[AE05] Assmann, B. and Eick, B., Computing polycyclic presentations for polycyclic rational matrix groups, J. Symbolic Comput., 40 (6) (2005), 1269--1284.
[Ass03] Assmann, B.,
Polycyclic presentations for matrix groups,
Diplomarbeit,
TU Braunschweig
(2003)
(http://www.icm.tu-bs.de/ag_algebra/software/assmann).
[Eic01] Eick, B., Algorithms for Polycyclic Groups, Habilitationsschrift, Gesamthochschule Kassel (2001).
[EN00] Eick, B. and Nickel, W.,
Polycyclic
(2000)
(GAP package).
[Ost96] Ostheimer, G., Algorithms for Polycyclic-by-finite groups, Ph.D. thesis, Rutgers University (1996).
[Sim94] Sims, C. C., Computation with finitely presented groups, Cambridge University Press (1994).
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polenta-1.3.11/doc/_entities.xml 000644 000766 000024 00000000057 14775712156 017021 0 ustar 00mhorn staff 000000 000000 Polenta'> polenta-1.3.11/doc/chap0_mj.html 000644 000766 000024 00000021453 14775712162 016663 0 ustar 00mhorn staff 000000 0000001.3.11
10 April 2025
Björn Assmann
© 2003-2007 by Björn Assmann
The Polenta package is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.
We appreciate very much all past and future comments, suggestions and contributions to this package and its documentation provided by GAP users and developers.
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polenta-1.3.11/doc/chap2.html 000644 000766 000024 00000046430 14775712162 016201 0 ustar 00mhorn staff 000000 000000Groups defined by polycyclic presentations are called PcpGroups in GAP. We refer to the Polycyclic manual [EN00] for further background.
Suppose that a collection X of matrices of GL(d,R) is given, where the ring R is either ℚ,ℤ or a finite field. Let G= ⟨ X ⟩. If the group G is polycyclic, then the following functions determine a PcpGroup isomorphic to G.
‣ PcpGroupByMatGroup( G ) | ( operation ) |
G is a subgroup of GL(d,R) where R=ℚ,ℤ or F_q. If G is polycyclic, then this function determines a PcpGroup isomorphic to G. If G is not polycyclic, then this function returns fail.
‣ IsomorphismPcpGroup( G ) | ( method ) |
G is a subgroup of GL(d,R) where R=ℚ,ℤ or F_q. If G is polycyclic, then this function determines an isomorphism onto a PcpGroup. If G is not polycyclic, then this function returns fail.
Note that the method IsomorphismPcpGroup, installed in this package, cannot be applied directly to a group given by the function AlmostCrystallographicGroup. Please use POL_AlmostCrystallographicGroup (with the same parameters as AlmostCrystallographicGroup) instead.
‣ ImagesRepresentative( map, elm ) | ( method ) |
‣ ImageElm( map, elm ) | ( method ) |
‣ ImagesSet( map, elms ) | ( method ) |
Here map is an isomorphism from a polycyclic matrix group G onto a PcpGroup H calculated by IsomorphismPcpGroup (2.1-2). These methods can be used to compute with such an isomorphism. If the input elm is an element of G, then the function ImageElm can be used to compute the image of elm under map. If elm is not contained in G then the function ImageElm returns fail. The input pcpelm is an element of H.
‣ IsSolvableGroup( G ) | ( method ) |
G is a subgroup of GL(d,R) where R=ℚ,ℤ or F_q. This function tests if G is solvable and returns true or false.
‣ IsTriangularizableMatGroup( G ) | ( property ) |
G is a subgroup of GL(d,ℚ). This function tests if G is triangularizable (possibly over a finite field extension) and returns true or false.
‣ IsPolycyclicGroup( G ) | ( method ) |
G is a subgroup of GL(d,R) where R=ℚ,ℤ or F_q. This function tests if G is polycyclic and returns true or false.
Let G be a finitely generated solvable subgroup of GL(d,ℚ). The vector space ℚ^d is a module for the algebra ℚ[G]. The following functions provide the possibility to compute certain module series of ℚ^d. Recall that the radical Rad_G(ℚ^d) is defined to be the intersection of maximal ℚ[G]-submodules of ℚ^d. Also recall that the radical series
0=R_n < R_{n-1} < \dots < R_1 < R_0=ℚ^d
is defined by R_i+1:= Rad_G(R_i).
‣ RadicalSeriesSolvableMatGroup( G ) | ( operation ) |
This function returns a radical series for the ℚ[G]-module ℚ^d, where G is a solvable subgroup of GL(d,ℚ).
A radical series of ℚ^d can be refined to a homogeneous series.
‣ HomogeneousSeriesAbelianMatGroup( G ) | ( function ) |
A module is said to be homogeneous if it is the direct sum of pairwise irreducible isomorphic submodules. A homogeneous series of a module is a submodule series such that the factors are homogeneous. This function returns a homogeneous series for the ℚ[G]-module ℚ^d, where G is an abelian subgroup of GL(d,ℚ).
‣ HomogeneousSeriesTriangularizableMatGroup( G ) | ( function ) |
A module is said to be homogeneous if it is the direct sum of pairwise irreducible isomorphic submodules. A homogeneous series of a module is a submodule series such that the factors are homogeneous. This function returns a homogeneous series for the ℚ[G]-module ℚ^d, where G is a triangularizable subgroup of GL(d,ℚ).
A homogeneous series can be refined to a composition series.
‣ CompositionSeriesAbelianMatGroup( G ) | ( function ) |
A composition series of a module is a submodule series such that the factors are irreducible. This function returns a composition series for the ℚ[G]-module ℚ^d, where G is an abelian subgroup of GL(d,ℚ).
‣ CompositionSeriesTriangularizableMatGroup( G ) | ( function ) |
A composition series of a module is a submodule series such that the factors are irreducible. This function returns a composition series for the ℚ[G]-module ℚ^d, where G is a triangularizable subgroup of GL(d,ℚ).
‣ SubgroupsUnipotentByAbelianByFinite( G ) | ( operation ) |
G is a subgroup of GL(d,R) where R=ℚ or ℤ. If G is polycyclic, then this function returns a record containing two normal subgroups T and U of G. The group T is unipotent-by-abelian (and thus triangularizable) and of finite index in G. The group U is unipotent and is such that T/U is abelian. If G is not polycyclic, then the algorithm returns fail.
‣ PolExamples( l ) | ( function ) |
Returns some examples for polycyclic rational matrix groups, where l is an integer between 1 and 24. These can be used to test the functions in this package. Some of the properties of the examples are summarised in the following table.
PolExamples number generators subgroup of Hirsch length
1 3 GL(4,Z) 6
2 2 GL(5,Z) 6
3 2 GL(4,Q) 4
4 2 GL(5,Q) 6
5 9 GL(16,Z) 3
6 6 GL(4,Z) 3
7 6 GL(4,Z) 3
8 7 GL(4,Z) 3
9 5 GL(4,Q) 3
10 4 GL(4,Q) 3
11 5 GL(4,Q) 3
12 5 GL(4,Q) 3
13 5 GL(5,Q) 4
14 6 GL(5,Q) 4
15 6 GL(5,Q) 4
16 5 GL(5,Q) 4
17 5 GL(5,Q) 4
18 5 GL(5,Q) 4
19 5 GL(5,Q) 4
20 7 GL(16,Z) 3
21 5 GL(16,Q) 3
22 4 GL(16,Q) 3
23 5 GL(16,Q) 3
24 5 GL(16,Q) 3
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polenta-1.3.11/doc/install.xml 000644 000766 000024 00000004411 14775712140 016473 0 ustar 00mhorn staff 000000 000000In this section we outline three example computations with functions from the previous chapter.
gap> mats := [ [ [ 1, 0, -1/2, 0 ], [ 0, 1, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 1/2, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 0, 1 ] ], [ [ 1, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, -1/2, -3, 7/6 ], [ 0, 1, -1, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ] ], [ [ -1, 3, 3, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ] ] ]; gap> G := Group( mats ); <matrix group with 5 generators> # calculate an isomorphism from G to a pcp-group gap> nat := IsomorphismPcpGroup( G );; gap> H := Image( nat ); Pcp-group with orders [ 2, 2, 3, 5, 5, 5, 0, 0, 0 ] gap> h := GeneratorsOfGroup( H ); [ g1, g2, g3, g4, g5, g6, g7, g8, g9] gap> mats2 := List( h, x -> PreImage( nat, x ) );; # take a random element of G gap> exp := [ 1, 1, 1, 1, 0, 0, 0, 0, 1 ];; gap> g := MappedVector( exp, mats2 ); [ [ -1, 17/2, -1, 233/6 ], [ 0, 1, 0, -2 ], [ 0, 1, -1, 2 ], [ 0, 0, 0, 1 ] ] # map g into the image of nat gap> i := ImageElm( nat, g ); g1*g2*g3*g4*g9 # exponent vector gap> Exponents( i ); [ 1, 1, 1, 1, 0, 0, 0, 0, 1 ] # compare the preimage with g gap> PreImagesRepresentative( nat, i ); [ [ -1, 17/2, -1, 233/6 ], [ 0, 1, 0, -2 ], [ 0, 1, -1, 2 ], [ 0, 0, 0, 1 ] ] gap> last = g; true
gap> gens := [ [ [ 1746/1405, 524/7025, 418/1405, -77/2810 ], [ 815/843, 899/843, -1675/843, 415/281 ], [ -3358/4215, -3512/21075, 4631/4215, -629/1405 ], [ 258/1405, 792/7025, 1404/1405, 832/1405 ] ], [ [ -2389/2810, 3664/21075, 8942/4215, -35851/16860 ], [ 395/281, 2498/2529, -5105/5058, 3260/2529 ], [ 3539/2810, -13832/63225, -12001/12645, 87053/50580 ], [ 5359/1405, -3128/21075, -13984/4215, 40561/8430 ] ] ]; gap> H := Group( gens ); <matrix group with 2 generators> gap> RadicalSeriesSolvableMatGroup( H ); [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 0, 0, 79/138 ], [ 0, 1, 0, -275/828 ], [ 0, 0, 1, -197/414 ] ], [ [ 1, 0, -3, 2 ], [ 0, 1, 55/4, -55/8 ] ], [ [ 1, 4/15, 2/3, 1/6 ] ], [ ] ]
gap> G := PolExamples(3); <matrix group with 2 generators> gap> GeneratorsOfGroup( G ); [ [ [ 73/10, -35/2, 42/5, 63/2 ], [ 27/20, -11/4, 9/5, 27/4 ], [ -3/5, 1, -4/5, -9 ], [ -11/20, 7/4, -2/5, 1/4 ] ], [ [ -42/5, 423/10, 27/5, 479/10 ], [ -23/10, 227/20, 13/10, 231/20 ], [ 14/5, -63/5, -4/5, -79/5 ], [ -1/10, 9/20, 1/10, 37/20 ] ] ] gap> subgroups := SubgroupsUnipotentByAbelianByFinite( G ); rec( T := <matrix group with 2 generators>, U := <matrix group with 4 generators> ) gap> GeneratorsOfGroup( subgroups.T ); [ [ [ 73/10, -35/2, 42/5, 63/2 ], [ 27/20, -11/4, 9/5, 27/4 ], [ -3/5, 1, -4/5, -9 ], [ -11/20, 7/4, -2/5, 1/4 ] ], [ [ -42/5, 423/10, 27/5, 479/10 ], [ -23/10, 227/20, 13/10, 231/20 ], [ 14/5, -63/5, -4/5, -79/5 ], [ -1/10, 9/20, 1/10, 37/20 ] ] ] # so G is triangularizable!
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polenta-1.3.11/doc/chap1_mj.html 000644 000766 000024 00000013327 14775712162 016665 0 ustar 00mhorn staff 000000 000000This package provides functions for computation with matrix groups. Let \(G\) be a subgroup of \(GL(d,R)\) where the ring \(R\) is either equal to \(ℚ,ℤ\) or a finite field \(\mathbb{F}_q\). Then:
We can test whether \(G\) is solvable.
We can test whether \(G\) is polycyclic.
If \(G\) is polycyclic, then we can determine a polycyclic presentation for \(G\).
A group \(G\) which is given by a polycyclic presentation can be largely investigated by algorithms implemented in the GAP-package Polycyclic [EN00]. For example we can determine if \(G\) is torsion-free and calculate the torsion subgroup. Further we can compute the derived series and the Hirsch length of the group \(G\). Also various methods for computations with subgroups, factor groups and extensions are available.
As a by-product, the Polenta package provides some functionality to compute certain module series for modules of solvable groups. For example, if \(G\) is a rational polycyclic matrix group, then we can compute the radical series of the natural \(ℚ[G]\)-module \(ℚ^d\).
A group \(G\) is called polycyclic if it has a finite subnormal series with cyclic factors. It is a well-known fact that every polycyclic group is finitely presented by a so-called polycyclic presentation (see for example Chapter 9 in [Sim94] or Chapter 2 in [EN00] ). In GAP, groups which are defined by polycyclic presentations are called polycyclically presented groups, abbreviated PcpGroups.
The overall idea of the algorithm implemented in this package was first introduced by Ostheimer in 1996 [Ost96]. In 2001 Eick presented a more detailed version [Eic01]. This package contains an implementation of Eick's algorithm. A description of this implementation together with some refinements and extensions can be found in [AE05] and [Ass03].
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polenta-1.3.11/doc/title.xml 000644 000766 000024 00000002114 14775712156 016153 0 ustar 00mhorn staff 000000 000000CompositionSeriesAbelianMatGroup 2.2-4 CompositionSeriesTriangularizableMatGroup 2.2-5 HomogeneousSeriesAbelianMatGroup 2.2-2 HomogeneousSeriesTriangularizableMatGroup 2.2-3 ImageElm 2.1-3 ImagesRepresentative 2.1-3 ImagesSet 2.1-3 InfoPolenta 5.1-1 IsomorphismPcpGroup 2.1-2 IsPolycyclicGroup 2.1-6 IsSolvableGroup 2.1-4 IsTriangularizableMatGroup 2.1-5 PcpGroupByMatGroup 2.1-1 PolExamples 2.4-1 RadicalSeriesSolvableMatGroup 2.2-1 SubgroupsUnipotentByAbelianByFinite 2.3-1
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polenta-1.3.11/doc/chap3_mj.html 000644 000766 000024 00000017152 14775712162 016667 0 ustar 00mhorn staff 000000 000000In this section we outline three example computations with functions from the previous chapter.
gap> mats := [ [ [ 1, 0, -1/2, 0 ], [ 0, 1, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 1/2, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 0, 1 ] ], [ [ 1, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, -1/2, -3, 7/6 ], [ 0, 1, -1, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ] ], [ [ -1, 3, 3, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ] ] ]; gap> G := Group( mats ); <matrix group with 5 generators> # calculate an isomorphism from G to a pcp-group gap> nat := IsomorphismPcpGroup( G );; gap> H := Image( nat ); Pcp-group with orders [ 2, 2, 3, 5, 5, 5, 0, 0, 0 ] gap> h := GeneratorsOfGroup( H ); [ g1, g2, g3, g4, g5, g6, g7, g8, g9] gap> mats2 := List( h, x -> PreImage( nat, x ) );; # take a random element of G gap> exp := [ 1, 1, 1, 1, 0, 0, 0, 0, 1 ];; gap> g := MappedVector( exp, mats2 ); [ [ -1, 17/2, -1, 233/6 ], [ 0, 1, 0, -2 ], [ 0, 1, -1, 2 ], [ 0, 0, 0, 1 ] ] # map g into the image of nat gap> i := ImageElm( nat, g ); g1*g2*g3*g4*g9 # exponent vector gap> Exponents( i ); [ 1, 1, 1, 1, 0, 0, 0, 0, 1 ] # compare the preimage with g gap> PreImagesRepresentative( nat, i ); [ [ -1, 17/2, -1, 233/6 ], [ 0, 1, 0, -2 ], [ 0, 1, -1, 2 ], [ 0, 0, 0, 1 ] ] gap> last = g; true
gap> gens := [ [ [ 1746/1405, 524/7025, 418/1405, -77/2810 ], [ 815/843, 899/843, -1675/843, 415/281 ], [ -3358/4215, -3512/21075, 4631/4215, -629/1405 ], [ 258/1405, 792/7025, 1404/1405, 832/1405 ] ], [ [ -2389/2810, 3664/21075, 8942/4215, -35851/16860 ], [ 395/281, 2498/2529, -5105/5058, 3260/2529 ], [ 3539/2810, -13832/63225, -12001/12645, 87053/50580 ], [ 5359/1405, -3128/21075, -13984/4215, 40561/8430 ] ] ]; gap> H := Group( gens ); <matrix group with 2 generators> gap> RadicalSeriesSolvableMatGroup( H ); [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 0, 0, 79/138 ], [ 0, 1, 0, -275/828 ], [ 0, 0, 1, -197/414 ] ], [ [ 1, 0, -3, 2 ], [ 0, 1, 55/4, -55/8 ] ], [ [ 1, 4/15, 2/3, 1/6 ] ], [ ] ]
gap> G := PolExamples(3); <matrix group with 2 generators> gap> GeneratorsOfGroup( G ); [ [ [ 73/10, -35/2, 42/5, 63/2 ], [ 27/20, -11/4, 9/5, 27/4 ], [ -3/5, 1, -4/5, -9 ], [ -11/20, 7/4, -2/5, 1/4 ] ], [ [ -42/5, 423/10, 27/5, 479/10 ], [ -23/10, 227/20, 13/10, 231/20 ], [ 14/5, -63/5, -4/5, -79/5 ], [ -1/10, 9/20, 1/10, 37/20 ] ] ] gap> subgroups := SubgroupsUnipotentByAbelianByFinite( G ); rec( T := <matrix group with 2 generators>, U := <matrix group with 4 generators> ) gap> GeneratorsOfGroup( subgroups.T ); [ [ [ 73/10, -35/2, 42/5, 63/2 ], [ 27/20, -11/4, 9/5, 27/4 ], [ -3/5, 1, -4/5, -9 ], [ -11/20, 7/4, -2/5, 1/4 ] ], [ [ -42/5, 423/10, 27/5, 479/10 ], [ -23/10, 227/20, 13/10, 231/20 ], [ 14/5, -63/5, -4/5, -79/5 ], [ -1/10, 9/20, 1/10, 37/20 ] ] ] # so G is triangularizable!
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polenta-1.3.11/doc/chapInd.txt 000644 000766 000024 00000001572 14775712156 016426 0 ustar 00mhorn staff 000000 000000 [1XIndex[101X [2XCompositionSeriesAbelianMatGroup[102X 2.2-4 [2XCompositionSeriesTriangularizableMatGroup[102X 2.2-5 [2XHomogeneousSeriesAbelianMatGroup[102X 2.2-2 [2XHomogeneousSeriesTriangularizableMatGroup[102X 2.2-3 [2XImageElm[102X 2.1-3 [2XImagesRepresentative[102X 2.1-3 [2XImagesSet[102X 2.1-3 [2XInfoPolenta[102X 5.1-1 Installation 4.0 [2XIsomorphismPcpGroup[102X 2.1-2 [2XIsPolycyclicGroup[102X 2.1-6 [2XIsSolvableGroup[102X 2.1-4 [2XIsTriangularizableMatGroup[102X 2.1-5 License 0.0-1 Loading the [5XPolenta[105X package 4.2 [2XPcpGroupByMatGroup[102X 2.1-1 Polenta 1.0 [2XPolExamples[102X 2.4-1 Polycyclic 1.0 [2XRadicalSeriesSolvableMatGroup[102X 2.2-1 [2XSubgroupsUnipotentByAbelianByFinite[102X 2.3-1 ------------------------------------------------------- polenta-1.3.11/doc/chap4.html 000644 000766 000024 00000012355 14775712162 016202 0 ustar 00mhorn staff 000000 000000The Polenta package is part of the standard distribution of GAP and so normally there should be no need to install it separately. If by any chance it is not part of your GAP distribution, then the standard method is to unpack the package into the pkg directory of your GAP distribution. This will create a polenta subdirectory. For other non-standard options please see Chapter Reference: Installing a GAP Package of the GAP Reference Manual.
Note that the GAP-Packages Alnuth and Polycyclic are needed for this package. Normally they should be contained in your distribution. If not, they can be obtained at https://www.gap-system.org/Packages/packages.html.
If the Polenta package is not already loaded then you have to request it explicitly. This can be done via the LoadPackage (Reference: LoadPackage) command.
Once the package is installed, it is possible to check the correct installation by running the test suite of the package.
gap> ReadPackage( "Polenta", "tst/testall.g" );
For more details on Test Files see Section Reference: Test Files of the GAP Reference Manual.
If the test suite runs into an error, even though the packages Polycyclic and Alnuth and their depdendencies have been correctly installed, then please send a message to mhorn@rptu.de including the error message.
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polenta-1.3.11/doc/nocolorprompt.css 000644 000766 000024 00000000313 14775712162 017733 0 ustar 00mhorn staff 000000 000000 /* colors for ColorPrompt like examples */ span.GAPprompt { color: #000000; font-weight: normal; } span.GAPbrkprompt { color: #000000; font-weight: normal; } span.GAPinput { color: #000000; } polenta-1.3.11/doc/lefttoc.css 000644 000766 000024 00000000474 14775712162 016466 0 ustar 00mhorn staff 000000 000000 /* leftmenu.css Frank Lübeck */ /* Change default CSS to show section menu on left side */ body { padding-left: 28%; } body.chap0 { padding-left: 2%; } div.ChapSects div.ContSect:hover div.ContSSBlock { left: 15%; } div.ChapSects { left: 1%; width: 25%; } polenta-1.3.11/doc/intro.xml 000644 000766 000024 00000004774 14775712140 016174 0 ustar 00mhorn staff 000000 000000Groups defined by polycyclic presentations are called PcpGroups in GAP. We refer to the Polycyclic manual [EN00] for further background.
Suppose that a collection \(X\) of matrices of \(GL(d,R)\) is given, where the ring \(R\) is either \(ℚ,ℤ\) or a finite field. Let \(G= \langle X \rangle\). If the group \(G\) is polycyclic, then the following functions determine a PcpGroup isomorphic to \(G\).
‣ PcpGroupByMatGroup( G ) | ( operation ) |
G is a subgroup of \(GL(d,R)\) where \(R=ℚ,ℤ \) or \(\mathbb{F}_q\). If G is polycyclic, then this function determines a PcpGroup isomorphic to G. If G is not polycyclic, then this function returns fail.
‣ IsomorphismPcpGroup( G ) | ( method ) |
G is a subgroup of \(GL(d,R)\) where \(R=ℚ,ℤ \) or \(\mathbb{F}_q\). If G is polycyclic, then this function determines an isomorphism onto a PcpGroup. If G is not polycyclic, then this function returns fail.
Note that the method IsomorphismPcpGroup, installed in this package, cannot be applied directly to a group given by the function AlmostCrystallographicGroup. Please use POL_AlmostCrystallographicGroup (with the same parameters as AlmostCrystallographicGroup) instead.
‣ ImagesRepresentative( map, elm ) | ( method ) |
‣ ImageElm( map, elm ) | ( method ) |
‣ ImagesSet( map, elms ) | ( method ) |
Here map is an isomorphism from a polycyclic matrix group G onto a PcpGroup H calculated by IsomorphismPcpGroup (2.1-2). These methods can be used to compute with such an isomorphism. If the input elm is an element of G, then the function ImageElm can be used to compute the image of elm under map. If elm is not contained in G then the function ImageElm returns fail. The input pcpelm is an element of H.
‣ IsSolvableGroup( G ) | ( method ) |
G is a subgroup of \(GL(d,R)\) where \(R=ℚ,ℤ \) or \(\mathbb{F}_q\). This function tests if G is solvable and returns true or false.
‣ IsTriangularizableMatGroup( G ) | ( property ) |
G is a subgroup of \(GL(d,ℚ)\). This function tests if G is triangularizable (possibly over a finite field extension) and returns true or false.
‣ IsPolycyclicGroup( G ) | ( method ) |
G is a subgroup of \(GL(d,R)\) where \(R=ℚ,ℤ \) or \(\mathbb{F}_q\). This function tests if G is polycyclic and returns true or false.
Let \(G\) be a finitely generated solvable subgroup of \(GL(d,ℚ)\). The vector space \(ℚ^d\) is a module for the algebra \(ℚ[G]\). The following functions provide the possibility to compute certain module series of \(ℚ^d\). Recall that the radical \(Rad_G(ℚ^d)\) is defined to be the intersection of maximal \(ℚ[G]\)-submodules of \(ℚ^d\). Also recall that the radical series
\[ 0=R_n < R_{n-1} < \dots < R_1 < R_0=ℚ^d \]
is defined by \(R_{i+1}:= Rad_G(R_i)\).
‣ RadicalSeriesSolvableMatGroup( G ) | ( operation ) |
This function returns a radical series for the \(ℚ[G]\)-module \(ℚ^d\), where G is a solvable subgroup of \(GL(d,ℚ)\).
A radical series of \(ℚ^d\) can be refined to a homogeneous series.
‣ HomogeneousSeriesAbelianMatGroup( G ) | ( function ) |
A module is said to be homogeneous if it is the direct sum of pairwise irreducible isomorphic submodules. A homogeneous series of a module is a submodule series such that the factors are homogeneous. This function returns a homogeneous series for the \(ℚ[G]\)-module \(ℚ^d\), where G is an abelian subgroup of \(GL(d,ℚ)\).
‣ HomogeneousSeriesTriangularizableMatGroup( G ) | ( function ) |
A module is said to be homogeneous if it is the direct sum of pairwise irreducible isomorphic submodules. A homogeneous series of a module is a submodule series such that the factors are homogeneous. This function returns a homogeneous series for the \(ℚ[G]\)-module \(ℚ^d\), where G is a triangularizable subgroup of \(GL(d,ℚ)\).
A homogeneous series can be refined to a composition series.
‣ CompositionSeriesAbelianMatGroup( G ) | ( function ) |
A composition series of a module is a submodule series such that the factors are irreducible. This function returns a composition series for the \(ℚ[G]\)-module \(ℚ^d\), where G is an abelian subgroup of \(GL(d,ℚ)\).
‣ CompositionSeriesTriangularizableMatGroup( G ) | ( function ) |
A composition series of a module is a submodule series such that the factors are irreducible. This function returns a composition series for the \(ℚ[G]\)-module \(ℚ^d\), where G is a triangularizable subgroup of \(GL(d,ℚ)\).
‣ SubgroupsUnipotentByAbelianByFinite( G ) | ( operation ) |
G is a subgroup of \(GL(d,R)\) where \(R=ℚ\) or \(ℤ\). If G is polycyclic, then this function returns a record containing two normal subgroups \(T\) and \(U\) of \(G\). The group \(T\) is unipotent-by-abelian (and thus triangularizable) and of finite index in G. The group \(U\) is unipotent and is such that \(T/U\) is abelian. If G is not polycyclic, then the algorithm returns fail.
‣ PolExamples( l ) | ( function ) |
Returns some examples for polycyclic rational matrix groups, where l is an integer between 1 and 24. These can be used to test the functions in this package. Some of the properties of the examples are summarised in the following table.
PolExamples number generators subgroup of Hirsch length
1 3 GL(4,Z) 6
2 2 GL(5,Z) 6
3 2 GL(4,Q) 4
4 2 GL(5,Q) 6
5 9 GL(16,Z) 3
6 6 GL(4,Z) 3
7 6 GL(4,Z) 3
8 7 GL(4,Z) 3
9 5 GL(4,Q) 3
10 4 GL(4,Q) 3
11 5 GL(4,Q) 3
12 5 GL(4,Q) 3
13 5 GL(5,Q) 4
14 6 GL(5,Q) 4
15 6 GL(5,Q) 4
16 5 GL(5,Q) 4
17 5 GL(5,Q) 4
18 5 GL(5,Q) 4
19 5 GL(5,Q) 4
20 7 GL(16,Z) 3
21 5 GL(16,Q) 3
22 4 GL(16,Q) 3
23 5 GL(16,Q) 3
24 5 GL(16,Q) 3
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polenta-1.3.11/doc/chap5.html 000644 000766 000024 00000025655 14775712162 016212 0 ustar 00mhorn staff 000000 000000It is possible to get informations about the status of the computation of the functions of Chapter 2 of this manual.
‣ InfoPolenta | ( info class ) |
is the Info class of the Polenta package (for more details on the Info mechanism see Section Reference: Info Functions of the GAP Reference Manual). With the help of the function SetInfoLevel(InfoPolenta,level) you can change the info level of InfoPolenta.
If InfoLevel( InfoPolenta ) is equal to 0 then no information messages are displayed.
If InfoLevel( InfoPolenta ) is equal to 1 then basic informations about the process are provided. For further background on the displayed informations we refer to [Ass03] (publicly available via the Internet address http://www.icm.tu-bs.de/ag_algebra/software/assmann/diploma.pdf).
If InfoLevel( InfoPolenta ) is equal to 2 then, in addition to the basic information, the generators of computed subgroups and module series are displayed.
gap> SetInfoLevel( InfoPolenta, 1 ); gap> PcpGroupByMatGroup( PolExamples(11) ); #I Determine a constructive polycyclic sequence for the input group ... #I #I Chosen admissible prime: 3 #I #I Determine a constructive polycyclic sequence for the image under the p-congruence homomorphism ... #I finished. #I Finite image has relative orders [ 3, 2, 3, 3, 3 ]. #I #I Compute normal subgroup generators for the kernel of the p-congruence homomorphism ... #I finished. #I #I Compute the radical series ... #I finished. #I The radical series has length 4. #I #I Compute the composition series ... #I finished. #I The composition series has length 5. #I #I Compute a constructive polycyclic sequence for the induced action of the kernel to the composition series ... #I finished. #I This polycyclic sequence has relative orders [ ]. #I #I Calculate normal subgroup generators for the unipotent part ... #I finished. #I #I Determine a constructive polycyclic sequence for the unipotent part ... #I finished. #I The unipotent part has relative orders #I [ 0, 0, 0 ]. #I #I ... computation of a constructive polycyclic sequence for the whole group finished. #I #I Compute the relations of the polycyclic presentation of the group ... #I Compute power relations ... #I ... finished. #I Compute conjugation relations ... #I ... finished. #I Update polycyclic collector ... #I ... finished. #I finished. #I #I Construct the polycyclic presented group ... #I finished. #I Pcp-group with orders [ 3, 2, 3, 3, 3, 0, 0, 0 ] gap> SetInfoLevel( InfoPolenta, 2 ); gap> PcpGroupByMatGroup( PolExamples(11) ); #I Determine a constructive polycyclic sequence for the input group ... #I #I Chosen admissible prime: 3 #I #I Determine a constructive polycyclic sequence for the image under the p-congruence homomorphism ... #I finished. #I Finite image has relative orders [ 3, 2, 3, 3, 3 ]. #I #I Compute normal subgroup generators for the kernel of the p-congruence homomorphism ... #I finished. #I The normal subgroup generators are #I [ [ [ 1, -3/2, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 3 ], [ 0, 0, 0, 1 ] ], [ [ 1, 0, 0, 24 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 3, 3, 15 ], [ 0, 1, 0, 6 ], [ 0, 0, 1, -6 ], [ 0, 0, 0, 1 ] ], [ [ 1, 3, 3, 9 ], [ 0, 1, 0, 6 ], [ 0, 0, 1, -6 ], [ 0, 0, 0, 1 ] ], [ [ 1, 3/2, 3/2, 3/2 ], [ 0, 1, 0, 3 ], [ 0, 0, 1, -3 ], [ 0, 0, 0, 1 ] ], [ [ 1, -3/2, 9/2, -69/2 ], [ 0, 1, 0, 9 ], [ 0, 0, 1, 3 ], [ 0, 0, 0, 1 ] ] , [ [ 1, 0, 0, -24 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, -3, -3, -9 ], [ 0, 1, 0, -6 ], [ 0, 0, 1, 6 ], [ 0, 0, 0, 1 ] ], [ [ 1, -3, -3, -15 ], [ 0, 1, 0, -6 ], [ 0, 0, 1, 6 ], [ 0, 0, 0, 1 ] ], [ [ 1, -3, 0, 9 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 6 ], [ 0, 0, 0, 1 ] ], [ [ 1, -3, -3, -9 ], [ 0, 1, 0, -6 ], [ 0, 0, 1, 6 ], [ 0, 0, 0, 1 ] ], [ [ 1, -3, 0, 9 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 6 ], [ 0, 0, 0, 1 ] ], [ [ 1, -3/2, -3/2, -9/2 ], [ 0, 1, 0, -3 ], [ 0, 0, 1, 3 ], [ 0, 0, 0, 1 ] ], [ [ 1, -3, -3, -12 ], [ 0, 1, 0, -6 ], [ 0, 0, 1, 6 ], [ 0, 0, 0, 1 ] ], [ [ 1, 3, -3/2, -21 ], [ 0, 1, 0, -3 ], [ 0, 0, 1, -6 ], [ 0, 0, 0, 1 ] ], [ [ 1, 3/2, 3/2, 9/2 ], [ 0, 1, 0, 3 ], [ 0, 0, 1, -3 ], [ 0, 0, 0, 1 ] ] ] #I #I Compute the radical series ... #I finished. #I The radical series has length 4. #I The radical series is #I [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 0, 0, 0, 1 ] ], [ ] ] #I #I Compute the composition series ... #I finished. #I The composition series has length 5. #I The composition series is #I [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 0, 0, 0, 1 ] ], [ ] ] #I #I Compute a constructive polycyclic sequence for the induced action of the kernel to the composition series ... #I finished. #I This polycyclic sequence has relative orders [ ]. #I #I Calculate normal subgroup generators for the unipotent part ... #I finished. #I The normal subgroup generators for the unipotent part are #I [ [ [ 1, -3/2, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 3 ], [ 0, 0, 0, 1 ] ], [ [ 1, 0, 0, 24 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 3, 3, 15 ], [ 0, 1, 0, 6 ], [ 0, 0, 1, -6 ], [ 0, 0, 0, 1 ] ], [ [ 1, 3, 3, 9 ], [ 0, 1, 0, 6 ], [ 0, 0, 1, -6 ], [ 0, 0, 0, 1 ] ], [ [ 1, 3/2, 3/2, 3/2 ], [ 0, 1, 0, 3 ], [ 0, 0, 1, -3 ], [ 0, 0, 0, 1 ] ], [ [ 1, -3/2, 9/2, -69/2 ], [ 0, 1, 0, 9 ], [ 0, 0, 1, 3 ], [ 0, 0, 0, 1 ] ] , [ [ 1, 0, 0, -24 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, -3, -3, -9 ], [ 0, 1, 0, -6 ], [ 0, 0, 1, 6 ], [ 0, 0, 0, 1 ] ], [ [ 1, -3, -3, -15 ], [ 0, 1, 0, -6 ], [ 0, 0, 1, 6 ], [ 0, 0, 0, 1 ] ], [ [ 1, -3, 0, 9 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 6 ], [ 0, 0, 0, 1 ] ], [ [ 1, -3, -3, -9 ], [ 0, 1, 0, -6 ], [ 0, 0, 1, 6 ], [ 0, 0, 0, 1 ] ], [ [ 1, -3, 0, 9 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 6 ], [ 0, 0, 0, 1 ] ], [ [ 1, -3/2, -3/2, -9/2 ], [ 0, 1, 0, -3 ], [ 0, 0, 1, 3 ], [ 0, 0, 0, 1 ] ], [ [ 1, -3, -3, -12 ], [ 0, 1, 0, -6 ], [ 0, 0, 1, 6 ], [ 0, 0, 0, 1 ] ], [ [ 1, 3, -3/2, -21 ], [ 0, 1, 0, -3 ], [ 0, 0, 1, -6 ], [ 0, 0, 0, 1 ] ], [ [ 1, 3/2, 3/2, 9/2 ], [ 0, 1, 0, 3 ], [ 0, 0, 1, -3 ], [ 0, 0, 0, 1 ] ] ] #I #I Determine a constructive polycyclic sequence for the unipotent part ... #I finished. #I The unipotent part has relative orders #I [ 0, 0, 0 ]. #I #I ... computation of a constructive polycyclic sequence for the whole group finished. #I #I Compute the relations of the polycyclic presentation of the group ... #I Compute power relations ... ..... #I ... finished. #I Compute conjugation relations ... .............................................. #I ... finished. #I Update polycyclic collector ... #I ... finished. #I finished. #I #I Construct the polycyclic presented group ... #I finished. #I Pcp-group with orders [ 3, 2, 3, 3, 3, 0, 0, 0 ]
generated by GAPDoc2HTML
polenta-1.3.11/doc/manual.lab 000644 000766 000024 00000005674 14775712162 016260 0 ustar 00mhorn staff 000000 000000 \GAPDocLabFile{polenta} \makelabel{polenta:Title page}{}{X7D2C85EC87DD46E5} \makelabel{polenta:Copyright}{}{X81488B807F2A1CF1} \makelabel{polenta:Acknowledgements}{}{X82A988D47DFAFCFA} \makelabel{polenta:Table of Contents}{}{X8537FEB07AF2BEC8} \makelabel{polenta:Introduction}{1}{X7DFB63A97E67C0A1} \makelabel{polenta:The package}{1.1}{X79DE59997FDCF767} \makelabel{polenta:Polycyclic groups}{1.2}{X86007B0083F60470} \makelabel{polenta:Methods for matrix groups}{2}{X829BA50B82FEC109} \makelabel{polenta:Polycyclic presentations of matrix groups}{2.1}{X826C51B3825A2789} \makelabel{polenta:Module series}{2.2}{X80D1E9E07DB87F97} \makelabel{polenta:Subgroups}{2.3}{X7BA181CA81D785BB} \makelabel{polenta:Examples}{2.4}{X7A489A5D79DA9E5C} \makelabel{polenta:An example application}{3}{X81CAD2F27B2066C4} \makelabel{polenta:Presentation for rational matrix groups}{3.1}{X7DAC33E37B977087} \makelabel{polenta:Modules series}{3.2}{X79CF643081B3FB26} \makelabel{polenta:Triangularizable subgroups}{3.3}{X7BA34CD28059D6CD} \makelabel{polenta:Installation}{4}{X8360C04082558A12} \makelabel{polenta:Installing this package}{4.1}{X81746D7285808409} \makelabel{polenta:Loading the Polenta package}{4.2}{X802ED64A87AA11DC} \makelabel{polenta:Running the test suite}{4.3}{X796DF52483B61C74} \makelabel{polenta:Information Messages}{5}{X85677704855596EB} \makelabel{polenta:Info Class}{5.1}{X7EBFC26F83EB9F72} \makelabel{polenta:Example}{5.2}{X85861B017AEEC50B} \makelabel{polenta:Bibliography}{Bib}{X7A6F98FD85F02BFE} \makelabel{polenta:References}{Bib}{X7A6F98FD85F02BFE} \makelabel{polenta:Index}{Ind}{X83A0356F839C696F} \makelabel{polenta:License}{}{X81488B807F2A1CF1} \makelabel{polenta:Polenta}{1}{X7DFB63A97E67C0A1} \makelabel{polenta:Polycyclic}{1}{X7DFB63A97E67C0A1} \makelabel{polenta:PcpGroupByMatGroup}{2.1.1}{X7A1BC4437FD92201} \makelabel{polenta:IsomorphismPcpGroup}{2.1.2}{X8771540F7A235763} \makelabel{polenta:ImagesRepresentative}{2.1.3}{X85ADB89B7C8DD7D0} \makelabel{polenta:ImageElm}{2.1.3}{X85ADB89B7C8DD7D0} \makelabel{polenta:ImagesSet}{2.1.3}{X85ADB89B7C8DD7D0} \makelabel{polenta:IsSolvableGroup}{2.1.4}{X809C78D5877D31DF} \makelabel{polenta:IsTriangularizableMatGroup}{2.1.5}{X7EE01C207C214C1F} \makelabel{polenta:IsPolycyclicGroup}{2.1.6}{X7D7456077D3D1B86} \makelabel{polenta:RadicalSeriesSolvableMatGroup}{2.2.1}{X84472FDC863322BD} \makelabel{polenta:HomogeneousSeriesAbelianMatGroup}{2.2.2}{X8524F992828B6A71} \makelabel{polenta:HomogeneousSeriesTriangularizableMatGroup}{2.2.3}{X87D9F67C7CBB1499} \makelabel{polenta:CompositionSeriesAbelianMatGroup}{2.2.4}{X86FB6E9B801A37D4} \makelabel{polenta:CompositionSeriesTriangularizableMatGroup}{2.2.5}{X78DE110C7E2A493C} \makelabel{polenta:SubgroupsUnipotentByAbelianByFinite}{2.3.1}{X79273B8581D15356} \makelabel{polenta:PolExamples}{2.4.1}{X7C7C3EFA7E49F932} \makelabel{polenta:Installation}{4}{X8360C04082558A12} \makelabel{polenta:Loading the Polenta package}{4.2}{X802ED64A87AA11DC} \makelabel{polenta:InfoPolenta}{5.1.1}{X809F2CFB87393CE0} polenta-1.3.11/doc/chapBib.txt 000644 000766 000024 00000002062 14775712156 016403 0 ustar 00mhorn staff 000000 000000 [1XReferences[101X [[20XAE05[120X] [16XAssmann, B. and Eick, B.[116X, [17XComputing polycyclic presentations for polycyclic rational matrix groups[117X, [18XJ. Symbolic Comput.[118X, [19X40[119X, 6 (2005), 1269--1284. [[20XAss03[120X] [16XAssmann, B.[116X, [17XPolycyclic presentations for matrix groups[117X, Diplomarbeit, TU Braunschweig (2003), (http://www.icm.tu-bs.de/ag_algebra/software/assmann). [[20XEic01[120X] [16XEick, B.[116X, [17XAlgorithms for Polycyclic Groups[117X, Habilitationsschrift, Gesamthochschule Kassel (2001). [[20XEN00[120X] [16XEick, B. and Nickel, W.[116X, [17XPolycyclic[117X (2000), (GAP package). [[20XOst96[120X] [16XOstheimer, G.[116X, [17XAlgorithms for Polycyclic-by-finite groups[117X, Ph.D. thesis, Rutgers University (1996). [[20XSim94[120X] [16XSims, C. C.[116X, [17XComputation with finitely presented groups[117X, Cambridge University Press (1994). [32X polenta-1.3.11/doc/rainbow.js 000644 000766 000024 00000005336 14775712162 016315 0 ustar 00mhorn staff 000000 000000 function randchar(str) { var i = Math.floor(Math.random() * str.length); while (i == str.length) i = Math.floor(Math.random() * str.length); return str[i]; } hexdigits = "0123456789abcdef"; function randlight() { return randchar("cdef")+randchar(hexdigits)+ randchar("cdef")+randchar(hexdigits)+ randchar("cdef")+randchar(hexdigits) } function randdark() { return randchar("012345789")+randchar(hexdigits)+ randchar("012345789")+randchar(hexdigits)+ randchar("102345789")+randchar(hexdigits) } document.write('\n'); polenta-1.3.11/doc/Polenta.xml 000644 000766 000024 00000000732 14775712156 016440 0 ustar 00mhorn staff 000000 000000 ] >'}br}1|\VjY铵O 7ģ,P>ؿ*ea@O5aWmRN>ЋI ѽI˒Y' {pc";@ -_;b[9R ),-G
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