Implementation of the SubwordComplex as defined by Knutson and Miller

[stumpc5]

This patch provides an implementation of the subword complex:

Fix a Coxeter system (W,S). Let Q = be a finite word in S and pi in W.

The subword complex Delta(Q,pi) is then defined to be the simplicial complex with vertices being {0,...,n-1}, (n = len(Q), one vertex for each letter in Q) and with facets given by all (indices of) subwords Q' of Q for which Q\Q' is a reduced expression for pi.

    sage: W = CoxeterGroup(['A',2],index_set=[1,2])
    sage: w = W.from_reduced_word([1,2,1])
    sage: C = SubwordComplex([2,1,2,1,2],w); C
    Subword complex of type ['A', 2] for Q = [2, 1, 2, 1, 2] and pi = 121
    sage: C.facets()
    {(1, 2), (3, 4), (0, 4), (2, 3), (0, 1)}

Component: combinatorics

Keywords: subword complex, simplicial complex

Author: Christian Stump

Branch/Commit: 17518c1

Reviewer: Frédéric Chapoton

Issue created by migration from https://trac.sagemath.org/ticket/11010

[tscrim]

comment:72

The first issue is the result of my quick hacking to get something working. The Coxeter matrices are equal for the Cartan types B and C, and the Coxeter matrix is the information passed to the UniqueRepresentation cache. There are a number of ways to go moving forward, but I think that deserves its own separate ticket.

For the second issue, yes, you are correct. The n-1 and n nodes have the order 4 relation between them, as this agrees with the corresponding B/C Cartan types (and the Coxeter matrix). You are welcome to relabel the Coxeter type, which should work, or go to the Coxeter graph and find the edge label4ed by 4 and use those nodes. Personally, I would use the latter option since it is independent of the labeling.

[stumpc5]

comment:73

I also have the problem that the plot only allows irreducible types. When playing with reducible types, I got to the types B vs. G bug in #19830...

[stumpc5]

Changed branch from u/chapoton/11010 to u/stumpc5/11010

[fchapoton]

comment:76

Is this "needs_review" ?

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New commits:

4864a64	Merge branch '11010-new' into u/stumpc5/11010-new
7722489	Merge branch 'u/chapoton/11010' of git://trac.sagemath.org/sage into u/stumpc5/11010-new
839b2e2	fixing the index set for the plot

[fchapoton]

Changed commit from 1b10c95 to 839b2e2

[stumpc5]

comment:77

Is this "needs_review" ?

Not quite, I wanted to add a few more doctests for the type B plotting with various indexing sets. But then saw that the indexing in CoxeterGroup is corrupted, see #19830. This is another issue, but I still want to add more examples/testing.

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

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

0f90c26

fixed another b/c bug + some cosmetics to the code

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

Changed commit from 839b2e2 to 0f90c26

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

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

e8e160c

fixed a docstring and adding one more test

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

Changed commit from 0f90c26 to e8e160c

[fchapoton]

Changed commit from e8e160c to 17518c1

[fchapoton]

comment:81

ok, good enough for me.

I have made two minor doc changes, and now set this again to positive review.

Any further changes will have to be in another ticket.

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New commits:

17518c1

trac #11010 two details

[fchapoton]

Changed branch from u/stumpc5/11010 to u/chapoton/11010

[vbraun]

Changed branch from u/chapoton/11010 to 17518c1