Version 0.5
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 [Ros10a] and [Ros10b], respectively.
This work is supported by the Research Frontiers Programme of Science Foundation Ireland.
Copyright © 2010 Tobias Rossmann.
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Download finn-0.5.tar.gz (released 23/06/2010) and extract it in some directory, $DIR say. From the Magma prompt, finn is then loaded using
AttachSpec("$DIR/finn.spec");
The previous version was finn-0.4.tar.gz (released 19/01/2010).
The following intrinsic provides irreducibility and primitivity testing of finite nilpotent linear 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 [Ros10a, §6] and [Ros10b, §8.3) using a norm equation solver, but
- 2 * AbsoluteDegree(Z) > NeqnThreshold.
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.
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 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 intrinsic provides compatibility with finn 0.4.
IsIrreducibleFiniteNilpotent(G : parameters): GrpMat -> BoolElt, Any
DecideOnly: BoolElt Default: falseVectorMode: BoolElt Default: falseCheapTests: 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.
IsPrimitiveFiniteNilpotent(G : parameters): GrpMat -> BoolElt, Any
DecideOnly: 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.
ExampleIrrednil(i : parameters): RngIntElt -> <GrpMat>
PaperOnly: BoolElt Default: falseFor 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 [Ros10a, §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 [Ros10b,§10]. Again, because of randomisations, the groups returned will most likely be different for each call.
[Ros10a] T. Rossmann. Irreducibility testing of finite nilpotent linear groups. J. Algebra (2010), doi:10.1016/j.jalgebra.2010.04.031. (Preprint)
[Ros10b] T. Rossmann. Primitivity testing of finite nilpotent linear groups. (Preprint)
Tobias Rossmann
School of Mathematics, Statistics and Applied Mathematics
National University of Ireland, Galway
University Road
Galway
Ireland
tobias.rossmann (at) googlemail.com