Exterior powers of free modules¶

Given a free module \(M\) of finite rank over a commutative ring \(R\) and a positive integer \(p\), the \(p\)-th exterior power of \(M\) is the set \(\Lambda^p(M)\) of all alternating contravariant tensors of degree \(p\) on \(M\), i.e. of all multilinear maps

\[\underbrace{M^*\times\cdots\times M^*}_{p\ \; \mbox{times}} \longrightarrow R\]

that vanish whenever any of two of their arguments are equal (\(M^*\) stands for the dual of \(M\)). Note that \(\Lambda^1(M) = M\). The exterior power \(\Lambda^p(M)\) is a free module of rank \(\binom{n}{p}\) over \(R\), where \(n\) is the rank of \(M\).

Similarly, the \(p\)-th exterior power of the dual of \(M\) is the set \(\Lambda^p(M^*)\) of all alternating forms of degree \(p\) on \(M\), i.e. of all multilinear maps

\[\underbrace{M\times\cdots\times M}_{p\ \; \mbox{times}} \longrightarrow R\]

that vanish whenever any of two of their arguments are equal. Note that \(\Lambda^1(M^*) = M^*\) (the dual of \(M\)). The exterior power \(\Lambda^p(M^*)\) is a free module of rank \(\binom{n}{p}\) over \(R\), where \(n\) is the rank of \(M\).

The class ExtPowerFreeModule implements \(\Lambda^p(M)\), while the class ExtPowerDualFreeModule implements \(\Lambda^p(M^*)\).

AUTHORS:

Eric Gourgoulhon: initial version, regarding \(\Lambda^p(M^*)\) only (2015); add class for \(\Lambda^p(M)\) (2017)

REFERENCES:

K. Conrad: Exterior powers [Con2013]
Chap. 19 of S. Lang: Algebra [Lan2002]

class sage.tensor.modules.ext_pow_free_module.ExtPowerDualFreeModule(fmodule, degree, name=None, latex_name=None)[source]¶

Bases: FiniteRankFreeModule_abstract

Exterior power of the dual of a free module of finite rank over a commutative ring.

Given a free module \(M\) of finite rank over a commutative ring \(R\) and a positive integer \(p\), the \(p\)-th exterior power of the dual of \(M\) is the set \(\Lambda^p(M^*)\) of all alternating forms of degree \(p\) on \(M\), i.e. of all multilinear maps

\[\underbrace{M\times\cdots\times M}_{p\ \; \mbox{times}} \longrightarrow R\]

that vanish whenever any of two of their arguments are equal. Note that \(\Lambda^1(M^*) = M^*\) (the dual of \(M\)).

\(\Lambda^p(M^*)\) is a free module of rank \(\binom{n}{p}\) over \(R\), where \(n\) is the rank of \(M\). Accordingly, the class ExtPowerDualFreeModule inherits from the class FiniteRankFreeModule_abstract.

This is a Sage parent class, whose element class is FreeModuleAltForm.

INPUT:

fmodule – free module \(M\) of finite rank, as an instance of FiniteRankFreeModule
degree – positive integer; the degree \(p\) of the alternating forms
name – (default: None) string; name given to \(\Lambda^p(M^*)\)
latex_name – (default: None) string; LaTeX symbol to denote \(\Lambda^p(M^*)\)

EXAMPLES:

2nd exterior power of the dual of a free \(\ZZ\)-module of rank 3:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e')
sage: from sage.tensor.modules.ext_pow_free_module import ExtPowerDualFreeModule
sage: A = ExtPowerDualFreeModule(M, 2) ; A
2nd exterior power of the dual of the Rank-3 free module M over the
 Integer Ring

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> e = M.basis('e')
>>> from sage.tensor.modules.ext_pow_free_module import ExtPowerDualFreeModule
>>> A = ExtPowerDualFreeModule(M, Integer(2)) ; A
2nd exterior power of the dual of the Rank-3 free module M over the
 Integer Ring

Instead of importing ExtPowerDualFreeModule in the global name space, it is recommended to use the module’s method dual_exterior_power():

Sage

sage: A = M.dual_exterior_power(2) ; A
2nd exterior power of the dual of the Rank-3 free module M over the
 Integer Ring
sage: latex(A)
\Lambda^{2}\left(M^*\right)

Python

>>> from sage.all import *
>>> A = M.dual_exterior_power(Integer(2)) ; A
2nd exterior power of the dual of the Rank-3 free module M over the
 Integer Ring
>>> latex(A)
\Lambda^{2}\left(M^*\right)

A is a module (actually a free module) over \(\ZZ\):

Sage

sage: A.category()
Category of finite dimensional modules over Integer Ring
sage: A in Modules(ZZ)
True
sage: A.rank()
3
sage: A.base_ring()
Integer Ring
sage: A.base_module()
Rank-3 free module M over the Integer Ring

Python

>>> from sage.all import *
>>> A.category()
Category of finite dimensional modules over Integer Ring
>>> A in Modules(ZZ)
True
>>> A.rank()
3
>>> A.base_ring()
Integer Ring
>>> A.base_module()
Rank-3 free module M over the Integer Ring

A is a parent object, whose elements are alternating forms, represented by instances of the class FreeModuleAltForm:

Sage

sage: a = A.an_element() ; a
Alternating form of degree 2 on the Rank-3 free module M over the
 Integer Ring
sage: a.display() # expansion with respect to M's default basis (e)
e^0∧e^1
sage: from sage.tensor.modules.free_module_alt_form import FreeModuleAltForm
sage: isinstance(a, FreeModuleAltForm)
True
sage: a in A
True
sage: A.is_parent_of(a)
True

Python

>>> from sage.all import *
>>> a = A.an_element() ; a
Alternating form of degree 2 on the Rank-3 free module M over the
 Integer Ring
>>> a.display() # expansion with respect to M's default basis (e)
e^0∧e^1
>>> from sage.tensor.modules.free_module_alt_form import FreeModuleAltForm
>>> isinstance(a, FreeModuleAltForm)
True
>>> a in A
True
>>> A.is_parent_of(a)
True

Elements can be constructed from A. In particular, 0 yields the zero element of A:

Sage

sage: A(0)
Alternating form zero of degree 2 on the Rank-3 free module M over the
 Integer Ring
sage: A(0) is A.zero()
True

Python

>>> from sage.all import *
>>> A(Integer(0))
Alternating form zero of degree 2 on the Rank-3 free module M over the
 Integer Ring
>>> A(Integer(0)) is A.zero()
True

while nonzero elements are constructed by providing their components in a given basis:

Sage

sage: e
Basis (e_0,e_1,e_2) on the Rank-3 free module M over the Integer Ring
sage: comp = [[0,3,-1],[-3,0,4],[1,-4,0]]
sage: a = A(comp, basis=e, name='a') ; a
Alternating form a of degree 2 on the Rank-3 free module M over the
 Integer Ring
sage: a.display(e)
a = 3 e^0∧e^1 - e^0∧e^2 + 4 e^1∧e^2

Python

>>> from sage.all import *
>>> e
Basis (e_0,e_1,e_2) on the Rank-3 free module M over the Integer Ring
>>> comp = [[Integer(0),Integer(3),-Integer(1)],[-Integer(3),Integer(0),Integer(4)],[Integer(1),-Integer(4),Integer(0)]]
>>> a = A(comp, basis=e, name='a') ; a
Alternating form a of degree 2 on the Rank-3 free module M over the
 Integer Ring
>>> a.display(e)
a = 3 e^0∧e^1 - e^0∧e^2 + 4 e^1∧e^2

An alternative is to construct the alternating form from an empty list of components and to set the nonzero components afterwards:

Sage

sage: a = A([], name='a')
sage: a.set_comp(e)[0,1] = 3
sage: a.set_comp(e)[0,2] = -1
sage: a.set_comp(e)[1,2] = 4
sage: a.display(e)
a = 3 e^0∧e^1 - e^0∧e^2 + 4 e^1∧e^2

Python

>>> from sage.all import *
>>> a = A([], name='a')
>>> a.set_comp(e)[Integer(0),Integer(1)] = Integer(3)
>>> a.set_comp(e)[Integer(0),Integer(2)] = -Integer(1)
>>> a.set_comp(e)[Integer(1),Integer(2)] = Integer(4)
>>> a.display(e)
a = 3 e^0∧e^1 - e^0∧e^2 + 4 e^1∧e^2

The exterior powers are unique:

Sage

sage: A is M.dual_exterior_power(2)
True

Python

>>> from sage.all import *
>>> A is M.dual_exterior_power(Integer(2))
True

The exterior power \(\Lambda^1(M^*)\) is nothing but \(M^*\):

Sage

sage: M.dual_exterior_power(1) is M.dual()
True
sage: M.dual()
Dual of the Rank-3 free module M over the Integer Ring
sage: latex(M.dual())
M^*

Python

>>> from sage.all import *
>>> M.dual_exterior_power(Integer(1)) is M.dual()
True
>>> M.dual()
Dual of the Rank-3 free module M over the Integer Ring
>>> latex(M.dual())
M^*

It also coincides with the module of type-\((0,1)\) tensors:

Sage

sage: M.dual_exterior_power(1) is M.tensor_module(0,1)
True

Python

>>> from sage.all import *
>>> M.dual_exterior_power(Integer(1)) is M.tensor_module(Integer(0),Integer(1))
True

For a degree \(p\geq 2\), there is a coercion map \(\Lambda^p(M^*)\rightarrow T^{(0,p)}(M)\):

Sage

sage: T02 = M.tensor_module(0,2) ; T02
Free module of type-(0,2) tensors on the Rank-3 free module M over the
 Integer Ring
sage: T02.has_coerce_map_from(A)
True
sage: A.has_coerce_map_from(T02)
False

Python

>>> from sage.all import *
>>> T02 = M.tensor_module(Integer(0),Integer(2)) ; T02
Free module of type-(0,2) tensors on the Rank-3 free module M over the
 Integer Ring
>>> T02.has_coerce_map_from(A)
True
>>> A.has_coerce_map_from(T02)
False

The coercion map \(\Lambda^2(M^*)\rightarrow T^{(0,2)}(M)\) in action:

Sage

sage: ta = T02(a) ; ta
Type-(0,2) tensor a on the Rank-3 free module M over the Integer Ring
sage: ta.display(e)
a = 3 e^0⊗e^1 - e^0⊗e^2 - 3 e^1⊗e^0 + 4 e^1⊗e^2 + e^2⊗e^0 - 4 e^2⊗e^1
sage: a.display(e)
a = 3 e^0∧e^1 - e^0∧e^2 + 4 e^1∧e^2
sage: ta.symmetries() # the antisymmetry is of course preserved
no symmetry;  antisymmetry: (0, 1)

Python

>>> from sage.all import *
>>> ta = T02(a) ; ta
Type-(0,2) tensor a on the Rank-3 free module M over the Integer Ring
>>> ta.display(e)
a = 3 e^0⊗e^1 - e^0⊗e^2 - 3 e^1⊗e^0 + 4 e^1⊗e^2 + e^2⊗e^0 - 4 e^2⊗e^1
>>> a.display(e)
a = 3 e^0∧e^1 - e^0∧e^2 + 4 e^1∧e^2
>>> ta.symmetries() # the antisymmetry is of course preserved
no symmetry;  antisymmetry: (0, 1)

Element[source]¶: alias of FreeModuleAltForm

base_module()[source]¶

Return the free module on which self is constructed.

OUTPUT:

instance of FiniteRankFreeModule representing the free module on which the exterior power is defined.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 5, name='M')
sage: A = M.dual_exterior_power(2)
sage: A.base_module()
Rank-5 free module M over the Integer Ring
sage: A.base_module() is M
True

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(5), name='M')
>>> A = M.dual_exterior_power(Integer(2))
>>> A.base_module()
Rank-5 free module M over the Integer Ring
>>> A.base_module() is M
True

construction()[source]¶

Return the functorial construction of self.

This implementation just returns None, as no functorial construction is implemented.

degree()[source]¶

Return the degree of self.

OUTPUT:

integer \(p\) such that self is the exterior power \(\Lambda^p(M^*)\)

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 5, name='M')
sage: A = M.dual_exterior_power(2)
sage: A.degree()
2
sage: M.dual_exterior_power(4).degree()
4

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(5), name='M')
>>> A = M.dual_exterior_power(Integer(2))
>>> A.degree()
2
>>> M.dual_exterior_power(Integer(4)).degree()
4

zero()[source]¶

Return the zero of self.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e')
sage: A = M.dual_exterior_power(2)
sage: A.zero()
Alternating form zero of degree 2 on the Rank-3 free module M over
 the Integer Ring
sage: A(0) is A.zero()
True

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> e = M.basis('e')
>>> A = M.dual_exterior_power(Integer(2))
>>> A.zero()
Alternating form zero of degree 2 on the Rank-3 free module M over
 the Integer Ring
>>> A(Integer(0)) is A.zero()
True

class sage.tensor.modules.ext_pow_free_module.ExtPowerFreeModule(fmodule, degree, name=None, latex_name=None)[source]¶

Bases: FiniteRankFreeModule_abstract

Exterior power of a free module of finite rank over a commutative ring.

\[\underbrace{M^*\times\cdots\times M^*}_{p\ \; \mbox{times}} \longrightarrow R\]

that vanish whenever any of two of their arguments are equal. Note that \(\Lambda^1(M) = M\).

\(\Lambda^p(M)\) is a free module of rank \(\binom{n}{p}\) over \(R\), where \(n\) is the rank of \(M\). Accordingly, the class ExtPowerFreeModule inherits from the class FiniteRankFreeModule_abstract.

This is a Sage parent class, whose element class is AlternatingContrTensor

INPUT:

fmodule – free module \(M\) of finite rank, as an instance of FiniteRankFreeModule
degree – positive integer; the degree \(p\) of the alternating elements
name – (default: None) string; name given to \(\Lambda^p(M)\)
latex_name – (default: None) string; LaTeX symbol to denote \(\Lambda^p(M)\)

EXAMPLES:

2nd exterior power of the dual of a free \(\ZZ\)-module of rank 3:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e')
sage: from sage.tensor.modules.ext_pow_free_module import ExtPowerFreeModule
sage: A = ExtPowerFreeModule(M, 2) ; A
2nd exterior power of the Rank-3 free module M over the
 Integer Ring

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> e = M.basis('e')
>>> from sage.tensor.modules.ext_pow_free_module import ExtPowerFreeModule
>>> A = ExtPowerFreeModule(M, Integer(2)) ; A
2nd exterior power of the Rank-3 free module M over the
 Integer Ring

Instead of importing ExtPowerFreeModule in the global name space, it is recommended to use the module’s method exterior_power():

Sage

sage: A = M.exterior_power(2) ; A
2nd exterior power of the Rank-3 free module M over the
 Integer Ring
sage: latex(A)
\Lambda^{2}\left(M\right)

Python

>>> from sage.all import *
>>> A = M.exterior_power(Integer(2)) ; A
2nd exterior power of the Rank-3 free module M over the
 Integer Ring
>>> latex(A)
\Lambda^{2}\left(M\right)

A is a module (actually a free module) over \(\ZZ\):

Sage

sage: A.category()
Category of finite dimensional modules over Integer Ring
sage: A in Modules(ZZ)
True
sage: A.rank()
3
sage: A.base_ring()
Integer Ring
sage: A.base_module()
Rank-3 free module M over the Integer Ring

Python

>>> from sage.all import *
>>> A.category()
Category of finite dimensional modules over Integer Ring
>>> A in Modules(ZZ)
True
>>> A.rank()
3
>>> A.base_ring()
Integer Ring
>>> A.base_module()
Rank-3 free module M over the Integer Ring

A is a parent object, whose elements are alternating contravariant tensors, represented by instances of the class AlternatingContrTensor:

Sage

sage: a = A.an_element() ; a
Alternating contravariant tensor of degree 2 on the Rank-3 free
 module M over the Integer Ring
sage: a.display() # expansion with respect to M's default basis (e)
e_0∧e_1
sage: from sage.tensor.modules.alternating_contr_tensor import AlternatingContrTensor
sage: isinstance(a, AlternatingContrTensor)
True
sage: a in A
True
sage: A.is_parent_of(a)
True

Python

>>> from sage.all import *
>>> a = A.an_element() ; a
Alternating contravariant tensor of degree 2 on the Rank-3 free
 module M over the Integer Ring
>>> a.display() # expansion with respect to M's default basis (e)
e_0∧e_1
>>> from sage.tensor.modules.alternating_contr_tensor import AlternatingContrTensor
>>> isinstance(a, AlternatingContrTensor)
True
>>> a in A
True
>>> A.is_parent_of(a)
True

Elements can be constructed from A. In particular, 0 yields the zero element of A:

Sage

sage: A(0)
Alternating contravariant tensor zero of degree 2 on the Rank-3
 free module M over the Integer Ring
sage: A(0) is A.zero()
True

Python

>>> from sage.all import *
>>> A(Integer(0))
Alternating contravariant tensor zero of degree 2 on the Rank-3
 free module M over the Integer Ring
>>> A(Integer(0)) is A.zero()
True

while nonzero elements are constructed by providing their components in a given basis:

Sage

sage: e
Basis (e_0,e_1,e_2) on the Rank-3 free module M over the Integer Ring
sage: comp = [[0,3,-1],[-3,0,4],[1,-4,0]]
sage: a = A(comp, basis=e, name='a') ; a
Alternating contravariant tensor a of degree 2 on the Rank-3
 free module M over the Integer Ring
sage: a.display(e)
a = 3 e_0∧e_1 - e_0∧e_2 + 4 e_1∧e_2

Python

>>> from sage.all import *
>>> e
Basis (e_0,e_1,e_2) on the Rank-3 free module M over the Integer Ring
>>> comp = [[Integer(0),Integer(3),-Integer(1)],[-Integer(3),Integer(0),Integer(4)],[Integer(1),-Integer(4),Integer(0)]]
>>> a = A(comp, basis=e, name='a') ; a
Alternating contravariant tensor a of degree 2 on the Rank-3
 free module M over the Integer Ring
>>> a.display(e)
a = 3 e_0∧e_1 - e_0∧e_2 + 4 e_1∧e_2

An alternative is to construct the alternating contravariant tensor from an empty list of components and to set the nonzero components afterwards:

Sage

sage: a = A([], name='a')
sage: a.set_comp(e)[0,1] = 3
sage: a.set_comp(e)[0,2] = -1
sage: a.set_comp(e)[1,2] = 4
sage: a.display(e)
a = 3 e_0∧e_1 - e_0∧e_2 + 4 e_1∧e_2

Python

>>> from sage.all import *
>>> a = A([], name='a')
>>> a.set_comp(e)[Integer(0),Integer(1)] = Integer(3)
>>> a.set_comp(e)[Integer(0),Integer(2)] = -Integer(1)
>>> a.set_comp(e)[Integer(1),Integer(2)] = Integer(4)
>>> a.display(e)
a = 3 e_0∧e_1 - e_0∧e_2 + 4 e_1∧e_2

The exterior powers are unique:

Sage

sage: A is M.exterior_power(2)
True

Python

>>> from sage.all import *
>>> A is M.exterior_power(Integer(2))
True

The exterior power \(\Lambda^1(M)\) is nothing but \(M\):

Sage

sage: M.exterior_power(1) is M
True

Python

>>> from sage.all import *
>>> M.exterior_power(Integer(1)) is M
True

For a degree \(p\geq 2\), there is a coercion \(\Lambda^p(M)\rightarrow T^{(p,0)}(M)\):

Sage

sage: T20 = M.tensor_module(2,0) ; T20
Free module of type-(2,0) tensors on the Rank-3 free module M
 over the Integer Ring
sage: T20.has_coerce_map_from(A)
True

Python

>>> from sage.all import *
>>> T20 = M.tensor_module(Integer(2),Integer(0)) ; T20
Free module of type-(2,0) tensors on the Rank-3 free module M
 over the Integer Ring
>>> T20.has_coerce_map_from(A)
True

Of course, there is no coercion in the reverse direction:

Sage

sage: A.has_coerce_map_from(T20)
False

Python

>>> from sage.all import *
>>> A.has_coerce_map_from(T20)
False

The coercion map \(\Lambda^2(M)\rightarrow T^{(2,0)}(M)\) in action:

Sage

sage: ta = T20(a) ; ta
Type-(2,0) tensor a on the Rank-3 free module M over the Integer Ring
sage: ta.display(e)
a = 3 e_0⊗e_1 - e_0⊗e_2 - 3 e_1⊗e_0 + 4 e_1⊗e_2 + e_2⊗e_0 - 4 e_2⊗e_1
sage: a.display(e)
a = 3 e_0∧e_1 - e_0∧e_2 + 4 e_1∧e_2
sage: ta.symmetries()  # the antisymmetry is of course preserved
no symmetry;  antisymmetry: (0, 1)
sage: ta == a  # equality as type-(2,0) tensors
True

Python

>>> from sage.all import *
>>> ta = T20(a) ; ta
Type-(2,0) tensor a on the Rank-3 free module M over the Integer Ring
>>> ta.display(e)
a = 3 e_0⊗e_1 - e_0⊗e_2 - 3 e_1⊗e_0 + 4 e_1⊗e_2 + e_2⊗e_0 - 4 e_2⊗e_1
>>> a.display(e)
a = 3 e_0∧e_1 - e_0∧e_2 + 4 e_1∧e_2
>>> ta.symmetries()  # the antisymmetry is of course preserved
no symmetry;  antisymmetry: (0, 1)
>>> ta == a  # equality as type-(2,0) tensors
True

Element[source]¶: alias of AlternatingContrTensor

base_module()[source]¶

Return the free module on which self is constructed.

OUTPUT:

instance of FiniteRankFreeModule representing the free module on which the exterior power is defined.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 5, name='M')
sage: A = M.exterior_power(2)
sage: A.base_module()
Rank-5 free module M over the Integer Ring
sage: A.base_module() is M
True

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(5), name='M')
>>> A = M.exterior_power(Integer(2))
>>> A.base_module()
Rank-5 free module M over the Integer Ring
>>> A.base_module() is M
True

construction()[source]¶

Return the functorial construction of self.

This implementation just returns None, as no functorial construction is implemented.

degree()[source]¶

Return the degree of self.

OUTPUT:

integer \(p\) such that self is the exterior power \(\Lambda^p(M)\)

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 5, name='M')
sage: A = M.exterior_power(2)
sage: A.degree()
2
sage: M.exterior_power(4).degree()
4

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(5), name='M')
>>> A = M.exterior_power(Integer(2))
>>> A.degree()
2
>>> M.exterior_power(Integer(4)).degree()
4

zero()[source]¶

Return the zero of self.

EXAMPLES:

Sage

sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
sage: e = M.basis('e')
sage: A = M.exterior_power(2)
sage: A.zero()
Alternating contravariant tensor zero of degree 2 on the Rank-3 free
 module M over the Integer Ring
sage: A(0) is A.zero()
True

Python

>>> from sage.all import *
>>> M = FiniteRankFreeModule(ZZ, Integer(3), name='M')
>>> e = M.basis('e')
>>> A = M.exterior_power(Integer(2))
>>> A.zero()
Alternating contravariant tensor zero of degree 2 on the Rank-3 free
 module M over the Integer Ring
>>> A(Integer(0)) is A.zero()
True