Support the generic Steenrod algebra at the prime 2

[cnassau]

For some applications one would like to work with the "odd primary Steenrod algebra at the prime 2". The attached patch introduces a new keyword "generic" that allows to do this:

  sage: EA=SteenrodAlgebra(p=2,generic=True) ; EA
  generic mod 2 Steenrod algebra, milnor basis
  sage: EA[8]
  Vector space spanned by (Q_0 Q_2, Q_0 Q_1 P(2), P(1,1), P(4)) over Finite Field of size 2
  sage: SteenrodAlgebra(p=2,generic=True,basis='adem')[7]
  Vector space spanned by (beta P^2 P^1, beta P^3, P^2 P^1 beta, P^3 beta) over Finite Field of size 2

[jhpalmieri]

New commits:

​8c59a4e

TSK16271 - support the generic Steenrod algebra at the prime 2

[git]

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

​c936a59	Merge branch 'develop' of trac.sagemath.org:sage into generic_steenrod
​7c8315e	merged 6.2.rc1

[git]

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

​5f8d71c

Merge branch 'develop' of git://github.com/sagemath/sage into generic_steenrod

[git]

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

​df429ca

more tests and minor improvements

[cnassau]

The code seems functional now. I've also added an is_generic method to allow inspection of the '_generic' attribute.

One thing that might be problematic about this patch is that a few routines have changed their signature now: 'serre_cartan_mono_to_string', for example, now expects 'generic' as second argument, not 'p'. We might instead go for

def serre_cartan_mono_to_string(data,p,generic=None)

which would save backward compatibility... I'm not sure if that is required, though.

[git]

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

​db6527e

Merge branch 'develop' of git://github.com/sagemath/sage into generic_steenrod

[jhpalmieri]

I'm not getting this to work. When I evaluate

sage: SteenrodAlgebra(p=2,generic=True)

I get an error:

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-1-7bd50008c6f7> in <module>()
----> 1 EA = SteenrodAlgebra(p=Integer(2),generic=True) ; EA

/Users/palmieri/Desktop/Sage_stuff/git/sage/local/lib/python2.7/site-packages/sage/algebras/steenrod/steenrod_algebra.pyc in SteenrodAlgebra(p, basis, generic, **kwds)
   4122         return SteenrodAlgebra_mod_two(p=2, basis=basis, **kwds)
   4123     else:
-> 4124         return SteenrodAlgebra_generic(p=p, basis=basis, generic=True, **kwds)
   4125 
   4126 

/Users/palmieri/Desktop/Sage_stuff/git/sage/local/lib/python2.7/site-packages/sage/misc/classcall_metaclass.so in sage.misc.classcall_metaclass.ClasscallMetaclass.__call__ (sage/misc/classcall_metaclass.c:1282)()

/Users/palmieri/Desktop/Sage_stuff/git/sage/local/lib/python2.7/site-packages/sage/algebras/steenrod/steenrod_algebra.pyc in __classcall__(self, p, basis, **kwds)
    525 
    526         std_basis = get_basis_name(basis, p, generic=std_generic)
--> 527         std_profile, std_type = normalize_profile(profile, precision=precision, truncation_type=truncation_type, p=p, generic=std_generic)
    528         return super(SteenrodAlgebra_generic, self).__classcall__(self, p=p, basis=std_basis, profile=std_profile,
    529                                                                   truncation_type=std_type, generic=std_generic)

/Users/palmieri/Desktop/Sage_stuff/git/sage/local/lib/python2.7/site-packages/sage/algebras/steenrod/steenrod_algebra_misc.pyc in normalize_profile(profile, precision, truncation_type, p, generic)
    550                     k = k[:-1]
    551             new_profile = (e, k)
--> 552         if is_valid_profile(new_profile, truncation_type, p):
    553             return new_profile, truncation_type
    554         else:

/Users/palmieri/Desktop/Sage_stuff/git/sage/local/lib/python2.7/site-packages/sage/algebras/steenrod/steenrod_algebra_misc.pyc in is_valid_profile(profile, truncation_type, p)
    258             if pro_r < Infinity:
    259                 for i in range(1,r):
--> 260                     if pro_r < min(pro[r-i-1] - i, pro[i-1]):
    261                         return False
    262     else:

Does is_valid_profile also need to accept a generic argument? Maybe something like this:

src/sage/algebras/steenrod/steenrod_algebra_misc.py

@@ -183 +183 @@
 ######################################################
 # profile functions
 
-def is_valid_profile(profile, truncation_type, p=2):
+def is_valid_profile(profile, truncation_type, p=2, generic=None):
     """
     True if ``profile``, together with ``truncation_type``, is a valid
     profile at the prime `p`.
@@ -250 +250 @@
         True
     """
     from sage.rings.infinity import Infinity
-    if p == 2:
+    if generic is None:
+        generic = False if p==2 else True
+    if p == 2 and not generic:
         pro = list(profile) + [truncation_type]*len(profile)
         r = 0
         for pro_r in pro:
@@ -549 +551 @@
                 while len(k) > 0 and k[-1] == 2:
                     k = k[:-1]
             new_profile = (e, k)
-        if is_valid_profile(new_profile, truncation_type, p):
+        if is_valid_profile(new_profile, truncation_type, p, generic=generic):
             return new_profile, truncation_type
         else:
             raise ValueError("Invalid profile")

Also, in the documentation for the function SteenrodAlgebra, I think you should say a bit more, for example what the degrees of the elements are. Maybe move the comment about what the dual looks like from is_generic to SteenrodAlgebra.

[git]

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

​14197a6

Merge branch 'develop' of git://github.com/sagemath/sage into generic_steenrod

[cnassau]

Replying to jhpalmieri:

I'm not getting this to work. When I evaluate

sage: SteenrodAlgebra(p=2,generic=True)

I get an error:

Very strange: it works for me, it works for the patchbot (selmer, sage 6.3.beta1) - yet your machine (and your patch) is definitely right. I do get this error when I try to construct a subalgebra:

sage: SteenrodAlgebra(p=2,generic=True,profile=([2,1],(2,2,2,)))
...
/waste/cn/sage-git/local/lib/python2.7/site-packages/sage/algebras/steenrod/steenrod_algebra_misc.pyc in is_valid_profile(profile, truncation_type, p)
    258             if pro_r < Infinity:
    259                 for i in range(1,r):
--> 260                     if pro_r < min(pro[r-i-1] - i, pro[i-1]):
    261                         return False
    262     else:

TypeError: unsupported operand type(s) for -: 'tuple' and 'int'

You're also right about the documentation, which should be improved. I'll post an updated patch when it is ready.

[git]

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

​ab43080	Merge branch 'develop' of git://github.com/sagemath/sage into generic_steenrod
​2bfe11d	Merge branch 'develop' of git://github.com/sagemath/sage into generic_steenrod

[git]

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

​1e7797c

Merge branch 'develop' of git://github.com/sagemath/sage into generic_steenrod

[git]

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

​be31a88

documentation improvements and a bugfix in is_valid_profile

[cnassau]

Replying to jhpalmieri:

Does is_valid_profile also need to accept a generic argument? Also, in the documentation for the function SteenrodAlgebra, I think you should say a bit more, for example what the degrees of the elements are. Maybe move the comment about what the dual looks like from is_generic to SteenrodAlgebra.

As suggested, I have added the generic argument to is_valid_profile and enhanced the documentation a (tiny) little bit. [My apologies that this took so long - I was busy with other matters.] I now refer to Voevodsky's motivic Steenrod algebra paper as another source that relates to the generic Steenrod algebra - other explicit references are not so easy to find.

It might be nice to also add support for the motivic Steenrod algebra, but I'll leave that for another ticket. Currently my cohomology programs can deal with the classical and the generic case, but not the motivic one, so my interest in the motivic case is not so urgent. It shouldn't be too difficult to implement the motivic version on top of the existing code, though...

[git]

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

​ef07a70

Merge branch 'develop' of git://github.com/sagemath/sage into generic_steenrod

[jhpalmieri]

I've made a few small changes:

• added some doctests explicitly involving the mod 2 generic Steenrod algebra. I guess this is mostly not necessary, since the generic code gets tested well at odd primes, but I added these anyway.

• top of the file: added # -*- coding: utf-8 -*- so the accent in "Études" wouldn't break docbuilding.

• fixed a pre-existing bug in _latex_term for the "pst" basis

• changed the __contains__ method so that elements in the ordinary mod 2 Steenrod algebra are not contained in the generic version. Is this the right thing to do?

If you're happy with these changes, it can get a positive review.

---

New commits:

​429a989

Generic Steenrod algebra: add some doctests

[cnassau]

Replying to jhpalmieri:

• changed the __contains__ method so that elements in the ordinary mod 2 Steenrod algebra are not contained in the generic version. Is this the right thing to do?

Yes, I think so, at least for the moment. It might eventually make sense to think of the generic variant as a quotient of the regular Steenrod algebra - but I'm hesitant to propose such a change right now, as the need for it is not clear to me. I'm also not sure whether one should then follow the Sage SubQuotient protocol or just provide some extra logic in the element constructor; currently, the latter sounds more resonable to me.

If you're happy with these changes, it can get a positive review.

I'm very happy - thanks for all the work you've put into this!