Version 0.57
by
Tobias Rossmann
The Magma-package finn provides functions for irreducibility and primitivity testing of finite nilpotent matrix groups defined over
For a reducible group, a proper submodule can be constructed; for an irreducible but imprimitive group, a system of imprimitivity can be obtained. The algorithms used for irreducibility and primitivity testing are described in [Ros10] and [Ros11], respectively.
The core functionality provided by finn has been included in Magma V2.17. The present version of finn provides finer control of the algorithms than the version contained in Magma.
This work is supported by the Research Frontiers Programme of Science Foundation Ireland, grant 08/RFP/MTH1331.
Copyright © 2010, 2011 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.
Download finn-0.57.tar.gz (released 03/03/2011) and extract it in some directory, $DIR say. From the Magma prompt, finn is then loaded using
AttachSpec("$DIR/finn.spec");
This will (temporarily) replace the version of finn built into Magma V2.17.
The following table lists all versions of finn released so far.
File | Released |
---|---|
finn-0.57.tar.gz | 03/03/2011 |
finn-0.5.tar.gz | 23/06/2010 |
finn-0.4.tar.gz | 19/01/2010 |
The following intrinsic provides low-level access to irreducibility and primitivity testing of finite nilpotent matrix groups.
IsIrrPrimFiniteNilpotent(G : parameters): GrpMat -> MonStgElt, Any
IrrTest: BoolElt Default: truePrimTest: BoolElt Default: trueDecideOnly: BoolElt Default: falseVectorMode: BoolElt Default: falseCheapTests: BoolElt Default: falseNeqnThreshold: 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.
If the verbose flag Finn is set, then IsIrrPrimFiniteNilpotent will print detailed information on the flow of the algorithm.
Case I: IrrTest = PrimTest = true
The first value returned by IsIrrPrimFiniteNilpotent is "red" if G is reducible, "imp" if G is irreducible but imprimitive, and "prim" if G is primitive.
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 IsIrrPrimFiniteNilpotent is either a proper submodule of GModule(G) (VectorMode=false) or a KG-module generator of such a submodule (VectorMode=true). If G is imprimitive, then the second return value is a system of imprimitivity for G, given as a sequence of subspaces of RSpace(G).
In the reducible or imprimitive case, a second value is always returned except when DecideOnly=true or the following exceptional conditions are satisfied:
In this situation, IsIrrPrimFiniteNilpotent will behave as if DecideOnly=true and thus not attempt to solve α^{2} + β^{2} = -1. To ensure termination of IsIrrPrimFiniteNilpotent within reasonable time in all cases, we recommend setting NeqnThreshold:=20.
- IsIrrPrimFiniteNilpotent would normally proceed to solve an equation α^{2} + β^{2} = -1 in an extension Z of K (see [Ros10, §6] and [Ros11, §8.3) using a norm equation solver, but
- 2 * AbsoluteDegree(Z) > NeqnThreshold.
If CheapTests = true then IsIrrPrimFiniteNilpotent 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.
Case II: IrrTest = true, PrimTest = false
In this case, primitivity will not be tested. Thus, the first value returned by IsIrrPrimFiniteNilpotent will be either "red" or "irr" and the second value (if any) will be as explained above.
Case III: IrrTest = false, PrimTest = true
If G is known to be irreducible, then this will only test primitivity; the return values and parameters are as described in case I.
The following intrinsics allow separate testing of irreducibility and primitivity.
IsIrreducibleFiniteNilpotent(G : parameters): GrpMat -> BoolElt, Any
DecideOnly: BoolElt Default: falseVectorMode: BoolElt Default: falseCheapTests: BoolElt Default: falseVerify: BoolElt Default: falseNeqnThreshold: RngIntElt Default: ∞This behaves like IsIrrPrimFiniteNilpotent with IrrTest:=true and PrimTest:=false, except that the first return value is true or false instead of "irr" or "red", respectively. Additionally, if Verify = true, then finiteness and nilpotency of G will be tested.
IsPrimitiveFiniteNilpotent(G : parameters): GrpMat -> BoolElt, Any
DecideOnly: BoolElt Default: falseCheapTests: BoolElt Default: falseVerify: BoolElt Default: falseNeqnThreshold: RngIntElt Default: ∞This behaves like IsIrrPrimFiniteNilpotent with IrrTest:=false and PrimTest:=true, except that the first return value is true or false instead of "prim" or "imp", respectively. Additionally, if Verify = true, then finiteness, nilpotency, and irreducibility of G will be tested.
ExampleIrrednil(i : parameters): RngIntElt -> <GrpMat>
PaperOnly: BoolElt Default: falseFor 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 ρ^{10} - ρ^{9} + 5ρ^{7} + 2ρ^{6} - 3ρ^{5} + 7ρ^{4} - 3ρ^{3} - 3ρ^{2} + 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.
ExamplePrimnil(i): RngIntElt -> GrpMat
For 1 ≤ i ≤ 14, return the group G_{i} from [Ros11, §10]. Again, because of randomisations, the groups returned will most likely be different for each call.
The exact groups used to obtain the run-times in [Ros11, §10] are available as a gzipped Magma-file (304k).
[Ros10] T. Rossmann. Irreducibility testing of finite nilpotent linear groups. Journal of Algebra 324(5): 1114-1124, 2010. (Preprint)
[Ros11] T. Rossmann. Primitivity testing of finite nilpotent linear groups. LMS Journal of Computation and Mathematics 14: 87-98, 2011. (Preprint)
Tobias Rossmann
School of Mathematics, Statistics and Applied Mathematics
National University of Ireland, Galway
University Road
Galway
Ireland
tobias.rossmann (at) googlemail.com