data for degree <var>n</var> (and any <var>q</var>) will be deleted. If <var>n</var> and <var>q</var> are given, only the data for <var>GL(n, q)</var> will be deleted from the <font face="Gill Sans,Helvetica,Arial">GAP</font> workspace. Use this function if you run out of <font face="Gill Sans,Helvetica,Arial">GAP</font> workspace. The data is automatically re-loaded when required. <p> <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP001.htm">Previous</a>] [<a href ="CHAP003.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <address>IRREDSOL manual<br>February 2007 </address></body></html>
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