Representation Theory#

This section describes some functions for creating \(R[G]\)-modules for a polycyclic group \(G\), which are unique for this category or have special properties when called for polycyclic groups. For a complete description of the functions available for creating and working with \(R[G]\)-modules we refer to Chapter Modules over an Algebra and Group Representations.

Note that the function GModuleAction can be used to extract the matrix representation associated to an \(R[G]\)-module.

EFAModuleMaps(G): GrpGPC -> [ModGrp]#

Every polycyclic group \(G\) has a normal series \(G = N_1 \triangleright N_2 \triangleright \dots \triangleright N_{r+1} = 1\), such that every quotient \(M_i := N_i / N_{i+1}\) is either free abelian or \(p\)-elementary abelian for some prime \(p\). The action of \(G\) by conjugation induces a \({\mathbb{Z}}[G]\)-module structure on \(M_i\) if \(M_i\) is free abelian and an \({\bf F}_{p}[G]\)-module structure if \(M_i\) is \(p\)-elementary abelian.

This function returns a sequence \([f_1,\dots,f_r]\), where \(f_i : N_i \rightarrow M_i\) is the natural epimorphism onto the additive group of an \(R_i[G]\)-module \(M_i\) (\(R_i\in\{{\bf F}_{p}, {\mathbb{Z}}\}\)), constructed as above.

The functions EFAModuleMaps and EFAModules use the normal series returned by the function EFASeries.

Note that the kernels of the epimorphisms \(f_i\) can be computed and hence it is possible to form preimages of submodules of \(M_i\), which are normal subgroups of \(G\) contained in \(N_i\) and containing \(N_{i+1}\).

EFAModules(G): GrpGPC -> [ModGrp]#

Every polycyclic group \(G\) has a normal series \(G = N_1 \triangleright N_2 \triangleright \dots \triangleright N_{r+1} = 1\), such that every quotient \(M_i := N_i / N_{i+1}\) is either free abelian or \(p\)-elementary abelian for some prime \(p\). The action of \(G\) by conjugation induces a \({\mathbb{Z}}[G]\)-module structure on \(M_i\) if \(M_i\) is free abelian and an \({\bf F}_{p}[G]\)-module structure if \(M_i\) is \(p\)-elementary abelian.

This function returns a sequence \([M_1,\dots,M_r]\) of \(R_i[G]\)-modules (where \(R_i\in\{{\bf F}_{p}, {\mathbb{Z}}\}\)), constructed as above.

The functions EFAModuleMaps and EFAModules use the normal series returned by the function EFASeries.

GModule(G, A, p): GrpGPC, GrpGPC, RngIntElt -> ModGrp, Map#

GModule(G, A): GrpGPC, GrpGPC -> ModGrp, Map#

Let \(A\) be a normal subgroup of the polycyclic group \(G\). If \(p = 0\), the function returns the \({\mathbb{Z}}[G]\)-module corresponding to the conjugation action of \(G\) on the maximal free abelian quotient of \(A\). If \(p\) is a prime, it returns the \({\bf F}_{p}[G]\)-module corresponding to the conjugation action of \(G\) on the maximal \(p\)-elementary abelian quotient of \(A\). The epimorphism \(\pi: A\rightarrow M\) onto the additive group of the constructed module \(M\) is returned as second return value. Note that the kernel of \(\pi\) can be computed and hence it is possible to form preimages of submodules of \(M\), which are normal subgroups of \(G\) contained in \(A\).

If the maximal abelian quotient \(A/A^\prime\) of \(A\) is either free abelian or \(p\)-elementary abelian for some prime \(p\), \(p\) can be omitted in the function call.

Note that it is the user’s responsibility to ensure that \(A\) is in fact normal in \(G\).

GModule(G, A, B, p): GrpGPC, GrpGPC, GrpGPC, RngIntElt -> ModGrp, Map#

GModule(G, A, B): GrpGPC, GrpGPC, GrpGPC -> ModGrp, Map#

Let \(A\) and \(B < A\) be normal subgroups of the polycyclic group \(G\). If \(p = 0\), the function returns the \({\mathbb{Z}}[G]\)-module corresponding to the conjugation action of \(G\) on the maximal free abelian quotient of \(A/B\). If \(p\) is a prime, it returns the \({\bf F}_{p}[G]\)-module corresponding to the conjugation action of \(G\) on the maximal \(p\)-elementary abelian quotient of \(A/B\). The epimorphism \(\pi: A\rightarrow M\) onto the additive group of the constructed module \(M\) is returned as second return value. Note that the kernel of \(\pi\) can be computed and hence it is possible to form preimages of submodules of \(M\), which are normal subgroups of \(G\) contained in \(A\) and containing \(B\).

If the maximal abelian quotient of \(A/B\) is either free abelian or \(p\)-elementary abelian for some prime \(p\), \(p\) can be omitted in the function call.

Note that it is the user’s responsibility to ensure that \(A\) and \(B\) are in fact normal in \(G\).

GModulePrimes(G, A): GrpGPC, GrpGPC -> SetMulti#: Let \(G\) be a polycyclic group and \(A\) a normal subgroup of \(G\). Given any prime \(p\), the maximal \(p\)-elementary abelian quotient of \(A\) can be viewed as an \({\bf F}_{p}[G]\)-module \(M_p\). The maximal free abelian quotient of \(A\) can be viewed as a \({\bf Z}[G]\)-module \(M_0\). This function determines those primes \(p\) for which the module \(M_p\) is non-trivial (i.e. not zero-dimensional) and the dimensions of the corresponding modules \(M_p\). The return value is a multiset \(S\). If \(0\notin S\), the maximal abelian quotient of \(A\) is finite and the multiplicity of \(p\) is the dimension of \(M_p\). If \(S\) contains 0 with multiplicity \(m\), the maximal abelian quotient of \(A\) contains \(m\) copies of \({\mathbb{Z}}\). In this case, the rank of \(M_0\) is \(m\) and \(M_p\) is non-trivial for every prime \(p\) and its rank is the sum of \(m\) and the multiplicity of \(p\) in \(S\).

GModulePrimes(G, A, B): GrpGPC, GrpGPC, GrpGPC -> SetMulti#: Let \(G\) be a polycyclic group, \(A\) a normal subgroup of \(G\) and \(B\) a normal subgroup of \(G\) contained in \(A\). Given any prime \(p\), the maximal \(p\)-elementary abelian quotient of \(A/B\) can be viewed as an \({\bf F}_{p}[G]\)-module \(M_p\). The maximal free abelian quotient of \(A/B\) can be viewed as a \({\bf Z}[G]\)-module \(M_0\). This function determines those primes \(p\) for which the module \(M_p\) is non-trivial (i.e. not zero-dimensional) and the dimensions of the corresponding modules \(M_p\). The return value is a multiset \(S\). If \(0\notin S\), the maximal abelian quotient of \(A/B\) is finite and the multiplicity of \(p\) is the dimension of \(M_p\). If \(S\) contains 0 with multiplicity \(m\), the maximal abelian quotient of \(A/B\) contains \(m\) copies of \({\mathbb{Z}}\). In this case, the rank of \(M_0\) is \(m\) and \(M_p\) is non-trivial for every prime \(p\) and its rank is the sum of \(m\) and the multiplicity of \(p\) in \(S\).

SemisimpleEFAModuleMaps(G): GrpGPC -> [ModGrp]#

Every polycyclic group \(G\) has a normal series \(G = N_1 \triangleright N_2 \triangleright \dots \triangleright N_{r+1} = 1\), such that every quotient \(M_i := N_i / N_{i+1}\) is either free abelian and semisimple as a \({\mathbb{Q}}[G]\)-module or \(p\)-elementary abelian and semisimple as an \({\bf F}_{p}[G]\)-module for some prime \(p\).

This function returns a sequence \([f_1,\dots,f_r]\), where \(f_i : N_i \rightarrow M_i\) is the natural epimorphism onto the additive group of an \(R_i[G]\)-module \(M_i\) (\(R_i\in\{{\bf F}_{p}, {\mathbb{Z}}\}\)), constructed as above.

The functions SemisimpleEFAModules and SemisimpleEFAModuleMaps use the normal series returned by the function SemisimpleEFASeries. Moreover, this normal series is a refinement of the normal series returned by the function EFASeries.

Note that the kernels of the epimorphisms \(f_i\) can be computed and hence it is possible to form preimages of submodules of \(M_i\), which are normal subgroups of \(G\) contained in \(N_i\) and containing \(N_{i+1}\).

SemisimpleEFAModules(G): GrpGPC -> [ModGrp]#

Every polycyclic group \(G\) has a normal series \(G = N_1 \triangleright N_2 \triangleright \dots \triangleright N_{r+1} = 1\), such that every quotient \(M_i := N_i / N_{i+1}\) is either free abelian and semisimple as a \({\mathbb{Q}}[G]\)-module or \(p\)-elementary abelian and semisimple as an \({\bf F}_{p}[G]\)-module for some prime \(p\).

This function returns a sequence \([M_1,\dots,M_r]\) of \(R_i[G]\)-modules (where \(R_i\in\{{\bf F}_{p}, {\mathbb{Z}}\}\)), constructed as above.

The functions SemisimpleEFAModules and SemisimpleEFAModuleMaps use the normal series returned by the function SemisimpleEFASeries. Moreover, this normal series is a refinement of the normal series returned by the function EFASeries.

Example: Representation Theory (ex-e27464)#

Consider the group \(G\) defined by the polycyclic presentation

\[\begin{split}\begin{aligned}\langle a,b,c,d,e\, |\, &c^6, e^3, d^c=de, e^c=e^2, b^a=b^{-1}, b^{a^{-1}}=b^{-1}, \\ &c^b=ce, c^{b^{-1}}=ce, d^b=d^{-1}, d^{b^{-1}}=d^{-1}, e^b=e^2, e^{b^{-1}}=e^2 \rangle.\end{aligned}\end{split}\]

(Trivial commutator relations have been omitted.)

> F<a,b,c,d,e> := FreeGroup(5);
> G<a,b,c,d,e> := quo< GrpGPC : F | c^6 = F!1, e^3 = F!1,
>                                   d^c=d*e, e^c=e^2,
>                                   b^a=b^-1,
>                                   b^(a^-1)=b^-1,
>                                   c^b=c*e,
>                                   c^(b^-1)=c*e,
>                                   d^b=d^-1,
>                                   d^(b^-1)=d^-1,
>                                   e^b=e^2,
>                                   e^(b^-1)=e^2 >;