5
$\begingroup$

Note: All Young diagrams are to use the English notation scheme.

Suppose I have two tableaux $T$ and $T'$ on the same Young diagram (we insert the numbers $1, 2, \ldots, n$ in two different ways in the diagram). We say that $T > T'$ if the first entry (while reading left to right, top to bottom) in which $T$ and $T'$ differs is such that the entry of $T$ is larger than that entry of $T'$.

Is it true that if $T > T'$, then there exist $a \neq b$, $\{a, b\} \subset \{1, 2, \ldots, n\}$ such that $\{a, b\}$ appears in the same row of $T$ and $\{a, b\}$ appears in the same column of $T'$?

  • 0
    If you are going to say «left to right, top to bottom» you need explain how you draw the diagrams, for there are *two* different conventions (the English and the French conventions...) which will result in two different orders.2012-02-23
  • 0
    Woah. Did not realize that there were English/French conventions. Thanks, edited.2012-02-23
  • 1
    I suspect you mean *standard* tableaux? If not, you might as well write $T\ne T'$, since the order wouldn't matter; also the statement would be evidently false, since you could have different tableaux on a diagram with only one row or only one column.2012-02-23

1 Answers 1