Symmetric group characters as bases of the symmetric functions

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This ticket implements two inhomogeneous bases of the symmetric functions, one called the irreducible_character and the other induced_trivial_character and shorthands st and ht. These bases play the roll for the symmetric group realized as permutation matrices that the Schur functions play to the character of the irreducible Gl_n representations.

In addition, two methods are added to the symmetric function element class. One eval_at_permutation_roots that evaluates a symmetric function at the roots of unity of a permutation matrix with a given cycle type and the other, character_to_frobenius_image, interprets a symmetric function as a symmetric group character and computes the Frobenius image.

Depends on #15536

CC: @tscrim @sagetrac-sage-combinat @alauve @saliola @darijgr

Component: combinatorics

Keywords: sf, sage-combinat

Author: Mike Zabrocki

Branch/Commit: 84f5917

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/19327

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New commits:

d152338	add character.py, character basis to sf, element methods to sfa
612248a	changes to powersum.py, sf/__init__.py and module_list.rst
c281a6c	implementation of irreducible character basis by power sum expansion

[d4d9e38a-6e64-40d7-a7f7-bd828eb9e0db]

Branch: u/zabrocki/sf/characterbases/19327

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Commit: c281a6c

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

c5a38dd	add missing import statements
f33cddb	additional documentation
d9b0942	additional documentation in the header of the file character.py

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Changed commit from c281a6c to d9b0942

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Changed commit from d9b0942 to 5ec1fff

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

5ec1fff

update the arxiv reference

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

0d0ad36

format incorrect on reference

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 5ec1fff to 0d0ad36

[tscrim]

comment:7

You might be interested in #15536 (symplectic and orthogonal bases for Sym based off Chari and Kleber's work) and #17096 (filtered algebras).

There is also a failing doctest according to the patchbot.

[d4d9e38a-6e64-40d7-a7f7-bd828eb9e0db]

Dependencies: #17096

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comment:8

This basis is similar to those found in #15536. In fact the orthogonal basis will have a positive expansion in the irreducible_symmetric_group_character basis (because permutation matrices are orthogonal). As a result I've added the filtered algebras ticket as a dependency. I checked and these bases also have an antipode problem and probably suffer from the same counit issue that the sp and o bases had at the start.

I merged #17096 into this branch on my computer and compilings starting failing. Given that I am on OSX 10.11 and Xcode 7 this means trouble for me because I have been in a precarious situation where compiling sage from scratch makes it unusable.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

b2d6919

removed target_basis as a method (not used), doc changes, TestSuite doc tests

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 0d0ad36 to b2d6919

[tscrim]

comment:10

Do you want me to make the changes to work with #17096 (similar to what I did in #15536) while compiling Sage for you as part of my review?

[d4d9e38a-6e64-40d7-a7f7-bd828eb9e0db]

comment:11

If you are willing. If I understand the changes, it is to merge with #17096, make sure that the counit works + put graded=False in the SFA_generic.__init__. The two TestSuite docs will fail until the counit is correct on the basis.

[tscrim]

Changed branch from u/zabrocki/sf/characterbases/19327 to public/combinat/sf/character_bases-19327

[tscrim]

comment:12

Okay, I've made both the counit and antipode work by coercion (although it seems that I don't need to do the former as h[[]] <-> ht[[]] <-> st[[]]). However, I'm still not getting the antipode test to pass. I would think any change of basis (as a filtered module) would preserve the Hopf algebra structure. I've checked that it is an isomorphism, but I don't get why it's not working. Thoughts?

---

Last 10 new commits:

4b2046f	Merge branch 'public/categories/super_categories-18044' into public/categories/filtered_algebras-17096
3f67b6b	Fixing trivial doctest due to changes in category heirarchy.
fa476dd	Fixing double-colon in INPUT block.
6cc8b84	Reviewer changes with Darij.
9bfd9e4	Merge branch 'public/categories/filtered_algebras-17096' into public/combinat/sf/sp_orth
57f92dd	Adding some doctests and making things use the filtered basis.
bd49490	Changes from Mike and some added tests.
a7ee857	Adding the Shimozono-Zabrocki paper as a reference.
530a953	Merge branch 'public/combinat/sf/sp_orth' of trac.sagemath.org:sage into public/combinat/sf/character_bases/19327
54a5af3	Initial round of revisions and trying to get the Hopf structure to work.

[tscrim]

Changed dependencies from #17096 to #15536

[tscrim]

Changed commit from b2d6919 to 54a5af3

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comment:13

When you say it doesn't work, are you finding that it isn't an algebra homomorphism? I think that was what was failing before.

[tscrim]

comment:14

No, it appears to be an algebra homomorphism:

sage: all(ht[la]*ht[mu] == ht(h(ht[la])*h(ht[mu])) for i in range(5) for la in Partitions(i) for j in range(5) for mu in Partitions(j))
True
sage: all(h[la]*h[mu] == h(ht(h[la])*ht(h[mu])) for i in range(5) for la in Partitions(i) for j in range(5) for mu in Partitions(j))
True
sage: ht.an_element() * ht.an_element() == ht(h(ht.an_element()) * h(ht.an_element()))
True

[d4d9e38a-6e64-40d7-a7f7-bd828eb9e0db]

comment:15

Here is the cause of the doc tests failing.

sage: 7*st[[]]*st[[]]
st[]

I have no idea where this error is coming from though.

[d4d9e38a-6e64-40d7-a7f7-bd828eb9e0db]

comment:16

OK, I found the error. It is in _other_to_self and you need to change the line

-            return self([])
+            return sexpr.coefficient([]) * self([])

I can't seem to push because I am ssh-ing to a remote non-OSX 10.11 Xcode 7 computer and git isn't set up properly. When I get that working I will be able to push a change.

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Changed commit from 54a5af3 to 54c08a6

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

54c08a6

Fixing transitions with the span of the unit element.

[tscrim]

Upstream: None of the above - read trac for reasoning.

[tscrim]

comment:18

Yep, that was the problem.

[tscrim]

Reviewer: Travis Scrimshaw

[tscrim]

Changed upstream from None of the above - read trac for reasoning. to none

[tscrim]

comment:20

So if you're happy with my changes, then I'm content to set a positive review.

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Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

cac1d88	Doing some surgery on the categories.
8263b10	Merge branch 'develop' into public/combinat/sf/sp_orth
b151195	Merge branch 'develop' into public/combinat/sf/sp_orth
8e690e2	Some hacks and FIXMEs.
c4c1416	Merge branch 'develop' into public/combinat/sf/sp_orth
d8d3eda	additions to NCSF/QSym/Sym/NCSym/NCSymD to ensure that MRO errors are not triggered
0513f6a	Merge branch 'public/combinat/sf/sp_orth' of trac.sagemath.org:sage into public/combinat/sf/sp_orth
5b492f3	Fixing non-ascii character.
b49a06e	Merge branch 'public/combinat/sf/sp_orth' into public/combinat/sf/character_bases-19327
84f5917	use the default implementation of antipode and counit, remove specifications of category in morphisms

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Changed commit from 54c08a6 to 84f5917

[tscrim]

comment:22

LGTM. Thanks.

[d4d9e38a-6e64-40d7-a7f7-bd828eb9e0db]

Description changed:

--- 
+++ 
@@ -1,3 +1,3 @@
 This ticket implements two inhomogeneous bases of the symmetric functions, one called the `irreducible_character` and the other `induced_trivial_character` and shorthands `st` and `ht`.  These bases play the roll for the symmetric group realized as permutation matrices that the Schur functions play to the character of the irreducible Gl_n representations.

-In addition, two methods are added to the symmetric function element class.  One `eval_at_permutation_roots` that evaluates a symmetric function at the roots of unity of a permutation matrix with a given cycle type and the other, `eval_at_permutation_roots`, interprets a symmetric function as a symmetric group character and computes the Frobenius image.
+In addition, two methods are added to the symmetric function element class.  One `eval_at_permutation_roots` that evaluates a symmetric function at the roots of unity of a permutation matrix with a given cycle type and the other, `character_to_frobenius_image`, interprets a symmetric function as a symmetric group character and computes the Frobenius image.

[d4d9e38a-6e64-40d7-a7f7-bd828eb9e0db]

comment:23

I've checked that all tests pass.

[tscrim]

comment:24

I thought I had set it to positive review. Whoops. :P

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comment:25

Thanks for the review and pushing forward on #15536 and this one. Without finding a solution there, this ticket was going nowhere.

[vbraun]

Changed branch from public/combinat/sf/character_bases-19327 to 84f5917