Opened 5 years ago
Closed 5 years ago
#18686 closed defect (fixed)
Tableaux: remove false theorem I stated about BenderKnuth involutions
Reported by:  darij  Owned by:  

Priority:  major  Milestone:  sage6.8 
Component:  combinatorics  Keywords:  tableaux, benderknuth involutions, sagecombinat 
Cc:  tscrim, aschilling, sagecombinat, nthiery, jkeitel, jpswanson, MariaMonks  Merged in:  
Authors:  Darij Grinberg  Reviewers:  Travis Scrimshaw 
Report Upstream:  N/A  Work issues:  
Branch:  7deac5c (Commits)  Commit:  7deac5cff145d9b4f79428331129727591a461ab 
Dependencies:  Stopgaps: 
Description
I claimed that (s_i s_{i+1})^6 = id
, where s_k
means the k
th BenderKnuth involution on semistandard skew tableaux. I even made a doctest that unfortunately used a hook shape, which renders the BenderKnuth involutions rather boring (though maybe it would make a nice exercise to check it in this case  although I don't know if it is always true there).
The claim is false. There is a counterexample for skew tableaux and i = 1
(or straight tableaux and i = 2
). The source where I had it from, a paper by Berenstein and Kirillov, only claimed it for straight tableaux and i = 1
(and that is indeed correct).
Thanks to Pavel Galashin for finding a counterexample!
Change History (7)
comment:1 Changed 5 years ago by
 Status changed from new to needs_review
comment:2 Changed 5 years ago by
 Commit changed from 33c3a7bb5243ba4244451255fb2bb007f12ed8a5 to cda0403517f0261efc355d0997eb303424242bd6
comment:3 Changed 5 years ago by
Could we use some for loops?
sage: p = lambda t, k: t.bender_knuth_involution(k).bender_knuth_involution(k + 1) sage: t = Tableau([[1,2,2],[3,4]]) sage: x = t sage: for i in range(6): x = p(x, 2) sage: x [[1, 2, 3], [2, 4]] sage: x == t False
If you agree and make these changes, you can set a positive review on my behalf.
comment:4 Changed 5 years ago by
 Commit changed from cda0403517f0261efc355d0997eb303424242bd6 to abbfda21f0eb9369dddb64358103e18ae6a0c6f0
Branch pushed to git repo; I updated commit sha1. New commits:
abbfda2  Rewrite doctests following Travis's suggestion

comment:5 Changed 5 years ago by
 Commit changed from abbfda21f0eb9369dddb64358103e18ae6a0c6f0 to 7deac5cff145d9b4f79428331129727591a461ab
Branch pushed to git repo; I updated commit sha1. New commits:
7deac5c  Fix a typo

comment:6 Changed 5 years ago by
 Keywords beginner removed
 Reviewers set to Travis Scrimshaw
 Status changed from needs_review to positive_review
comment:7 Changed 5 years ago by
 Branch changed from u/darij/benderknuth to 7deac5cff145d9b4f79428331129727591a461ab
 Resolution set to fixed
 Status changed from positive_review to closed
Branch pushed to git repo; I updated commit sha1. New commits:
more sensible doc, and another doctest