# Modules over Basic Algebras A module $M$ over a basic algebra $B$ is presented as a sequence of matrices, one for each generator of the algebra. ## Indecomposable Projective Modules The indecomposable projective modules are defined from the structure of the algebra and have associated path trees that solve the homomorphism lifting problem. ### `ProjectiveModule(B, i): AlgBas, RngIntElt -> ModRng` The $i^{th}$ projective module of the basic algebra $B$. ### `PathTree(B, i): AlgBas, RngIntElt -> ModRng` The path tree of the $i^{th}$ projective module of the basic algebra $B$. ### `ActionGenerator(B, i): AlgBas, RngIntElt -> SeqEnum` The sequence of matrices for the generators of the basic algebra $B$ acting on the $i^{th}$ projective module of $B$. ### `IdempotentActionGenerators(B, i): AlgBas, RngIntElt -> SeqEnum` The sequence of matrices for the idempotent generators of the basic algebra $B$ acting on the $i^{th}$ projective module of $B$. ### `NonIdempotentActionGenerators(B, i): AlgBas, RngIntElt -> SeqEnum` The sequence of matrices for the nonidempotent generators of the basic algebra $B$ acting on the $i^{th}$ projective module of $B$. ### `Injection(B, i, v): AlgBas, RngIntElt, ModRngElt -> AlgBasElt` Given a vector $v$ in in the $i^{th}$ projective module of the basic algebra $B$, the function returns the image of inclusion of $v$ into $B$. ## Creation ### `AModule(B, Q): AlgBas, SeqEnum[AlgMatElt] -> ModRng` Given a basic algebra $B$ and a sequence $Q$ of elements in a matrix algebra the function returns the $B$-module $M$ on which the generators of $B$ act by multiplication by the corresponding elements of $Q$. ### `ProjectiveModule(B, S): AlgBas, SeqEnum[RngIntElt] -> ModAlg, SeqEnum, SeqEnum` Given a sequence $S = [s_1,s_2, ... ]$, the function returns a projective module which is the direct sum of $s_1$ copies of the first projective of the algebra $B$, $s_2$ copies of the second, etc. It also returns the sequence of inclusions and projections from and to the indecomposable projective modules. ### `IrreducibleModule(B, i): AlgBas, RngIntElt -> ModAlg` ### `SimpleModule(B, i): AlgBas, RngIntElt -> ModAlg` The $i^{th}$ irreducible module of the algebra $B$. The module is the quotient of the $i^{th}$ projective module by its radical. ### `ZeroModule(B): AlgBas -> ModAlg` The zero $B$-module. ### `RightRegularModule(B): AlgBas -> ModAlg` The algebra $B$ as a right module over itself. The module is the direct sum of the projectives modules of $B$. ### `RegularRepresentation(v): AlgBasElt -> AlgMatElt` If $v$ is an element of a basic algebra given as a vector in the underlying space, then the function computes the matrix of the action by right multiplication of the element on the algebra. ### `Restriction(M, B, xi): ModAlgBas, AlgBas, ModMatFldElt -> ModAlgBas` If $B$ is a subalgebra of the basic algebra $A$, $\xi$ is the embedding of $B$ into $A$, and $M$ is an $A$-module, then the function returns the restriction of M to a B-module. ### `ChangeAlgebra(M, B, xi): ModAlgBas, AlgBas, Map -> ModAlgBas` ### `ChangeAlgebra(M, B, xi): ModAlgBas, AlgBas, ModMatFldElt -> ModAlgBas` Given a module $M$ over an algebra $A$ and an algebra homomorphism $\xi$ from $B$ to $A$, the function returns the module $M$ as a $B$-module. ### `JacobsonRadical(M): ModAlg -> ModAlg` The Jacobson radical of the module $M$. ### `Socle(M): ModAlg -> ModAlg` The socle of the module $M$. The sum of the simple submodules of $M$. ## Access Functions ### `Algebra(M): ModAlg -> AlgBas` Given a module $M$ over a basic algebra $B$, the function returns $B$. ### `Dimension(M): ModAlg -> RngIntElt` The dimension of the module $M$ over its base ring. ### `Action(M): ModAlg -> AlgMat` The matrix algebra of the action of the algebra of $M$ on $M$. ### `IsomorphismTypesOfRadicalLayers(M): ModAlgBas -> SeqEnum` Given a module $M$ over a basic algebra, returns the sequence of isomorphism types of simple composition factors in each layer of the radical filtration of $M$. ### `IsomorphismTypesOfSocleLayers(M): ModAlgBas -> SeqEnum` Given a module $M$ over a basic algebra, returns a sequence of isomorphism types of simple composition factors in each socle layer with reversed order, *i. e.* isomorphism types of the socle of $M$ will appear last. ### `IsomorphismTypesOfBasicAlgebraSequence(S): SeqEnum -> SeqEnum` Given a sequence of irreducible modules $S$ for a basic algebra $A$, return a sequence of isomorphism types comparing with the simple modules of $A$. ### `Example: Restriction To Center (ex-6be5c5)` We show the restriction of a module over an algebra $A$ to a subalgebra of $A$. ```magma > G := SmallGroup(32,7); > A := BasicAlgebra(G); > C, mu := Center(A); > X := RightRegularModule(A); > Z := JacobsonRadical(X); > L := Restriction(Z,C,mu); > L; AModule L of dimension 31 over GF(2) > A eq Algebra(L); True > IndecomposableSummands(L); [ AModule of dimension 1 over GF(2), AModule of dimension 30 over GF(2) ] > Dimension(Socle(L)); 16 ``` Next we show how to pull back modules along a quotient map. We use the same algebra $A$. ```magma > U := ideal; > Q, theta := quo; > X := ProjectiveModule(Q,1); > Y := ChangeAlgebras(X,A,theta); > Y; AModule Y of dimension 16 over GF(2) ``` ### `Example: ChangeAlgebras 2 (ex-5bc3d0)` Here is another example of pulling back a module along a quotient map. This one involves algebras with more than one idempotent. ```magma > load m11; Loading "/usr/local/dmagma/libs/pergps/m11" M11 - Mathieu group on 11 letters - degree 11 Order 7 920 = 2^4 * 3^2 * 5 * 11; Base 1,2,3,4 Group: G > A:= BasicAlgebraOfPrincipalBlock(G,GF(2)); > A; Basic algebra of dimension 22 over GF(2) Number of projective modules: 3 Number of generators: 9 > DimensionsOfProjectiveModules(A); [ 8, 8, 6 ] > I := ideal; > B, mu := quo; > B; Basic algebra of dimension 6 over GF(2) Number of projective modules: 2 Number of generators: 5 > P := ProjectiveModule(B,1); > P; AModule P of dimension 3 over GF(2) > Q := ChangeAlgebras(P,A,mu); > Algebra(Q) eq A; true ``` ### `Example: Radical Layers (ex-e824ff)` In this example, we investigate the structure of the projective modules of a basic algebra. ```magma > G := PSL(3,3); > N := Normalizer(G,Sylow(G,2)); > A := BasicAlgebraOfHeckeAlgebra(G,N,GF(2)); > DimensionsOfProjectiveModules(A); [ 1, 2, 3, 9, 9, 1, 1, 1, 1 ] > IsomorphismTypesOfRadicalLayers(ProjectiveModule(A,4)); [ [ 4 ], [ 2, 3, 4, 5 ], [ 4, 4, 5 ], [ 5 ] ] > IsomorphismTypesOfSocleLayers(ProjectiveModule(A,4)); [ [ 4 ], [ 3, 4, 5 ], [ 2, 4, 5 ], [ 4, 5 ] ] ``` So we see that, unlike a group algebra, a Hecke algebra can have indecomposable projective modules whose socles are not simple. ## Predicates The following functions return a boolean value. ### `IsSemisimple(M): ModAlg -> BoolElt, SeqEnum` Returns `true` if the module $M$ is a semisimple module and `false` otherwise. If `true`, then the function also returns a list of the ranks of the primitive idempotents of the algebra. This is also a list of the multiplicities of the simple modules of the algebra as composition factors in a composition series for the module. ### `IsProjective(M): ModAlg -> BoolElt, SeqEnum` Returns `true` if the module $M$ is projective. The function also returns a sequence of multiplicities of the standard projective modules as direct summands of the projective cover of $M$. ### `IsInjective(M): ModAlg -> BoolElt, SeqEnum` Returns `true` if the module $M$ is injective. The function also returns a sequence of multiplicities of the standard injective modules as direct summands of the injective hull of $M$. ## Elementary Operations ### `m * b: ModAlgElt, AlgBasElt -> ModAlgElt` Given an element $b$ in a basic algebra $B$ and an element $m$ in a module $M$ over $B$, $m*b$ is the product. ### `Example: A Modules (ex-bb4604)` We obtain the dimensions of the radical layers of the group algebra of an extra special group of order 243 over a field of characteristic 3. ```magma > G := ExtraSpecialGroup(3,2); > G; Permutation group G acting on a set of cardinality 243 > ff := GF(3); > A := BasicAlgebra(G,ff); > A; Basic algebra of dimension 243 over GF(3) Number of projective modules: 1 Number of generators: 6 > P := ProjectiveModule(A,1); > P; AModule P of dimension 243 over GF(3) > R := JacobsonRadical(P); > R; AModule R of dimension 242 over GF(3) > while Dimension(R) ne 0 do > T := JacobsonRadical(R); > print Dimension(R) - Dimension(T); > R := T; > end while; 4 11 20 30 36 39 36 30 20 11 4 1 ``` ### `Example: AModules 2 (ex-2f301a)` We consider the mod-2 group algebra of an extraspecial group of order 128, and construct the module induced from the trivial module on the subgroups of order 4 generated by the first generator of the group. ```magma > G := ExtraSpecialGroup(2,3); > G; Permutation group G acting on a set of cardinality 128 > F := GF(2); > A := BasicAlgebra(G,F); > A; Basic algebra of dimension 128 over GF(2) Number of projective modules: 1 Number of generators: 8 > A.1; (1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0) ``` Note that `A.1` is the unique idempotent (identity element) in the group algebra, whereas `A.2` is `G.1 - 1` where `G.1` is the first generator of the group. ```magma > A.2; (0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0) > g := A.1+A.2; ``` Now we check the order of `g`. ```magma > g^2; (1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0) > g^4; (1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0) > g^4 eq A!1; true ``` So `g` has order 4. ```magma > P := ProjectiveModule(A,1); > P; AModule P of dimension 128 over GF(2) ``` Note that `P` is generated by `P.1` which corresponds to the identity element of `A` if we think of `P` as the algebra `A` as a module over itself. Now we create the induced module as the submodule generated by $(\hbox{\tt g}-1)^3$, since $(\hbox{\tt g}-1)^4 = 0$. ```magma > U := sub; > U; AModule U of dimension 32 over GF(2) ``` Because the dimension is a quarter of the order of the group we can be sure that we have the right thing by just checking that `U` is generated by a `g` fixed point. ```magma > U.1*g eq U.1; true ```