Opened 5 years ago

Closed 5 years ago

#18686 closed defect (fixed)

Tableaux: remove false theorem I stated about Bender-Knuth involutions

Reported by: darij Owned by:
Priority: major Milestone: sage-6.8
Component: combinatorics Keywords: tableaux, bender-knuth involutions, sage-combinat
Cc: tscrim, aschilling, sage-combinat, 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 Bender-Knuth involution on semistandard skew tableaux. I even made a doctest that unfortunately used a hook shape, which renders the Bender-Knuth 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 darij

  • Status changed from new to needs_review

comment:2 Changed 5 years ago by git

  • Commit changed from 33c3a7bb5243ba4244451255fb2bb007f12ed8a5 to cda0403517f0261efc355d0997eb303424242bd6

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

cda0403more sensible doc, and another doctest

comment:3 Changed 5 years ago by tscrim

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 git

  • Commit changed from cda0403517f0261efc355d0997eb303424242bd6 to abbfda21f0eb9369dddb64358103e18ae6a0c6f0

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

abbfda2Rewrite doctests following Travis's suggestion

comment:5 Changed 5 years ago by git

  • Commit changed from abbfda21f0eb9369dddb64358103e18ae6a0c6f0 to 7deac5cff145d9b4f79428331129727591a461ab

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

7deac5cFix a typo

comment:6 Changed 5 years ago by darij

  • Keywords beginner removed
  • Reviewers set to Travis Scrimshaw
  • Status changed from needs_review to positive_review

Travis: thank you!


New commits:

7deac5cFix a typo

New commits:

7deac5cFix a typo

comment:7 Changed 5 years ago by vbraun

  • Branch changed from u/darij/benderknuth to 7deac5cff145d9b4f79428331129727591a461ab
  • Resolution set to fixed
  • Status changed from positive_review to closed
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