# New Modules from Old Given one or more existing modules, standard constructions are available to construct new modules. ## Direct Sums and Tensor Products These direct sums and tensor products for $K[G]$-modules should work for any field $K$ with the exception of the GTensorProduct intrinsics which require $K$ to be a finite field. ### `DirectSum(M, N): ModRng, ModRng -> ModRng, Map, Map, Map, Map` Given $A$-modules $M$ and $N$, construct the direct sum $D$ of $M$ and $N$ as an $A$-module. The embedding maps from $M$ into $D$ and from $N$ into $D$ respectively and the projection maps from $D$ onto $M$ and from $D$ onto $N$ respectively, are also returned. ### `DirectSum(Q): [ ModRng ] -> ModRng, [ Map ], [ Map ]` Given a sequence $Q$ of $A$-modules, construct the direct sum $D$ of these modules. The embedding maps from each of the elements of $Q$ into $D$ and the projection maps from $D$ onto each of the elements of $Q$ are also returned. ### `TensorProduct(M, N): ModMat, ModMat -> ModMat` Given A-modules $M$ and $N$, construct their tensor product, $M \otimes N$. ### `TensorProduct(M, N): ModGrp, ModGrp -> ModGrp` Let $M$ and $N$ be two $K[G]$-modules, where $K$ is a field. This function constructs the tensor product, $M \otimes_K N$, with action of $G$ defined by $(m \otimes n)*g := (m*g) \otimes (n*g)$ for $m,n \in M$ and $g \in G$. ### `TensorPower(M, n): ModMat, RngIntElt -> ModMat` Given an $A$-module $M$ and an integer $n \geq 1$, construct the $n$-th tensor power of $M$. ### `ExteriorSquare(M): ModMat -> ModMat` Given an $A$–module $M$, construct the $A$-submodule of $M \otimes_A M$ consisting of the skew tensors. ### `SymmetricSquare(M): ModMat -> ModMat` Given an $A$-module $M$, construct the $A$-submodule of $M \otimes_A M$ consisting of the symmetric tensors. ### `GTensorProduct(M, N): ModGrp, ModGrp -> ModGrp, Map` Let $M$ and $N$ be two $K[G]$ modules where $K$ is a finite field. This function constructs the tensor product, $M \otimes_{KG} N$. It is equal to `FixDual(TensorProduct(M,N))`. ### `GTensorProduct(M, N, H): ModGrp, ModGrp, Grp -> ModGrp, Map` Let $M$ and $N$ be two $K[G]$ modules where $K$ is a finite field and let $H$ be a subgroup of $G$. This function constructs the tensor product, $M \otimes_{KH} N$ as a $N_G(H)$-module. It is equal to `FixDualMod(TensorProduct(M,N), H)`. ## Induction, Restriction and Inflation for $K[G]$-Modules ### `Dual(M): ModGrp -> ModGrp` Given a $K[G]$-module $M$, where $K$ is a field, construct the $K[G]$-module which is the $K$-dual, ${\operatorname{Hom}}_K(M, K)$, of $M$. ### `Induction(M, G): ModGrp, Grp -> ModGrp` Given a $K[H]$-module $M$, where $K$ is a field, and a supergroup $G$ of $H$, construct the $K[G]$-module obtained by inducing $M$ up to $G$. ### `Induction(R, G): Map, Grp -> Map` Given a representation $R$ of a subgroup of $G$, where $R$ is defined over a field $K$, construct the representation of $G$ obtained by inducing $R$ up to $G$. ### `Restriction(M, H): ModGrp, Grp -> ModGrp` Given a $K[G]$-module $M$, where $K$ is a field, and a subgroup $H$ of $G$, form the $K[H]$-module corresponding to the restriction of $M$ to the subgroup $H$. ### `Inflation(M, h): ModGrp, Map -> ModGrp` Given a $K[H]$-module $M$, where $K$ is a field, and a group homomorphism $h:G \to H$, return $M$ as a $K[G]$-module by inflation using $h$. ### `Example: G Modules1 (ex-6e84be)` Starting with the permutation module $M$ over ${\bf F}_{2}$ for the Mathieu group $M_{22}$, we apply the induction and restriction functions to find new irreducible modules for $M_{22}$. ```magma > SetSeed(1); > G := PermutationGroup< 22 | > (1,2,4,8,16,9,18,13,3,6,12)(5,10,20,17,11,22,21,19,15,7,14), > (1,18,4,2,6)(5,21,20,10,7)(8,16,13,9,12)(11,19,22,14,17), > (1,18,2,4)(3,15)(5,9)(7,16,21,8)(10,12,20,13)(11,17,22,14) >; > M := PermutationModule(G, GaloisField(2)); > M; GModule M of dimension 22 with base ring GF(2) > CM := Constituents(M); > CM; [ GModule of dimension 1 over GF(2), GModule of dimension 10 over GF(2), GModule of dimension 10 over GF(2) ] ``` We compute the restriction $N$ of the module $M$ to the stabilizer $L34$ of a point in $M_{22}$ and then induce two constituents of $N$ back up to $G$. ```magma > L34 := Stabilizer(G, 1); > N := Restriction(M, L34); > N; GModule N of dimension 22 with base ring GF(2) > CN := Constituents(N); > CN; [ GModule of dimension 1 over GF(2), GModule of dimension 9 over GF(2), GModule of dimension 9 over GF(2) ] > Ind1 := Induction(CN[1], G); > Ind1; GModule Ind1 of dimension 22 over GF(2) > Constituents(Ind1); [ GModule of dimension 1 over GF(2), GModule of dimension 10 over GF(2), GModule of dimension 10 over GF(2) ] > Ind2 := Induction(CN[2], G); > Ind2; GModule Ind2 of dimension 198 over GF(2) > Constituents(Ind2); [ GModule of dimension 1 over GF(2), GModule of dimension 10 over GF(2), GModule of dimension 10 over GF(2), GModule of dimension 34 over GF(2), GModule of dimension 98 over GF(2) ] ``` Thus, inducing up the $1$-dimensional constituent of $N$ gives us irreducible modules for $G$ having the same dimensions as those appearing as constituents of $M$. However, inducing up a $9$-dimensional module gives us irreducible modules of new dimensions: $34$ and $98$. Hence starting out with only the permutation module for $M_{22}$ over ${\bf F}_{2}$, we have found $5$ irreducible modules for the group. ## The Fixed-point Spaces for a $K[G]$-Module ### `Fix(M): Mod -> Mod` Given a $K[G]$-module $M$, $K$ a field, construct the largest submodule of $M$ on which $G$ acts trivially, i.e. the fixed-point space of $M$. This is equal to the cohomology group $H^0(G,M)$. ### `FixMod(M, H): ModGrp, Grp -> Mod` Given a $K[G]$-module $M$, $K$ a field, and a subgroup $H$ of $G$, return `Fix(M_H)` as a $N_G(H)$-module, where $M_H$ is the restriction of $M$ to $H$. ### `FixDual(M): ModGrp -> Modgrp, Map` Given a $K[G]$-module $M$, $K$ a field, construct the largest quotient module of $M$ on which $G$ acts trivially, i.e. $M/N$ with $N := \langle m*g-m \vert m \in M, g \in G \rangle$. This is equal to the homology group $H_0(G,M)$. The natural map from $M$ to $M/N$ is also returned. ### `FixDualMod(M, H): ModGrp, Grp -> Mod` Given a $K[G]$-module $M$, $K$ a field, and a subgroup $H$ of $G$, return `FixDual(M_H)` as a $N_G(H)$-module, where $M_H$ is the restriction of $M$ to $H$. The natural map from $M$ to $M/N$ is also returned. ## Change Ring and Base Change ### `ChangeRing(M, S): ModRng, Rng -> ModRng, Map` Given an $A$-module $M$ with base ring $R$, together with a ring $S$, such that there is a natural homomorphism from $R$ to $S$, construct the module $N$ with base ring $S$ where $N$ is obtained from $M$ by coercing the components of the vectors of $M$ into $N$. The corresponding homomorphism from $M$ to $N$ is returned as a second value. ### `ChangeRing(M, S, f): ModRng, Rng, Map -> ModRng, Map` Given an $A$-module $M$ with base ring $R$, together with a ring $S$, and a homomorphism $f: R \rightarrow S$, construct the module $N$ with base ring $S$, where $N$ is obtained from $M$ by applying $f$ to the components of the vectors of $M$. The corresponding homomorphism from $M$ to $N$ is returned as a second value. ### `M ^ T: ModGrp, AlgMatElt -> ModGrp` Given a $K[G]$-module $M$ of dimension $n$, where $K$ is a field, and a nonsingular $n\times n$ matrix $T$ over $K$, construct the $K[G]$-module $N$ which corresponds to taking the rows of $T$ as a basis for $M$. ## Writing a $K[G]$-Module over a Smaller Field ### Rewriting Over a Smaller Degree Finite Field #### `IsRealisableOverSmallerField(M): ModGrp -> BoolElt, ModGrp` Given a $K[G]$-module $M$, where $K$ is a finite field, return true if $M$ can be realised over a proper subfield $F$ of $K$. The equivalent $F[G]$-module is also returned. The Glasby-Howlett algorithm is used to determine the smallest field over which $M$ can be realised. #### `IsRealisableOverSubfield(M, F): ModGrp, FldFin -> BoolElt, ModGrp` Let $M$ be a $K[G]$-module, where $K$ is a finite field of characteristic $p$, and let $F$ be a finite field also of characteristic $p$. If it is possible to realise $M$ over the subfield $F$ of $K$, return true and the equivalent $F[G]$-module. #### `WriteOverSmallerField(M, F): ModGrp, FldFin -> ModGrp, Map` Given a module $M$ of dimension $d$ over a finite field $E$ having degree $e$ and a subfield $F$ of $E$ having degree $f$, write the action of $M$ as $d*e/f$ by $d*e/f$ matrices over $F$ and return the module and the isomorphism. #### `AbsoluteModuleOverMinimalField(M, F): ModGrp, FldFin -> ModGrp` Let $M$ be a $K[G]$-module, where $K$ is a finite field of characteristic $p$, and let $F$ be a finite field also of characteristic $p$. This function returns the module obtained by writing $M$ over the smallest possible field containing $F$ subject to the condition that the dimension of $M$ does not increase. The Glasby-Howlett algorithm is used to determine the smallest field over which $M$ can be realised. #### `AbsoluteModuleOverMinimalField(M): ModGrp -> ModGrp` ```magma verbose: Reduce Default: Verbose : 2 verbose: Cohomology Default: Verbose : 2 verbose: GrunwaldWang Default: Verbose : 2 ``` Let $M$ be a $K[G]$-module, where $K$ is a finite field of characteristic $p$. This function returns the module obtained by writing $M$ over a field of smallest possible degree subject to the condition that the dimension of $M$ does not increase. For modules over finite fields, a field of smallest degree is always a subfield of $K$, in this case, the Glasby-Howlet algorithm is used. #### `AbsoluteModulesOverMinimalField(Q, F): [ ModGrp ], FldFin -> [ ModGrp ]` Let $Q$ be a sequence of $K[G]$-modules, where $K$ is a finite field of characteristic $p$, and let $F$ be a finite field also of characteristic $p$. This function returns the sequence of modules obtained by writing each module $M$ of $Q$ over the smallest possible field containing $F$ subject to the condition that the dimension of $M$ does not increase. Thus, the effect of the function is to apply the function `AbsoluteModuleOverMinimalField` to each module of $Q$. The Glasby-Howlett algorithm is used to determine the smallest field over which the modules $M$ of $Q$ can be realised. #### `ModuleOverSmallerField(M, F): ModGrp, FldFin -> ModGrp` Let $M$ be a $K[G]$-module of dimension $d$, where $K$ is a finite field of characteristic $p$, and let $F$ be a subfield of $K$ of index $n$. This function returns the $F[G]$-module $N$ obtained by writing the action of $M$ as $dn \times dn$ matrices over $F$. #### `ModulesOverSmallerField(Q, F): SeqEnum, FldFin -> ModGrp` Let $Q$ be a sequence of $K[G]$-modules, where $K$ is a finite field of characteristic $p$, and let $F$ be a subfield of $K$ of index $n$. This function returns the sequence $R$ of $F[G]$-modules obtained by applying the function `ModuleOverSmallerField` to each term of $Q$. That is, each term $N$ of $R$ is formed by writing the action of the corresponding term of $Q$ as $dn \times dn$ matrices over $F$. #### `ModulesOverCommonField(M, N): ModGrp, ModGrp -> ModGrp, ModGrp` Given $K[G]$-modules $M$ and $N$ defined over finite fields of characteristic $p$, change their base fields to $L$, where $L$ is the smallest field containing the base fields of $M$ and $N$. ### Rewriting Over a Smaller Degree Number Field #### `AbsoluteModuleOverMinimalField(M): ModGrp -> ModGrp` ```magma verbose: Reduce Default: Verbose : 2 verbose: Cohomology Default: Verbose : 2 verbose: GrunwaldWang Default: Verbose : 2 ``` Let $M$ be a $K[G]$-module, where $K$ is a a number field. This function returns the module obtained by writing $M$ over a field $F$ of smallest possible degree subject to the condition that the dimension of $M$ does not increase. The field $F$ is found using a combination of a method due to W. Plesken and a constructive version of the Grunwald-Wang theorem. #### `Minimize(M): ModGrp -> ModGrp` #### `Minimize(R): Map -> Map` ```magma All : BoolElt Default: false Char : AlgChtrElt Default: false FindSmallest: BoolElt Default: false verbose : Reduce Default: Verbose : 2 verbose : Cohomology Default: Verbose : 2 verbose : GrunwaldWang Default: Verbose : 2 ``` Given an absolutely irreducible $KG$-module $M$ or an absolutely irreducible representation $R : G \rightarrow {\operatorname{GL}}(n, K)$, where $K$ is a number field, this function tries to find minimal subfields $k$ of $K$ that afford $M$ ($R$), i.e. it tries to write the module (representation) over a smaller field. In general however, there might be a number field $k$ not contained in $K$ of smaller degree that affords $M$ ($R$). If `All` is set to `true`, then instead of returning a single module (representation) over a minimal degree subfield of $K$, a list of modules (representations) over all minimal subfields of $K$ is returned instead. If `Char` is given, it should be set to the character of the module (representation). If `FindSmallest` is given, the field $K$ will be extended by some auxiliary field $A$ such that $KA$ will contain a minimal degree field affording $R$. This involves a constructive version of the Grunwald-Wang theorem and can be computationally expensive if the degree of $KA$ is too large. #### `WriteGModuleOver(M, K): ModGrp, FldAlg -> ModGrp` #### `WriteRepresentationOver(R, K): Map, FldAlg -> Map` ```magma Char : AlgChtrElt Default: false Subfield: BoolElt Default: false verbose : Reduce Default: Verbose : 2 verbose : Cohomology Default: Verbose : 2 verbose : GrunwaldWang Default: Verbose : 2 ``` Given an absolutely irreducible $L[G]$-module $M$ or an absolutely irreducible representation $R : G \rightarrow GL(n, L)$, where $L$ is a number field, and a number field $K$ that is a normal extension, try to write $M$ ($R$) over $K$. If `Char` is specified, it should be set to the character of this module (representation). If `Subfield` is set to `true`, the module (representation) will be rewritten over a minimal degree subfield of $K$. #### `Example: Minimal Field (ex-217bd4)` We will work with the $G$-module and character of the unique $2$-dimensional character of $Q_8$. It is well known that, while the character is defined over $Q$, the corresponding representation can only be defined over fields where $-1$ is the sum of $2$ squares. ```magma > _ := PolynomialRing(Rationals()); > G := TransitiveGroup(8, 5); > TransitiveGroupDescription(G); Q_8(8) > R := AbsolutelyIrreducibleModules(G, Rationals()); > R; [ GModule of dimension 1 over Rational Field, GModule of dimension 1 over Rational Field, GModule of dimension 1 over Rational Field, GModule of dimension 1 over Rational Field, GModule of dimension 2 over Cyclotomic Field of order 4 and degree 2 ] > R := R[5]; > W := WriteGModuleOver(R, CyclotomicField(5)); > W; GModule W of dimension 2 over Cyclotomic Field of order 5 and degree 4 ``` So $Q(\zeta_5)$ is an example of a field affording the module but having no minimal degree subfield (of degree 2 here) affording $R$!. ```magma > M := AbsoluteModuleOverMinimalField(W); > M; GModule M of dimension 2 over Number Field with defining polynomial x^2 + 10 over the Rational Field ``` Note that the base field returned here is of degree $2$ but different from $Q(\zeta_4)$ that was found initially. In general there are infinitely many minimal degree splitting fields, and the specific one returned depends on the algorithm’s internal choices. If we try to realize $R$ over a field where $-1$ cannot be written as a sums of two squares we get an error: ```magma > WriteGModuleOver(R, QuadraticField(3)); >> WriteGModuleOver(R, QuadraticField(3)); ^ Runtime error in 'WriteGModuleOver': The G-module cannot be realised over K ``` We can try to find a minimal field containing $Q(\sqrt 3)$ by computing the local Schur-indices and then obtain a splitting field: ```magma > k := QuadraticField(3); > si := SchurIndices(Character(R), k); > si; [ <1st place at infinity, 2>, <2nd place at infinity, 2> ] > A := SplittingField(si); > A; FldAb, defined by (<3>, [1 2]) of structure: Z/2 ``` So the splitting field is returned as an abelian extension. We can see that $A$ is of degree $2$ over $k$ and will be ramified at most at $3$ and both infinite places. In order to use it to rewrite the module, we need to convert to a number field over $Q$ first: ```magma > A := NumberField(A); > A; Number Field with defining polynomial $.1^2 + 1 over k > A := AbsoluteField(A); > A; Number Field A with defining polynomial x^4 - 4*x^2 + 16 over the Rational Field > M := WriteGModuleOver(R, A); > M; GModule M of dimension 2 over A > M := WriteGModuleOver(R, A : Subfield); > M; GModule M of dimension 2 over Number Field with defining polynomial x^2 + 1 over the Rational Field ```