Implement symmetric functions in super space

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

Symmetric functions in superspace is the Hopf algebra of super partitions (ticket #25132). It has the symmetric functions as a sub-algebra.

• [DLM06] P. Desrosiers, L. Lapointe, P. Mathieu, Classical symmetric functions in superspace, J. Algebr Comb. (2006) 24:209--238, :arXiv:0509408

• [JL16] M. Jones, L. Lapointe, Pieri rules for Schur functions in superspace, :arXiv:1608.08577

Depends on #25132 Depends on #25133

CC: @alauve @darijgr @zabrocki @tscrim

Component: combinatorics

Keywords: CHAs, sf, super partitions, IMA coding sprint

Author: Mike Zabrocki

Branch/Commit: public/symsuperspace/25163 @ 0c5eb09

Issue created by migration from https://trac.sagemath.org/ticket/25163

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

Changed dependencies from #25132 to #25132, #25133

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

Commit: 38280f7

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

Branch: public/symsuperspace/25163

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

comment:1

A first implementation. I merged #25133 so that the CHAS directory is there.

---

Last 10 new commits:

d41f19c	Corrected doc tests, moved theorem_10 as a method, removed super_compositions
de0f90b	doc changes and rename one pieri rule
139b594	add superpartition to the doc
0e7c0c9	doc corrections, added to list of catalog_partitions and enumerated_sets, pyflakes
89a1764	Refactored class to use ClonableArray. Other reviewer changes.
c83f8da	corrected reference
9b7cac1	Merge branch 'public/superpartitions' into public/symsuperspace/25163
4184561	first implementation
de1c9c4	remove spaces
38280f7	import statements, documentation lists, minor correction

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

Changed commit from 38280f7 to e7d352b

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

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

f901766	import statements, doc test corrections
2389b91	change to superpartition.py to fix bug
26ad8c2	fixed doc tests
f68ebfd	documentation of SFSS
e72edee	progress on documentation
e7d352b	introduction documentation

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

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

bc276dd	Merge branch 'public/combinat/implement_wqsym-25133' of git://trac.sagemath.org/sage into public/combinat/implement_wqsym-25133
c9cacf9	Merge branch 'public/symsuperspace/25163' of git://trac.sagemath.org/sage into public/symsuperspace/25163

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

Changed commit from e7d352b to c9cacf9

[tscrim]

comment:4

Just did a trivial rebase over the trivial conflict in #25133.

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

Changed commit from c9cacf9 to e3d942f

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

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

03c6e39	general documentation
e3d942f	Merge branch 'public/symsuperspace/25163' of trac.sagemath.org:sage into public/symsuperspace/25163

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

Changed commit from e3d942f to fef7563

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

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

680bd70	Partitions are SuperPartitions, correct typo in options
fef7563	document m-product functions

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

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

2978e35	more doc tests for Schur bases
633987b	for consistency -* is -star
fc3b2b2	cleanup of documentation
e581b38	remove unimplemented method

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

Changed commit from fef7563 to e581b38

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

comment:8

I think that this is ready to go.

[darijgr]

comment:9

Some bikeshedding on the module-level doc:

Let `P_{\Theta_n, X_n}` be the polynomial ring in two sets of variables
`\Theta_n = \theta_1, \theta_2, \ldots, \theta_n` and
`X_n = x_1, x_2, \ldots, x_n`

The "n" here clashes with the "n" in the later definitions of p_{n;} et al; maybe use a capital N instead? Or work with infinitely many variables right away, if this is possible?

where the first set of variables anti-commute

If this includes squaring to 0, then please say so.

The symmetric group
acts diagonally on this polynomial ring and the symmetric functions in
superspace are isomorphic to the invariants in this polynomial ring.

I'd put a comma before the "and" so it's not misread as "on this polynomial ring and the symmetric functions".

    p_{(;i)} p_{(;j)} = p_{(;j)} p_{(;i)} \qquad
    p_{(i;)} p_{(;j)} = p_{(;j)} p_{(i;)} \qquad
    p_{(i;)} p_{(j;)} = - p_{(j;)} p_{(i;)}

And p_{(i;)}^2 = 0, I assume.

    p_{(; n)} = p_n \quad e_{(; n)} = e_n \quad h_{(; n)} = h_n

Please use , \quad, not just \quad.

from `Theta_n`

Don't forget the backslash. (And again, that's an "N", not an "n".)

bosnic degree

The adjective is "Bosnian".

    h_{(n; )} = \sum_{\Lambda \in SPar(n|1)} m_\Lambda \qquad

Again, use a comma before the \qquad.

`\left<\left< p_\Lambda, p_\Gamma \right>\right> = \delta_{\Lambda\Gamma} z_{\Lambda^s}`
where `z_{\Lambda^s}` is the usual constant `z_\lambda` which is the size of

What is \Lambda^s?

in pairs.  For all `\Lambda, \Gamma in SPar(n|m)` ,

in -> \in

As with the space of symmetric functions, there is an involution `\omega`
that for all super partitions `\Lambda`, `\omega(h_\Lambda) = e_\Lambda`.

add "satisfies".

Hopf algebra and the coproduct is defined bby declaring that the power sum

"by".

I don't quite get: How is e_\Lambda defined for a strict partition \Lambda? Just by multiplying e_{n;} over all fermionic entries n and e_{;n} over all bosonic entries n? In what order?

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

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

4c26ad5

Darij's suggestions for documentation

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

Changed commit from e581b38 to 4c26ad5

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

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

e905504

n -> N in two places

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

Changed commit from 4c26ad5 to e905504

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

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

2c67a04

mostly missing periods

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

Changed commit from e905504 to 2c67a04

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

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

82f92cc

minor corrections to doc

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

Changed commit from 2c67a04 to 82f92cc

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

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

58fb4a4

correction of a few typos

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

Changed commit from 82f92cc to 58fb4a4

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

Changed commit from 58fb4a4 to 9966e88

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

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

9966e88

manual merge with develop

[darijgr]

comment:16

+    class Schur_sb(CombinatorialFreeModule, BindableClass):
+        r"""
+        The Schur basis arising from `q=t=0` specialization of Macdonald.

+    class Schur_s(CombinatorialFreeModule, BindableClass):
+        r"""
+        The Schur-star basis arising from `q=t=0` specialization of Macdonald.

+    class Schur_b(CombinatorialFreeModule, BindableClass):
+        r"""
+        The Schur-bar basis as the `q=t=\infty` specialization of Macdonald.

+    class Schur(CombinatorialFreeModule, BindableClass):
+        r"""
+        The Schur-bar basis as the `q=t=\infty` specialization of Macdonald.

Can it be that these descriptions are messed up? Why is the "Schur" doc claiming to be Schur-bar? Shouldn't Schur be from q=t=0, not from q=t=\infty?

Also, is there no simpler way to define these bases than by Macdonald specialization? I'd expect some Berezinian analogue of the Jacobi-Trudi formulas?

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

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

832dca6

change the description of the Schur bases

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

Changed commit from 9966e88 to 832dca6

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

comment:18

You are right the descriptions were confused there. The Schur-star and Schur-star-bar are related to these by duality and I looked in the paper and noticed that they are not listed as a specialization so I cut that description.

Also, is there no simpler way to define these bases than by Macdonald specialization? I'd expect some Berezinian analogue of the Jacobi-Trudi formulas?

That would be welcome. Here they are implemented by Pieri rules which are given in [JL2016].

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

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

b710430

added reference [BFM2015] and examples from that reference

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

Changed commit from 832dca6 to b710430

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

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

d50bdb3

missing def for monomial basis, mark test long

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

Changed commit from b710430 to d50bdb3

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

Changed commit from d50bdb3 to f8e9a96

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

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

f8e9a96

doc test for anti-homomorphism; delete not particularly useful method

[darijgr]

comment:22

More random comments:

+            The product of monomial basis elements is calculated for a
+            fixed fermionic partition.

What does that mean?

What is a BindableClass (no, I don't get the doc at https://doc.sagemath.org/html/en/reference/misc/sage/misc/bindable_class.html ) and why is it being used here?

+where `\Lambda` is a super partition with femionic sector `m`.

feRmionic

Is this a Hopf algebra or a super-Hopf algebra? I.e., is there a twist involved in the bialgebra axiom? My suspicion is that it is, because otherwise the square-zero relations p_{(i;)}^2 = 0 would contradict the primitivity of the p_{(i;)} wrt the coproduct. But do you ever tell this to Sage? If you don't, I doubt that it's correctly computing coproducts of non-generators! Also, of course, this should be said in the doc. (Is the Hopf structure anywhere in the literature?)

+                Test if super partitions an element all have the same bi-degree.

Probably should be "if the super-partitions in the support of self have the same bi-degree".

Maybe decide between "symmetric generators" and "bosonic generators" -- currently you seem to be using both languages (or do they mean different things?).

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

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

0b7c97f

corrections to documentation

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

Changed commit from f8e9a96 to 0b7c97f

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

comment:24

I used BindableClass because that seems to be what all bases of chas need to inherit (see WQSymBasis_abstract in wqsym.py, FSymBasis_abstract in fsym.py and bases in ncsf.py and qsym.py). I tried deleting the BindableClass and the command h = self.Complete() raised a TypeError: __init__() takes exactly 2 arguments (1 given). And, no, I do not understand that documentation either.

About the Hopf algebra structure:

My suspicion is that it is, because otherwise the square-zero relations p_{(i;)}^2 = 0 would contradict the primitivity of the p_{(i;)} wrt the coproduct.

As far as I can tell this is a Hopf algebra and not a super-Hopf algebra. There is no posted paper that I can point to, but I do have a non-posted preprint. However I will continue to check this carefully because I am reviewing the paper. Can you explain your suspicion more carefully? Perhaps there is an error in the paper.

[darijgr]

comment:25

In characteristic 0, any nilpotent primitive element of a (non-super) Hopf algebra must be 0. More strongly: If a is a primitive element of a Hopf algebra in characteristic 0, then the powers of a are linearly independent.

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

comment:26

Well then. Something is not right with the universe. No test I've performed has identified a problem with the Hopf structure. I'll need to check it against the proof. Can you provide me a reference?

[alauve]

comment:27

How about Appendix A here: https://arxiv.org/pdf/1105.5572.pdf

[darijgr]

comment:28

You are probably not testing enough. The error should materialize when you take the coproduct of a product of p{n;}s. Basically, the coproduct of p{n;}p_{m;} should switch signs if you swap n with m; but if I compute it naively in a Hopf algebra, I get

p{n;} p{m;} \otimes 1 + p{n;} \otimes p{m;} + p{m;} \otimes p{n;} + 1 \otimes p{n;} p{m;},

which does not swap signs.

Further typos: "Summetric", "fermonionic".

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

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

0c5eb09

typos in documentation

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

Changed commit from 0b7c97f to 0c5eb09

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

comment:30

What do you mean by swap signs? The sign does arise in the product. Does this seem like it is not correct?

sage: p[-4,-2].coproduct()
p[; ] # p[4, 2; ] + p[2; ] # p[4; ] + p[4; ] # p[2; ] + p[4, 2; ] # p[; ]
sage: (p[-2]*p[-4]).coproduct()
-p[; ] # p[4, 2; ] - p[2; ] # p[4; ] - p[4; ] # p[2; ] - p[4, 2; ] # p[; ]

Aaron, looking at Proposition 36, "Let K denote the associated graded Hopf algebra with respect to the coradical filtration of H." In order to compare to that proof, do I need to know what the coradical filtration of symmetric functions in super space is?