Coverage for local/lib/python2.7/site-packages/sage/algebras/group_algebra.py : 55%

r""" 

Group algebras 

This functionality has been moved to :mod:`sage.categories.algebra_functor`. 

TESTS: 

Check that unpicking old group algebra classes work:: 

sage: G = loads(b"x\x9cM\xceM\n\xc20\x10\x86a\xac\xff\xf1$n\xb2\xf1\x04\x82" 

....: b"\xe8>\xe0:\xc4fL\x83i\xda\x99$K\xc1M\xf5\xdaj\x1a\xc1\xdd<" 

....: b"\xf0\xbd0\x8f\xaa\x0e\xca\x00\x0f\x91R\x1d\x13\x01O\xdeb\x02I" 

....: b"\xd0\x13\x04\xf0QE\xdby\x96<\x81N50\x9c\x8c\x81r\x06.\xa4\x027" 

....: b"\xd4\xa5^\x16\xb2\xd3W\xfb\x02\xac\x9a\xb2\xce\xa3\xc0{\xa0V" 

....: b"\x9ar\x8c\xa1W-hv\xb0\rhR.\xe7\x0c\xa7cE\xd6\x9b\xc0\xad\x8f`" 

....: b"\x80X\xabn \x7f\xc0\xd9y\xb2\x1b\x04\xce\x87\xfb\x0b\x17\x02" 

....: b"\x97\xff\x05\xe5\x9f\x95\x93W\x0bN3Qx\xcc\xc2\xd5V\xe0\xfa\xf9" 

....: b"\xc9\x98\xc0\r\x7f\x03\x9d\xd7^'") 

sage: G 

Algebra of Dihedral group of order 6 as a permutation group over Rational Field 

sage: type(G) 

<class 'sage.algebras.group_algebra.GroupAlgebra_class_with_category'> 

""" 

#***************************************************************************** 

# Copyright (C) 2008 William Stein <wstein@gmail.com> 

# 2008 David Loeffler <d.loeffler.01@cantab.net> 

# 2009 Martin Raum <mraum@mpim-bonn.mpg.de> 

# 2011 John Palmieri <palmieri@math.washington.edu> 

# 

# This program is free software: you can redistribute it and/or modify 

# it under the terms of the GNU General Public License as published by 

# the Free Software Foundation, either version 2 of the License, or 

# (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from sage.rings.all import IntegerRing 

from sage.categories.all import Rings 

from sage.categories.magmas import Magmas 

from sage.categories.additive_magmas import AdditiveMagmas 

from sage.categories.sets_cat import Sets 

from sage.categories.morphism import SetMorphism 

from sage.combinat.free_module import CombinatorialFreeModule 

def GroupAlgebra(G, R=IntegerRing()): 

""" 

Return the group algebra of `G` over `R`. 

INPUT: 

- `G` -- a group 

- `R` -- (default: `\ZZ`) a ring 

EXAMPLES: 

The *group algebra* `A=RG` is the space of formal linear 

combinations of elements of `G` with coefficients in `R`:: 

sage: G = DihedralGroup(3) 

sage: R = QQ 

sage: A = GroupAlgebra(G, R); A 

Algebra of Dihedral group of order 6 as a permutation group over Rational Field 

sage: a = A.an_element(); a 

() + 4*(1,2,3) + 2*(1,3) 

This space is endowed with an algebra structure, obtained by extending 

by bilinearity the multiplication of `G` to a multiplication on `RG`:: 

sage: A in Algebras 

True 

sage: a * a 

5*() + 8*(2,3) + 8*(1,2) + 8*(1,2,3) + 16*(1,3,2) + 4*(1,3) 

:func:`GroupAlgebra` is just a short hand for a more general 

construction that covers, e.g., monoid algebras, additive group 

algebras and so on:: 

sage: G.algebra(QQ) 

Algebra of Dihedral group of order 6 as a permutation group over Rational Field 

sage: GroupAlgebra(G,QQ) is G.algebra(QQ) 

True 

sage: M = Monoids().example(); M 

An example of a monoid: 

the free monoid generated by ('a', 'b', 'c', 'd') 

sage: M.algebra(QQ) 

Algebra of An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd') 

over Rational Field 

See the documentation of :mod:`sage.categories.algebra_functor` 

for details. 

TESTS:: 

sage: GroupAlgebra(1) 

Traceback (most recent call last): 

... 

ValueError: 1 is not a magma or additive magma 

sage: GroupAlgebra(GL(3, GF(7))) 

Algebra of General Linear Group of degree 3 over Finite Field of size 7 

over Integer Ring 

sage: GroupAlgebra(GL(3, GF(7)), QQ) 

Algebra of General Linear Group of degree 3 over Finite Field of size 7 

over Rational Field 

""" 

if not (G in Magmas() or G in AdditiveMagmas()): 

raise ValueError("%s is not a magma or additive magma"%G) 

if not R in Rings(): 

raise ValueError("%s is not a ring"%R) 

return G.algebra(R) 

class GroupAlgebra_class(CombinatorialFreeModule): 

def _coerce_map_from_(self, S): 

r""" 

Return a coercion from `S` or ``None``. 

INPUT: 

- ``S`` -- a parent 

Let us write ``self`` as `R[G]`. This method handles 

the case where `S` is another group/monoid/...-algebra 

`R'[H]`, with R coercing into `R'` and `H` coercing 

into `G`. In that case it returns the naturally 

induced coercion between `R'[H]` and `R[G]`. Otherwise 

it returns ``None``. 

EXAMPLES:: 

sage: A = GroupAlgebra(SymmetricGroup(4), QQ) 

sage: B = GroupAlgebra(SymmetricGroup(3), ZZ) 

sage: A.has_coerce_map_from(B) 

True 

sage: B.has_coerce_map_from(A) 

False 

sage: A.has_coerce_map_from(ZZ) 

True 

sage: A.has_coerce_map_from(CC) 

False 

sage: A.has_coerce_map_from(SymmetricGroup(5)) 

False 

sage: A.has_coerce_map_from(SymmetricGroup(2)) 

True 

sage: H = CyclicPermutationGroup(3) 

sage: G = DihedralGroup(3) 

sage: QH = H.algebra(QQ) 

sage: ZH = H.algebra(ZZ) 

sage: QG = G.algebra(QQ) 

sage: ZG = G.algebra(ZZ) 

sage: ZG.coerce_map_from(H) 

Coercion map: 

From: Cyclic group of order 3 as a permutation group 

To: Algebra of Dihedral group of order 6 as a permutation group over Integer Ring 

sage: QG.coerce_map_from(ZG) 

Generic morphism: 

From: Algebra of Dihedral group of order 6 as a permutation group over Integer Ring 

To: Algebra of Dihedral group of order 6 as a permutation group over Rational Field 

sage: QG.coerce_map_from(QH) 

Generic morphism: 

From: Algebra of Cyclic group of order 3 as a permutation group over Rational Field 

To: Algebra of Dihedral group of order 6 as a permutation group over Rational Field 

sage: QG.coerce_map_from(ZH) 

Generic morphism: 

From: Algebra of Cyclic group of order 3 as a permutation group over Integer Ring 

To: Algebra of Dihedral group of order 6 as a permutation group over Rational Field 

As expected, there is no coercion when restricting the 

field:: 

sage: ZG.coerce_map_from(QG) 

This coercion when restricting the group is unexpected:: 

sage: QH.coerce_map_from(QG) 

Generic morphism: 

From: Algebra of Dihedral group of order 6 as a permutation group over Rational Field 

To: Algebra of Cyclic group of order 3 as a permutation group over Rational Field 

but is induced by the partial coercion at the level of 

the groups:: 

sage: H.coerce_map_from(G) 

Call morphism: 

From: Dihedral group of order 6 as a permutation group 

To: Cyclic group of order 3 as a permutation group 

There is no coercion for additive groups since ``+`` could mean 

both the action (i.e., the group operation) or adding a term:: 

sage: G = groups.misc.AdditiveCyclic(3) 

sage: ZG = G.algebra(ZZ, category=AdditiveMagmas()) 

sage: ZG.has_coerce_map_from(G) 

False 

""" 

G = self.basis().keys() 

K = self.base_ring() 

if G.has_coerce_map_from(S): 

from sage.categories.groups import Groups 

# No coercion for additive groups because of ambiguity of + 

# being the group action or addition of a new term. 

return self.category().is_subcategory(Groups().Algebras(K)) 

if S in Sets.Algebras: 

S_K = S.base_ring() 

S_G = S.basis().keys() 

hom_K = K.coerce_map_from(S_K) 

hom_G = G.coerce_map_from(S_G) 

if hom_K is not None and hom_G is not None: 

return SetMorphism(S.Hom(self, category=self.category() | S.category()), 

lambda x: self.sum_of_terms( (hom_G(g), hom_K(c)) for g,c in x )) 

from sage.structure.sage_object import register_unpickle_override 

register_unpickle_override('sage.algebras.group_algebras', 'GroupAlgebra', GroupAlgebra_class)