Implement NCSym

[tscrim]

Implement the Hopf algebra of symmetric functions in non-commuting variables in the following bases:

• monomial m
• elementary e
• homogeneous h
• power-sum p
• x
• q

as well as the dual basis w.

Apply: attachment: trac_15150-ncsym-ts.patch​

Depends on #15143
Depends on #15164

CC: @sagetrac-sage-combinat @zabrocki @saliola @darijgr

Component: combinatorics

Author: Travis Scrimshaw

Reviewer: Mike Zabrocki, Darij Grinberg

Merged: sage-5.13.rc0

Issue created by migration from https://trac.sagemath.org/ticket/15150

[saliola]

comment:1

[Answered my own question.]

[tscrim]

comment:2

Here's what I currently have. The multiplication in the dual basis isn't working correctly yet. I know there's the hopf_algebra_of_supercharacters-fs.patch patch in the queue which seems like its targeted more for base rings of ZZ[q] (or like Hall-Littlewoods are to Sym), and I was hoping to link them together at some point (please correct me if I'm wrong). I'd be happy to work with you on this patch.

[saliola]

comment:3

The only problem is that I don't have much time these days to work on this. I've been meaning to implement NCSym. I'll try to take a look at what you have and see what I can offer, if anything.

As usual, thank you for your work, Travis!

[darijgr]

comment:4

Travis, what sources are you using for this? I'm asking because I haven't read anything about NCSym so far. I have a vague impression it has been introduced in one of Rota's Foundations papers, but not one that I have. A good introduction would simplify my job reviewing this by a lot.

[tscrim]

comment:5

Here's one of the references (Bergeron, Zabrocki): http://arxiv.org/pdf/math/0509265.pdf. I forget the other references off-hand (it's on my other computer tho), and I'll be adding all of the references to the doc as I finish it up.

[tscrim]

Description changed:

--- 
+++ 
@@ -1,11 +1,11 @@
 Implement the Hopf algebra of symmetric functions in non-commuting variables in the following bases:
 
-- monomial
-- elementary
-- homogeneous
-- power-sum
-- x
-- q
+- monomial **m**
+- elementary **e**
+- homogeneous **h**
+- power-sum **p**
+- **x**
+- **q**
 
-as well as the dual basis w.
+as well as the dual basis **w**.
 

[tscrim]

comment:6

Also I just fixed the w basis multiplication.

[tscrim]

comment:7

Okay, the patch is now ready for first review. It has full doctest coverage and all tests pass for me. I've removed some of the coercions in the old version since I was worried about possibly breaking coercion commutative diagrams.

Some of the things I'm not perfectly happy about (but can live with currently):

• A descriptive name of the q, x, and w bases.
• No antipode for the w basis.
• Creating the SymmetricFunctions base object when creating SymmetricFunctionNonCommutingVariables in order to set up the coercion.
• Is it safe to put the natural maps Sym -> NCSym and NSym -> NCSym as coercions?

The references I used are included in the documentation now too.

[zabrocki]

comment:8

Hi Travis, One initial comment about the NSym -> NCSym map. It is dangerous to call it "the natural" map from NSym to NCSym. It is one of many. There does not seem to be a reason to favor the map which sends the complete generators to the complete generators. If you were to send the elementary generators to the elementary generators you get a map which is equally compelling and (unfortunately) they are not the same map.