Flips on standard Young tableaux and descent sets

Question

Consider $T$ to be a standard Young tableau of shape $\lambda$ (in English notation). The descent set of $T$, $Des(T)$, is defined to the set of all positive integers $i$ such that $i+1$ lies strictly south (and weakly west) of $i$ in $T$.

A flip on $T$ is defined to be a move where we exchange the positions of $i$ and $i+1$ in $T$ if $i\in Des(T)$, provided that $i$ and $i+1$ are not the same column. Let $T'$ be another standard Young tableau obtained by doing a sequence of flips starting from $T$.

$\textbf{Question}$: Is it true that $Des(T)\neq Des(T')$?

It seems to be true from the examples I worked out, and seemed to be sort of result that would be present in literature. If this is indeed the case, I'd appreciate any reference that states this.

Thanks.

Answer 1

Here is a counterexample copied out of a Sage session:

sage: T = StandardTableau([[1,2,3,7,9],[4,5,8],[6,11],[10]]); T.pp()
  1  2  3  7  9
  4  5  8
  6 11
 10
sage: T.standard_descents()                                         
[3, 5, 7, 9]
sage: T = StandardTableau(T.bender_knuth_involution(9)); T.pp()     
  1  2  3  7 10
  4  5  8
  6 11
  9
sage: T.standard_descents()                                    
[3, 5, 7, 8, 10]
sage: T = StandardTableau(T.bender_knuth_involution(10)); T.pp()    
  1  2  3  7 11
  4  5  8
  6 10
  9
sage: T.standard_descents()
[3, 5, 7, 8]
sage: T = StandardTableau(T.bender_knuth_involution(8)); T.pp()     
  1  2  3  7 11
  4  5  9
  6 10
  8
sage: T.standard_descents()                                    
[3, 5, 7, 9]

Some comments on the syntax:

T.pp() is short for "pretty-print of T"; this prints the tableau T as a table rather than as a list of lists.

T.standard_descents() gives the descent set of T. Why is it not called T.descents() ? Because T.descents() computes something different (some kind of measure for the deviation of T from semistandardness -- hence, completely useless for standard tableaux). It is an artefact of history that it is the latter method, not the former, which has the shorter name.

Calling the Bender-Knuth involution method is a bit of an overkill here since all we are actually doing is switching two entries; an alternative would be to use T.symmetric_group_action_on_values(p) for a permutation p. Either way one has to explicitly cast the result into the StandardTableau class in order to apply the standard_descents() method, because T.bender_knuth_involution(i) returns a SemistandardTableau, while T.symmetric_group_action_on_values(p) returns just a Tableau.

If you find a descent statistic that is actually strictly semiinvariant under these flips, I'm curious to know.

Answer 2

To the contrary, it can be shown that any two SYT of the same shape are related by a sequence of flips. An explicit negative answer is given by $$ \begin{array}{ll}1246 & 1246\\ 35 & 37\\ 7 & 5 \end{array}. $$