Finite dimensional modules with basis#
- class sage.categories.finite_dimensional_modules_with_basis.FiniteDimensionalModulesWithBasis(base_category)#
Bases:
CategoryWithAxiom_over_base_ringThe category of finite dimensional modules with a distinguished basis
EXAMPLES:
sage: C = FiniteDimensionalModulesWithBasis(ZZ); C Category of finite dimensional modules with basis over Integer Ring sage: sorted(C.super_categories(), key=str) [Category of finite dimensional modules over Integer Ring, Category of modules with basis over Integer Ring] sage: C is Modules(ZZ).WithBasis().FiniteDimensional() True
- class ElementMethods#
Bases:
object- dense_coefficient_list(order=None)#
Return a list of all coefficients of
self.By default, this list is ordered in the same way as the indexing set of the basis of the parent of
self.INPUT:
order– (optional) an ordering of the basis indexing set
EXAMPLES:
sage: # needs sage.modules sage: v = vector([0, -1, -3]) sage: v.dense_coefficient_list() [0, -1, -3] sage: v.dense_coefficient_list([2,1,0]) [-3, -1, 0] sage: sorted(v.coefficients()) [-3, -1]
- class MorphismMethods#
Bases:
object- image()#
Return the image of
selfas a submodule of the codomain.EXAMPLES:
sage: SGA = SymmetricGroupAlgebra(QQ, 3) # needs sage.groups sage.modules sage: f = SGA.module_morphism(lambda x: SGA(x**2), codomain=SGA) # needs sage.groups sage.modules sage: f.image() # needs sage.groups sage.modules Free module generated by {0, 1, 2} over Rational Field
- image_basis()#
Return a basis for the image of
selfin echelon form.EXAMPLES:
sage: SGA = SymmetricGroupAlgebra(QQ, 3) # needs sage.groups sage.modules sage: f = SGA.module_morphism(lambda x: SGA(x**2), codomain=SGA) # needs sage.groups sage.modules sage: f.image_basis() # needs sage.groups sage.modules ([1, 2, 3], [2, 3, 1], [3, 1, 2])
- kernel()#
Return the kernel of
selfas a submodule of the domain.EXAMPLES:
sage: # needs sage.groups sage.modules sage: SGA = SymmetricGroupAlgebra(QQ, 3) sage: f = SGA.module_morphism(lambda x: SGA(x**2), codomain=SGA) sage: K = f.kernel() sage: K Free module generated by {0, 1, 2} over Rational Field sage: K.ambient() Symmetric group algebra of order 3 over Rational Field
- kernel_basis()#
Return a basis of the kernel of
selfin echelon form.EXAMPLES:
sage: SGA = SymmetricGroupAlgebra(QQ, 3) # needs sage.groups sage.modules sage: f = SGA.module_morphism(lambda x: SGA(x**2), codomain=SGA) # needs sage.groups sage.modules sage: f.kernel_basis() # needs sage.groups sage.modules ([1, 2, 3] - [3, 2, 1], [1, 3, 2] - [3, 2, 1], [2, 1, 3] - [3, 2, 1])
- matrix(base_ring=None, side='left')#
Return the matrix of this morphism in the distinguished bases of the domain and codomain.
INPUT:
base_ring– a ring (default:None, meaning the base ring of the codomain)side– “left” or “right” (default: “left”)
If
sideis “left”, this morphism is considered as acting on the left; i.e. each column of the matrix represents the image of an element of the basis of the domain.The order of the rows and columns matches with the order in which the bases are enumerated.
See also
Modules.WithBasis.ParentMethods.module_morphism()EXAMPLES:
sage: # needs sage.modules sage: X = CombinatorialFreeModule(ZZ, [1,2]); x = X.basis() sage: Y = CombinatorialFreeModule(ZZ, [3,4]); y = Y.basis() sage: phi = X.module_morphism(on_basis={1: y[3] + 3*y[4], ....: 2: 2*y[3] + 5*y[4]}.__getitem__, ....: codomain=Y) sage: phi.matrix() [1 2] [3 5] sage: phi.matrix(side="right") [1 3] [2 5] sage: phi.matrix().parent() # needs sage.modules Full MatrixSpace of 2 by 2 dense matrices over Integer Ring sage: phi.matrix(QQ).parent() # needs sage.modules Full MatrixSpace of 2 by 2 dense matrices over Rational Field
The resulting matrix is immutable:
sage: phi.matrix().is_mutable() # needs sage.modules False
The zero morphism has a zero matrix:
sage: Hom(X, Y).zero().matrix() # needs sage.modules [0 0] [0 0]
Todo
Add support for morphisms where the codomain has a different base ring than the domain:
sage: Y = CombinatorialFreeModule(QQ, [3,4]); y = Y.basis() # needs sage.modules sage: phi = X.module_morphism(on_basis={1: y[3] + 3*y[4], # needs sage.modules ....: 2: 2*y[3] + 5/2*y[4]}.__getitem__, ....: codomain=Y) sage: phi.matrix().parent() # not implemented # needs sage.modules Full MatrixSpace of 2 by 2 dense matrices over Rational Field
This currently does not work because, in this case, the morphism is just in the category of commutative additive groups (i.e. the intersection of the categories of modules over \(\ZZ\) and over \(\QQ\)):
sage: phi.parent().homset_category() # needs sage.modules Category of commutative additive semigroups sage: phi.parent().homset_category() # not implemented, needs sage.modules Category of finite dimensional modules with basis over Integer Ring
- class ParentMethods#
Bases:
object- annihilator(S, action=<built-in function mul>, side='right', category=None)#
Return the annihilator of a finite set.
INPUT:
S– a finite setaction– a function (default:operator.mul)side– ‘left’ or ‘right’ (default: ‘right’)category– a category
Assumptions:
actiontakes elements ofselfas first argument and elements ofSas second argument;The codomain is any vector space, and
actionis linear on its first argument; typically it is bilinear;If
sideis ‘left’, this is reversed.
OUTPUT:
The subspace of the elements \(x\) of
selfsuch thataction(x,s) = 0for all \(s\in S\). Ifsideis ‘left’ replace the above equation byaction(s,x) = 0.If
selfis a ring,actionan action ofselfon a module \(M\) and \(S\) is a subset of \(M\), we recover the Wikipedia article Annihilator_%28ring_theory%29. Similarly this can be used to compute torsion or orthogonals.See also
annihilator_basis()for lots of examples.EXAMPLES:
sage: # needs sage.modules sage: F = FiniteDimensionalAlgebrasWithBasis(QQ).example(); F An example of a finite dimensional algebra with basis: the path algebra of the Kronecker quiver (containing the arrows a:x->y and b:x->y) over Rational Field sage: x, y, a, b = F.basis() sage: A = F.annihilator([a + 3*b + 2*y]); A Free module generated by {0} over Rational Field sage: [b.lift() for b in A.basis()] [-1/2*a - 3/2*b + x]
The category can be used to specify other properties of this subspace, like that this is a subalgebra:
sage: # needs sage.modules sage: center = F.annihilator(F.basis(), F.bracket, ....: category=Algebras(QQ).Subobjects()) sage: (e,) = center.basis() sage: e.lift() x + y sage: e * e == e True
Taking annihilator is order reversing for inclusion:
sage: # needs sage.modules sage: A = F.annihilator([]); A .rename("A") sage: Ax = F.annihilator([x]); Ax .rename("Ax") sage: Ay = F.annihilator([y]); Ay .rename("Ay") sage: Axy = F.annihilator([x,y]); Axy.rename("Axy") sage: P = Poset(([A, Ax, Ay, Axy], attrcall("is_submodule"))) # needs sage.combinat sage.graphs sage: sorted(P.cover_relations(), key=str) # needs sage.combinat sage.graphs [[Ax, A], [Axy, Ax], [Axy, Ay], [Ay, A]]
- annihilator_basis(S, action=<built-in function mul>, side='right')#
Return a basis of the annihilator of a finite set of elements.
INPUT:
S– a finite set of objectsaction– a function (default:operator.mul)side– ‘left’ or ‘right’ (default: ‘right’): on which side ofselfthe elements of \(S\) acts.
See
annihilator()for the assumptions and definition of the annihilator.EXAMPLES:
By default, the action is the standard \(*\) operation. So our first example is about an algebra:
sage: # needs sage.graphs sage.modules sage: F = FiniteDimensionalAlgebrasWithBasis(QQ).example(); F An example of a finite dimensional algebra with basis: the path algebra of the Kronecker quiver (containing the arrows a:x->y and b:x->y) over Rational Field sage: x,y,a,b = F.basis()
In this algebra, multiplication on the right by \(x\) annihilates all basis elements but \(x\):
sage: x*x, y*x, a*x, b*x # needs sage.graphs sage.modules (x, 0, 0, 0)
So the annihilator is the subspace spanned by \(y\), \(a\), and \(b\):
sage: F.annihilator_basis([x]) # needs sage.graphs sage.modules (y, a, b)
The same holds for \(a\) and \(b\):
sage: x*a, y*a, a*a, b*a # needs sage.graphs sage.modules (a, 0, 0, 0) sage: F.annihilator_basis([a]) # needs sage.graphs sage.modules (y, a, b)
On the other hand, \(y\) annihilates only \(x\):
sage: F.annihilator_basis([y]) # needs sage.graphs sage.modules (x,)
Here is a non trivial annihilator:
sage: F.annihilator_basis([a + 3*b + 2*y]) # needs sage.graphs sage.modules (-1/2*a - 3/2*b + x,)
Let’s check it:
sage: (-1/2*a - 3/2*b + x) * (a + 3*b + 2*y) # needs sage.graphs sage.modules 0
Doing the same calculations on the left exchanges the roles of \(x\) and \(y\):
sage: # needs sage.graphs sage.modules sage: F.annihilator_basis([y], side="left") (x, a, b) sage: F.annihilator_basis([a], side="left") (x, a, b) sage: F.annihilator_basis([b], side="left") (x, a, b) sage: F.annihilator_basis([x], side="left") (y,) sage: F.annihilator_basis([a + 3*b + 2*x], side="left") (-1/2*a - 3/2*b + y,)
By specifying an inner product, this method can be used to compute the orthogonal of a subspace:
sage: # needs sage.graphs sage.modules sage: x,y,a,b = F.basis() sage: def scalar(u,v): ....: return vector([sum(u[i]*v[i] for i in F.basis().keys())]) sage: F.annihilator_basis([x + y, a + b], scalar) (x - y, a - b)
By specifying the standard Lie bracket as action, one can compute the commutator of a subspace of \(F\):
sage: F.annihilator_basis([a + b], action=F.bracket) # needs sage.graphs sage.modules (x + y, a, b)
In particular one can compute a basis of the center of the algebra. In our example, it is reduced to the identity:
sage: F.annihilator_basis(F.algebra_generators(), action=F.bracket) # needs sage.graphs sage.modules (x + y,)
But see also
FiniteDimensionalAlgebrasWithBasis.ParentMethods.center_basis().
- echelon_form(elements, row_reduced=False, order=None)#
Return a basis in echelon form of the subspace spanned by a finite set of elements.
INPUT:
elements– a list or finite iterable of elements ofselfrow_reduced– (default:False) whether to compute the basis for the row reduced echelon formorder– (optional) either something that can be converted into a tuple or a key function
OUTPUT:
A list of elements of
selfwhose expressions as vectors form a matrix in echelon form. Ifbase_ringis specified, then the calculation is achieved in this base ring.EXAMPLES:
sage: # needs sage.modules sage: X = CombinatorialFreeModule(QQ, range(3), prefix="x") sage: x = X.basis() sage: V = X.echelon_form([x[0]-x[1], x[0]-x[2], x[1]-x[2]]); V [x[0] - x[2], x[1] - x[2]] sage: matrix(list(map(vector, V))) [ 1 0 -1] [ 0 1 -1]
sage: # needs sage.modules sage: F = CombinatorialFreeModule(ZZ, [1,2,3,4]) sage: B = F.basis() sage: elements = [B[1]-17*B[2]+6*B[3], B[1]-17*B[2]+B[4]] sage: F.echelon_form(elements) [B[1] - 17*B[2] + B[4], 6*B[3] - B[4]]
sage: F = CombinatorialFreeModule(QQ, ['a','b','c']) # needs sage.modules sage: a,b,c = F.basis() # needs sage.modules sage: F.echelon_form([8*a+b+10*c, -3*a+b-c, a-b-c]) # needs sage.modules [B['a'] + B['c'], B['b'] + 2*B['c']]
sage: R.<x,y> = QQ[] sage: C = CombinatorialFreeModule(R, range(3), prefix='x') # needs sage.modules sage: x = C.basis() # needs sage.modules sage: C.echelon_form([x[0] - x[1], 2*x[1] - 2*x[2], x[0] - x[2]]) # needs sage.modules sage.rings.function_field [x[0] - x[2], x[1] - x[2]]
sage: M = MatrixSpace(QQ, 3, 3) # needs sage.modules sage: A = M([[0, 0, 2], [0, 0, 0], [0, 0, 0]]) # needs sage.modules sage: M.echelon_form([A, A]) # needs sage.modules [ [0 0 1] [0 0 0] [0 0 0] ]
- from_vector(vector, order=None, coerce=True)#
Build an element of
selffrom a vector.EXAMPLES:
sage: # needs sage.modules sage: p_mult = matrix([[0,0,0], [0,0,-1], [0,0,0]]) sage: q_mult = matrix([[0,0,1], [0,0,0], [0,0,0]]) sage: A = algebras.FiniteDimensional( ....: QQ, [p_mult, q_mult, matrix(QQ, 3, 3)], 'p,q,z') sage: A.from_vector(vector([1,0,2])) p + 2*z
- gens()#
Return the generators of
self.OUTPUT:
A tuple containing the basis elements of
self.EXAMPLES:
sage: F = CombinatorialFreeModule(ZZ, ['a', 'b', 'c']) # needs sage.modules sage: F.gens() # needs sage.modules (B['a'], B['b'], B['c'])
- invariant_module(S, action=<built-in function mul>, action_on_basis=None, side='left', **kwargs)#
Return the submodule of
selfinvariant under the action ofS.For a semigroup \(S\) acting on a module \(M\), the invariant submodule is given by
\[M^S = \{m \in M : s \cdot m = m,\, \forall s \in S\}.\]INPUT:
S– a finitely-generated semigroupaction– a function (default:operator.mul)side–'left'or'right'(default:'right'); which side ofselfthe elements ofSactsaction_on_basis– (optional) define the action ofSon the basis ofself
OUTPUT:
EXAMPLES:
We build the invariant module of the permutation representation of the symmetric group:
sage: # needs sage.groups sage.modules sage: G = SymmetricGroup(3); G.rename('S3') sage: M = FreeModule(ZZ, [1,2,3], prefix='M'); M.rename('M') sage: action = lambda g, x: M.term(g(x)) sage: I = M.invariant_module(G, action_on_basis=action); I (S3)-invariant submodule of M sage: I.basis() Finite family {0: B[0]} sage: [I.lift(b) for b in I.basis()] [M[1] + M[2] + M[3]] sage: G.rename(); M.rename() # reset the names
We can construct the invariant module of any module that has an action of
S. In this example, we consider the dihedral group \(G = D_4\) and the subgroup \(H < G\) of all rotations. We construct the \(H\)-invariant module of the group algebra \(\QQ[G]\):sage: # needs sage.groups sage: G = groups.permutation.Dihedral(4) sage: H = G.subgroup(G.gen(0)) sage: H Subgroup generated by [(1,2,3,4)] of (Dihedral group of order 8 as a permutation group) sage: H.cardinality() 4 sage: # needs sage.groups sage.modules sage: A = G.algebra(QQ) sage: I = A.invariant_module(H) sage: [I.lift(b) for b in I.basis()] [() + (1,2,3,4) + (1,3)(2,4) + (1,4,3,2), (2,4) + (1,2)(3,4) + (1,3) + (1,4)(2,3)] sage: all(h * I.lift(b) == I.lift(b) ....: for b in I.basis() for h in H) True
- twisted_invariant_module(G, chi, action=<built-in function mul>, action_on_basis=None, side='left', **kwargs)#
Create the isotypic component of the action of
Gonselfwith irreducible character given bychi.See also
INPUT:
G– a finitely-generated groupchi– a list/tuple of character values or an instance ofClassFunction_gapaction– a function (default:operator.mul)action_on_basis– (optional) define the action ofgon the basis ofselfside–'left'or'right'(default:'right'); which side ofselfthe elements ofSacts
OUTPUT:
EXAMPLES:
sage: # needs sage.groups sage.modules sage: M = CombinatorialFreeModule(QQ, [1,2,3]) sage: G = SymmetricGroup(3) sage: def action(g,x): return(M.term(g(x))) # permute coordinates sage: T = M.twisted_invariant_module(G, [2,0,-1], ....: action_on_basis=action) sage: import __main__; __main__.action = action sage: TestSuite(T).run()
- class TensorProducts(category, *args)#
Bases:
TensorProductsCategory- extra_super_categories()#
Implement the fact that a (finite) tensor product of finite dimensional modules is a finite dimensional module.
EXAMPLES:
sage: C = ModulesWithBasis(ZZ).FiniteDimensional().TensorProducts() sage: C.extra_super_categories() [Category of finite dimensional modules with basis over Integer Ring] sage: C.FiniteDimensional() Category of tensor products of finite dimensional modules with basis over Integer Ring