All of the functions described here assume that the base ring is a finite field.
Meataxe(M) : ModRng -> ModRng, ModRng, AlgMatElt
Given an R-module M with base ring a finite field and action algebra A, attempt to find a proper submodule N of M or else prove that M is irreducible. If a splitting of M is found, three values are returned:
- An R-module N corresponding to the induced action of A on N;
- An R-module P corresponding to the induced action of A on the quotient space M/N;
- A matrix T belonging to the matrix ring B defining the action of A on M which conjugates the matrices of A into reduced form. Let x be an element of G. Then M, N, P and T are related as follows:
T * M(x) * T^-1 = [ S(x) 0 ] [ * T(x) ]If M is proved to be irreducible, the function notes this fact in M and simply returns M.
> O53 := PermutationGroup<45 | > (2,3)(4,6)(7,9)(8,11)(12,16)(13,14)(15,19)(18,22)(20,25)(21,26)(27,33) > (28,35) (29,34)(31,38)(36,43)(39,41), > (1,2,4,7,10,14,16,3,5)(6,8,12,17,21,27,34,41,44)(9,13,18,23,29,37,33,40,43) > (11,15,20)(19,24,30,25,31,22,28,36,38)(26,32,39)(35,42,45)>; > > P := PermutationModule(O53, GaloisField(2)); > P; GModule P of dimension 45 with base ring GF(2) > P1, P2 := Meataxe(P); > P1; GModule P1 of dimension 44 with base ring GF(2) > P2; GModule P2 of dimension 1 with base ring GF(2) > P11, P12 := Meataxe(P1); > P11; GModule P11 of dimension 20 with base ring GF(2) > P12; GModule P12 of dimension 24 with base ring GF(2) > P111, P112 := Meataxe(P11); > P111; GModule P111 of dimension 14 with base ring GF(2) > P112; GModule P111 of dimension 6 with base ring GF(2) > P1111, P1112 := Meataxe(P111); > P1111; GModule P1111 of dimension 14 with base ring GF(2) > // Since the dimension of P1111 is the same as P111, it is irreducible > P1121, P1122 := Meataxe(P112); > P1121; GModule P1121 of dimension 6 with base ring GF(2) > // Again we note that P112 is irreducible
The functions given here are currently only available for modules
whose base ring is a finite field.
AbsoluteRepresentation(M) : ModRng -> ModRng
Given an irreducible R-module M, construct the isomorphic reduced-degree absolute representation of M, which is over the absolute field of M and is absolutely irreducible.
True iff the R-module M is absolutely irreducible. Return also the (matrix algebra) generator of the endomorphism algebra E of M (which is always a field), as well as the dimension of E.
Given a R-module M, return true if M is decomposable and false otherwise. If M is decomposable, the function also returns proper submodules A and B of M such that M = A direct-sum B.
True iff the R-module M is irreducible. If M is not irreducible, a proper submodule S of M and the corresponding quotient module Q = M/S is also returned.
True if the R-modules M and N are isomorphic, false otherwise. If M and N are isomorphic, the function also returns a matrix T such that M^(T) = N.
The minimal field to which M can be restricted (a subfield of the coefficient field of M).
The functions defined here are currently only available for modules whose base ring is a finite field.
CompositionFactors(M) : ModRng -> [ ModRng ]
Compute the composition factors for the R-module M by repeatedly applying the meataxe. The composition factors are returned in the form of a sequence of R-modules in the order determined by a composition series for M. If M is irreducible, the function returns a sequence containing M alone.
Construct a composition series for the R-module M by repeatedly applying the meataxe. The function returns three values:
- The composition series as a sequence of R-modules;
- The composition factors as a sequence of R-modules in the order determined by the composition series (a).
- A transformation matrix T such that T * M * T^(-1) is in reduced form.
The constituents of M, i.e. a sequence of representatives for the isomorphism classes of composition factors of M.
The non-isomorphic constituents of M, together with their multiplicities.
Given a R-module M, return a sequence Q of indecomposable summands of M. Each element of Q is an indecomposable submodule of M and the (direct) sum of the elements of Q is M.
> O53 := PermutationGroup<45 |
> (2,3)(4,6)(7,9)(8,11)(12,16)(13,14)(15,19)(18,22)(20,25)(21,26)(27,33)
> (28,35) (29,34)(31,38)(36,43)(39,41),
> (1,2,4,7,10,14,16,3,5)(6,8,12,17,21,27,34,41,44)(9,13,18,23,29,37,33,40,43)
> (11,15,20)(19,24,30,25,31,22,28,36,38)(26,32,39)(35,42,45)>;
>
> P := PermutationModule(O53, GaloisField(2));
> Constituents(P);
[
GModule of dimension 1 with base ring GF(2),
GModule of dimension 6 with base ring GF(2),
GModule of dimension 8 with base ring GF(2),
GModule of dimension 14 with base ring GF(2)
]
> ConstituentsWithMultiplicities(P);
[
<GModule of dimension 1 with base ring GF(2), 3>,
<GModule of dimension 6 with base ring GF(2), 1>,
<GModule of dimension 8 with base ring GF(2), 1>,
<GModule of dimension 14 with base ring GF(2), 2>
]
> S, F := CompositionSeries(P);
> S, F;
[
GModule of dimension 14 with base ring GF(2),
GModule of dimension 20 with base ring GF(2),
GModule of dimension 21 with base ring GF(2),
GModule of dimension 22 with base ring GF(2),
GModule of dimension 30 with base ring GF(2),
GModule of dimension 31 with base ring GF(2),
GModule P of dimension 45 with base ring GF(2)
]
[
GModule of dimension 14 with base ring GF(2),
GModule of dimension 6 with base ring GF(2),
GModule of dimension 1 with base ring GF(2),
GModule of dimension 1 with base ring GF(2),
GModule of dimension 8 with base ring GF(2),
GModule of dimension 1 with base ring GF(2),
GModule of dimension 14 with base ring GF(2)
]
> IndecomposableSummands(P);
[
GModule of dimension 1 with base ring GF(2),
GModule of dimension 44 with base ring GF(2)
]
The functions described here are currently only available for modules whose base ring is a finite field.
MaximalSubmodules(M) : ModRng -> [ ModRng ], BoolElt
Given a R-module M, return a sequence containing the maximal submodules of M.
Limit: RngIntElt Default: 0If a limit L is provided, only up L submodules are calculated, and the second return value indicates whether all of the submodules are returned.
Given a R-module M, return a sequence containing the minimal submodules of M.
Limit: RngIntElt Default: 0If a limit L is provided, only up L submodules are calculated, and the second return value indicates whether all of the submodules are returned.
Given a R-module M, return a sequence containing the minimal submodules of M which are isomorphic to the irreducible module F.
Limit: RngIntElt Default: 0If a limit L is provided, only up L submodules are calculated, and the second return value indicates whether all of the submodules are returned.
The Jacobson radical of the R-module M.
Given a R-module M, return its socle, i.e. the sum of the minimal submodules of M.
A socle series S for M (not including the trivial module but including M), together with the socle factors corresponding to the terms of S and a matrix T giving the transformation of M into (semi-simple) reduced form.
The factors corresponding to the terms of a socle series for M. The factors are returned in the form of a sequence of R-modules in the order determined by a socle series for M. If M is irreducible, the function returns a sequence containing M.
> O53 := PermutationGroup<45 |
> (2,3)(4,6)(7,9)(8,11)(12,16)(13,14)(15,19)(18,22)(20,25)(21,26)(27,33)
> (28,35) (29,34)(31,38)(36,43)(39,41),
> (1,2,4,7,10,14,16,3,5)(6,8,12,17,21,27,34,41,44)(9,13,18,23,29,37,33,40,43)
> (11,15,20)(19,24,30,25,31,22,28,36,38)(26,32,39)(35,42,45)>;
>
> P := PermutationModule(O53, FiniteField(2));
> MaximalSubmodules(P);
[
GModule of dimension 31 with base ring GF(2),
GModule of dimension 44 with base ring GF(2)
]
> JacobsonRadical(P);
GModule of dimension 30 with base ring GF(2)
> MinimalSubmodules(P);
[
GModule of dimension 1 with base ring GF(2),
GModule of dimension 14 with base ring GF(2)
]
> Soc := Socle(P);
> Soc: Maximal;
GModule Soc of dimension 15 with base ring GF(2)
> SocleSeries(P);
[
GModule of dimension 15 with base ring GF(2),
GModule of dimension 22 with base ring GF(2),
GModule of dimension 30 with base ring GF(2),
GModule of dimension 31 with base ring GF(2),
GModule P of dimension 45 with base ring GF(2)
]
> SocleFactors(P);
[
GModule of dimension 15 with base ring GF(2),
GModule of dimension 7 with base ring GF(2),
GModule of dimension 8 with base ring GF(2),
GModule of dimension 1 with base ring GF(2),
GModule of dimension 14 with base ring GF(2)
]
Given a G-module M and a submodule S of M, determine whether S has a G-invariant complement in M. If so return true and a submodule T of M such that M = S direct-sum T; otherwise return false.
Given a G-module M and a submodule S of M, return all G-invariant complements of S in M.[Next] [Prev] [Right] [Left] [Up] [Index] [Root]