finn — computing with finite nilpotent linear groups

Contents

1. Description
2. License
3. Download and installation
4. Using finn
   1. Irreducibility testing
   2. Creation of examples
5. References
6. Contact

1. Description

The Magma-package finn provides irreducibility testing for finite nilpotent matrix groups defined over

• number fields or
• rational function fields over number fields.

For a reducible group, a proper submodule can be constructed. The algorithm used is described in [Ros10].

This work is supported by the Research Frontiers Programme of Science Foundation Ireland.

2. License

Copyright © 2010 Tobias Rossmann.

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

The Software is provided “as is”, without warranty of any kind, express or implied, including but not limited to the warranties of merchantability, fitness for a particular purpose and noninfringement. In no event shall the authors or copyright holders be liable for any claim, damages or other liability, whether in an action of contract, tort or otherwise, arising from, out of or in connection with the Software or the use or other dealings in the Software.

3. Download and installation

Download finn-0.4.tar.gz and extract it in some directory, $DIR say. From the Magma prompt, finn is then loaded using

    AttachSpec("$DIR/finn.spec");

4. Using finn

4.1 Irreducibility testing

The following intrinsic for irreducibility testing is provided.

IsIrreducibleFiniteNilpotent(G : parameters): GrpMat -> BoolElt, Any

    DecideOnly: BoolElt                 Default: false

    VectorMode: BoolElt                 Default: false

    CheapTests: BoolElt                 Default: false

    NeqnThreshold: RngIntElt            Default: ∞

The matrix group G over K=BaseRing(G) has to be finite and nilpotent. Moreover, K has to be either a number field or a rational function field (of type FldFunRat) such that BaseRing(K) is a number field.

The first value returned by IsIrreducibleFiniteNilpotent is true if G is irreducible and false otherwise. If DecideOnly=true, then no second value is returned at all.

Suppose that DecideOnly=false. If G is reducible, then the second value returned by IsIrreducibleFiniteNilpotent is either a proper submodule of GModule(G) (VectorMode=false) or a KG-module generator of such a submodule (VectorMode=true). A second value is always returned by IsIrreducibleFiniteNilpotent except when the following exceptional conditions are satisfied:

1. IsIrreducibleFiniteNilpotent would normally proceed to construct a proper submodule (or a generator) by solving an equation α² + β² = -1 in an extension Z of K (see [Ros10, §6]) using a norm equation solver, but
2. 2 * AbsoluteDegree(Z) > NeqnThreshold.

In this situation, IsIrreducibleFiniteNilpotent will behave as if DecideOnly=true and thus not attempt to solve α² + β² = -1. (By default, NeqnThreshold=∞ so this behaviour is turned off.) To force termination of IsIrreducibleFiniteNilpotent within reasonable time in all cases, we recommend setting NeqnThreshold:=20.

If CheapTests=true, then IsIrreducibleFiniteNilpotent may perform some tests which may or may not succeed in proving irreducibility or reducibility of G. These tests should be “cheap” under all circumstances, causing very little extra run-times or memory requirements even if they fail.

If the verbose flag Irrednil is set via

    SetVerbose("Irrednil",1);

then IsIrreducibleFiniteNilpotent will print detailed information on the flow of the algorithm.

4.2 Creation of examples

ExampleIrrednil(i : parameters): RngIntElt -> <GrpMat>

    PaperOnly: BoolElt                  Default: false

For 1 ≤ i ≤ 14, return a tuple < G_i, G_i,γ, G_i,X, G_i,ρ, G_i,ρ,X> of groups. The first three of these groups are described in [Ros10, §9]. G_i,ρ and G_{i, ρ, X} are conjugates of G_i over Q(ρ) and Q(ρ, X), respectively, where ρ¹⁰ - ρ⁹ + 5ρ⁷ + 2ρ⁶ - 3ρ⁵ + 7ρ⁴ - 3ρ³ - 3ρ² + 2ρ + 1 = 0. Some randomisations are employed so that the groups returned will (probably) be different for each call.

If PaperOnly=true, then only < G_i, G_i,γ, G_i,X> is returned.

5. References

[Ros10] T. Rossmann. Irreducibility testing of finite nilpotent linear groups. Preprint. (PDF)

6. Contact

Tobias Rossmann
School of Mathematics, Statistics and Applied Mathematics
National University of Ireland, Galway
University Road
Galway
Ireland

tobias.rossmann (at) googlemail.com