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.
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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
This will (temporarily) replace the version of finn built into Magma V2.17.
The following table lists all versions of finn released so far.
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 α2 + β2 = -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 α2 + β2 = -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: false
For 1 ≤ i ≤ 14, return a tuple < Gi, Gi,γ, Gi,X, Gi,ρ, Gi,ρ,X> of groups. The first three of these groups are described in [Ros10, §9]. Gi,ρ and Gi, ρ, X are conjugates of Gi 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 < Gi, Gi,γ, Gi,X> is returned.
ExamplePrimnil(i): RngIntElt -> GrpMat
For 1 ≤ i ≤ 14, return the group Gi 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)
School of Mathematics, Statistics and Applied Mathematics
National University of Ireland, Galway
tobias.rossmann (at) googlemail.com