Magma - Constructing All Irreducible K[G]-Modules

Constructing All Irreducible K[G]-Modules

Constructing Irreducible K[G]-Modules (Characteristic p)

In this section we describe how to obtain irreducible K[G]-modules for a group G where K is a finite field. In this case, it is currently necessary to construct all irreducible modules in order to obtain a particular irreducible. This section gives brief descriptions of the more useful intrinsics. Much more information is given in the section Enumerating All Irreducible Modules.

IrreducibleModules(G, K : parameters) : Grp, Fld -> SeqEnum

Let G be a permutation group, a matrix group or a group defined by a power-commutator presentation and let K be a finite field. This intrinsic constructs all of the irreducible K[G]-modules for G over K. Thus, some of the irreducibles returned may not be absolutely irreducible. Much more information about the enumeration of all irreducibles can be found in Section Enumerating All Irreducible Modules.

AbsolutelyIrreducibleModules(G, K : parameters) : Grp, Fld -> SeqEnum

Let G be a permutation group, a matrix group or a group defined by a power-commutator presentation and let K be a finite field. This intrinsic constructs all of the absolutely irreducible K[G]-modules. Each irreducible is returned over a field which is the minimal field for which it is absolutely irreducible. Much more information about the enumeration of all irreducibles can be found in Section Enumerating All Irreducible Modules.

AbsolutelyIrreducibleModulesSoluble(G, p) : Grp, RngIntElt -> SeqEnum

Given a finite soluble group G and a prime p dividing the order of G, return the absolutely irreducible K[G]-modules for G in characteristic p. The method works its way up a central series finding the modules at each level using induction and restriction.

AbsolutelyIrreducibleModulesSchur(G, K) : GrpPC, Fld -> List

Let G be a finite soluble group defined by a pc-presentation and let K be a finite field. The function constructs all absolutely irreducible representations of G over appropriate extensions or subfields of the field K and returns them as a list. The Glasby-Howlett algorithm is used to determine a minimal field over which an irreducible module may be realised.

Example `ModAlg_Irreducibles (H99E21)`

The irreducible modules for PSL(3, 4) in characteristic 3 are constructed.

> G := PSL(3, 4);
> irrs := IrreducibleModules(G, GF(3));
> irrs;
[
    GModule of dimension 1 over GF(3),
    GModule of dimension 15 over GF(3),
    GModule of dimension 15 over GF(3),
    GModule of dimension 15 over GF(3),
    GModule of dimension 19 over GF(3),
    GModule of dimension 90 over GF(3),
    GModule of dimension 126 over GF(3)
]

Constructing Irreducible K[G]-Modules (Characteristic Zero)

In this section we describe how to obtain irreducible K[G]-modules for a group G where K is a characteristic zero field. In contrast to the positive characteristic case, Magma can construct just the irreducible that affords a given character. This section gives brief descriptions of the more useful intrinsics. Much more information is given the section Enumerating All Irreducible Modules.

GModule(x) : Chtr -> ModGrp

Let G be a finite group and let x either a rational character or a complex character. This intrinsic constructs the irreducible K[G]-module that affords the character x.

IrreducibleModules(G, K) : Grp, Fld -> SeqEnum

Let G be a finite group and let K be the rational field or a simple number field (including quadratic and cyclotomic fields). This intrinsic constructs all of the irreducible K[G]-modules for G over K. Thus, some of the irreducibles returned may not be absolutely irreducible. If the absolute irreducibles over a finite field are required then the intrinsic AbsolutelyIrreducibleModules should be used.

AbsolutelyIrreducibleModules(G) : Grp -> SeqEnum

Let G be a finite group. This intrinsic constructs all of the absolutely irreducible G-modules in characteristic zero, which are written over the rational field or a simple number field. Each irreducible is also returned over a field which is the minimal field for it to be absolutely irreducible.

AbsolutelyIrreducibleModulesSchur(G, K) : GrpPC, Fld -> List

Let G be a finite soluble group defined by a pc-presentation and let K be the rational field or a cyclotomic field. The order of a cyclotomic field must divide the exponent of G. The function constructs all absolutely irreducible representations of G over appropriate extensions or subfields of the field K and returns them as a list. The field over which an irreducible module is given may not be minimal. See Section Enumerating All Irreducible Modules for more information.

Example `ModAlg_Irreducibles (H99E22)`

The irreducible modules for G=PSL(3, 4) over the field Q are constructed. Then the absolutely irreducible modules for G are constructed.

> G := PSL(3, 4);
> irrs := IrreducibleModules(G, Rationals());          
> irrs;
[
    GModule of dimension 1 over Rational Field,
    GModule of dimension 20 over Rational Field,
    GModule of dimension 35 over Rational Field,
    GModule of dimension 35 over Rational Field,
    GModule of dimension 35 over Rational Field,
    GModule of dimension 64 over Rational Field,
    GModule of dimension 90 over Rational Field,
    GModule of dimension 126 over Rational Field
]
> absirrs := AbsolutelyIrreducibleModules(G);
> absirrs;
[
    GModule of dimension 1 over Rational Field,
    GModule of dimension 20 over Rational Field,
    GModule of dimension 35 over Rational Field,
    GModule of dimension 35 over Rational Field,
    GModule of dimension 35 over Rational Field,
    GModule of dimension 45 over Number Field with defining 
    polynomial x^2 - x + 2 over the Rational Field,
    GModule of dimension 45 over Number Field with defining 
    polynomial x^2 - x + 2 over the Rational Field,
    GModule of dimension 63 over Number Field with defining 
    polynomial x^2 - x - 1 over the Rational Field,
    GModule of dimension 63 over Number Field with defining 
    polynomial x^2 - x - 1 over the Rational Field,
    GModule of dimension 64 over Rational Field
]

The Cartan Matrix

CartanMatrix(G, K) : Grp, FldFin -> AlgMatElt

Let G be a finite group and let K be a finite field. This intrinsic constructs the Cartan matrix C for G over K. Let k be the number of irreducible K[G]-modules. The Cartan matrix C for G over K is a k x k matrix of integers, in which the entry C_ij is equal to the number of times that the j-th irreducible K[G]-module is a constituent of the i-th projective indecomposable K[G]-module. This can be computed quickly from the Brauer characters of the irreducible K[G]-modules. (Note that, unlike the absolute Cartan matrix discussed below, C need not be symmetric.)

AbsoluteCartanMatrix(G, K) : Grp, FldFin -> AlgMatElt

Let G be a finite group and let K be a finite field. This intrinsic constructs the Cartan matrix C for G over an extension L of K that is large enough to ensure that all irreducible L[G]-modules are absolutely irreducible. Its rows and columns correspond to the K[G]-modules returned by AbsolutelyIrreducibleModules(G,K). The matrix C is symmetric and has integer entries. It is equal to the Cartan matrix for G in the characteristic p of K, as defined in textbooks on modular representation theory.

DecompositionMatrix(G, K) : Grp, FldFin -> AlgMatElt

Let G be a finite group and let K be a finite field. This intrinsic constructs the decomposition matrix C for G in the characteristic p of K, The entry D_ij is equal to the number of times that the j-th absolutely irreducible K[G]-module occurs as a constituent of the i-th ordinary irreducible G-module over the complex numbers reduced modulo p. Note that D^T D is equal to the absolute Cartan matrix.

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