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

darijgr commented 9 years ago

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!

CC: @tscrim @anneschilling @sagetrac-sage-combinat @nthiery @sagetrac-jkeitel @sagetrac-jpswanson @MariaMonks

Component: combinatorics

Keywords: tableaux, bender-knuth involutions, sage-combinat

Author: Darij Grinberg

Branch/Commit: 7deac5c

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/18686

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 9 years ago

Changed commit from 33c3a7b to cda0403

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 9 years ago

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

cda0403 more sensible doc, and another doctest

tscrim commented 9 years ago

comment:3

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.

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 9 years ago

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

abbfda2 Rewrite doctests following Travis's suggestion

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 9 years ago

Changed commit from cda0403 to abbfda2

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 9 years ago

Changed commit from abbfda2 to 7deac5c

7ed8c4ca-6d56-4ae9-953a-41e42b4ed313 commented 9 years ago

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

7deac5c Fix a typo

darijgr commented 9 years ago

comment:6

Travis: thank you!

New commits:

7deac5c Fix a typo

New commits:

7deac5c Fix a typo

darijgr commented 9 years ago

Reviewer: Travis Scrimshaw

vbraun commented 9 years ago

Changed branch from u/darij/benderknuth to 7deac5c

sagemath / sage

Tableaux: remove false theorem I stated about Bender-Knuth involutions #18686