principal and exponential specializations for symmetric functions

[mantepse]

With this patch, you can now do:

sage: s = SymmetricFunctions(QQ).s()
sage: x = s[2,2,1]
sage: x.principal_specialization()
-q^4/(q^11 - 2*q^10 + q^8 + 2*q^6 - 2*q^5 - q^3 + 2*q - 1)
sage: x.principal_specialization(n=4)
q^11 + 2*q^10 + 3*q^9 + 4*q^8 + 4*q^7 + 3*q^6 + 2*q^5 + q^4
sage: x.exponential_specialization()
1/24*t^5
sage: x.exponential_specialization(q=QQ["q"].gen())
(q^4/(q^6 + 3*q^5 + 5*q^4 + 6*q^3 + 5*q^2 + 3*q + 1))*t^5

Implement the principal and exponential specializations for symmetric functions as given in Stanley, Enumerative Combinatorics, Section 7.8.

CC: @jbandlow @zabrocki @tscrim @darijgr

Component: combinatorics

Keywords: principal specialization, exponential specialization, symmetric functions

Author: Martin Rubey

Branch/Commit: cf9e0f2

Reviewer: Darij Grinberg, Mike Zabrocki

Issue created by migration from https://trac.sagemath.org/ticket/10930

[mantepse]

patch for principal and exponential specialization

[mantepse]

comment:1

Attachment: sf_principal_specialization-mr.patch.gz

[mantepse]

Description changed:

--- 
+++ 
@@ -1,8 +1,16 @@
-This patch implements the principal and exponential specialisations for symmetric functions as given in Stanley, Enumerative Combinatorics, Section 7.8.
+This patch -- in the combinat queue:
+
+http://combinat.sagemath.org/hgwebdir.cgi/patches/file/tip/sf_principal_specialization-mr.patch
+
+implements the principal and exponential specialisations for symmetric functions as given in Stanley, Enumerative Combinatorics, Section 7.8.
 
 Unfortunately, the patch currently has several problems:
+
 1) it seems that always the default code in sf/sfa.py is called, I don't know why
+
 2) I do not know how to give the desired default argument 1 the correct type in exponential_specialization
+
 3) I guess one should rather implement this for quasi-symmetric functions, but I do not know enough about these currently
+
 4) the documentation and sensible tests are still missing
 

[jbandlow]

comment:4

Hi Martin,

Thanks submitting this! A quick response to your points above. I'm not sure what you mean by point (1)... can you elaborate?

For point (2), you can use None as the default value, and then do

    if q is None:
        q = self.parent().base_ring().one()

as the first line of your code.

For point (3), quasisymmetric functions are still somewhat immature--in particular they are not in Sage proper. So this is not too big of a concern.

For point (4), yes, there should be more doc and tests. In particular, I find tests like

    all( e[mu].principal_specialization(4) == e[mu].expand(4)(1,q,q^2,q^3) for mu in Partitions(4) )

particularly convincing.

[mantepse]

comment:5

Hi Jason!

Many thanks for your quick comments.

1. Using trace I find

sage: S = SymmetricFunctions(QQ); s=S.s(); f = s[2,1]
sage: trace("f.principal_specialization()")
> <string>(1)<module>()

ipdb> s
--Call--
> /home/martin/SAGE/local/lib/python2.6/site-packages/sage/combinat/sf/sfa.py(1653)principal_specialization()
   1652 
-> 1653     def principal_specialization(self, n=infinity, q=var('q')):
   1654         r"""

ipdb> s
> /home/martin/SAGE/local/lib/python2.6/site-packages/sage/combinat/sf/sfa.py(1681)principal_specialization()
   1680         """
-> 1681         from sage.combinat.sf.sf import SymmetricFunctions
   1682         p = SymmetricFunctions(self.parent().base_ring()).p()

ipdb> 

but I was hoping that the principal specialisation from schur.py would be called.

2. Well, currently the actual value of 1 is not used at all (I test q==1 and call principal_specialization without passing q). So my question really is: some day somebody might implement something where the q is actually used. Is it better then if the default is None and the doc says, None should always mean one?

Thanks again!

[mantepse]

comment:6

I fixed indenting, bugs and added documentation and tests. Feedback welcome!

[jbandlow]

comment:7

Hi Martin, a few more points:

1. If I'm not mistaken, the function f in the exponential_specialization code for Schur functions is returning the product of the hooks of partition. This can also be done with Partition(partition).hook_product(1)

2. You do not give the result of the test in line 275 of the Schur function code.

3. See this and make sure that your patch satisfies these criteria. Please ask the sage-combinat list if you have any questions about these.

4. Once you have completed all of this, mark the patch as 'Needs Review' (by clicking the button at the bottom of the page). Then the 'Patchbot' will automatically apply and test your code.

Again, many thanks for your good work!

[mantepse]

comment:8

Replying to @jbandlow:

Jason, might it be that you were looking on a different version of the patch? There is no exponential_specialization for Schur functions (I use the generic version) and there is no test in line 275...

On the other hand, I just noticed several other problems with the code. Eg., s[1].exponential_specialization() doesn't work right now. I also should include tests for calling without any arguments...