Adding category options to Representations

[tkarn]

Add the option to give a specific choice of category for the Representation class.

[tkarn]

New commits:

​fafed31

Add _mul_ to allow for algebras and add ability for Representation class to take category as a keyword argument.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​e767248

Fix multiplication and add tests

[tscrim]

Looks good overall, but I don't think your representation of C₃ on the exterior algebra has the group acting by ring automorphims. In particular, I would expect g (x1 x2) = (g x1) (g x2) = x2 x0 = -x0 x2, but your code yields x0 x2. First there is a more fundamental issue I had to fix:

-sage: on_basis = lambda g,m: E.monomial(tuple(g(i+1)-1 for i in m[0])) #cyclically permute generators
+sage: on_basis = lambda g,m: M.monomial(tuple(g(i+1)-1 for i in m)) #cyclically permute generators

then I get

sage: g = G.gen()
sage: r = R.an_element(); r
1 + 2*x0 + x0*x1 + 3*x1
sage: g * r
1 + 2*x1 + x1*x2 + 3*x2
sage: g^2 * r
1 + 3*x0 + 2*x2 + x2*x0

So a little more care is needed with this doctest.

Also, PEP8 spacing:

-sage: R = Representation(G, M, on_basis, category = Algebras(QQ).WithBasis().FiniteDimensional())
+sage: R = Representation(G, M, on_basis, category=Algebras(QQ).WithBasis().FiniteDimensional())

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​374207f

Fix cyclic action on generators of ExteriorAlgebra, add tests for the multiplication of group elements on the group algebra as a representation

[tscrim]

This change should be reverted for PEP8 spacing:

-            sage: R = Representation(G, A, action, 'left', category=category)
+            sage: R = Representation(G, A, action, 'left', category = category)

You can also simplify tuple([g(j+1)-1]) -> (g(j+1)-1,)

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​5351ebb

Fix PEP8 spacing issue, simplify tuple

[tscrim]

LGTM.