# The Unapologetic Mathematician

## The Dominance Lemma for Tabloids

If $k, and $k$ appears in a lower row than $l$ in the Young tabloid $\{t\}$, then $(k\,l)\{t\}$ dominates $\{t\}$. That is, swapping two entries of $\{t\}$ so as to move the lower number to a higher row moves the tabloid up in the dominance relations.

Let the composition sequences of $\{t\}$ and $(k\,l)\{t\}$ be $\lambda^i$ and $\mu^i$, respectively. For $i and $i\geq l$ we automatically have $\lambda^i=\mu^i$. For $k\leq i there is a difference between the two: the entry $k$ has been added in a different place. Let $k$ and $l$ be in rows $q$ and $r$ of $\{t\}$, respectively. In $\lambda^i$, the entry $k$ is added to row $q$, while in $\mu^i$ it’s been added to row $r$. That is, $\lambda^i$ is the same as $\mu^i$ with part $q$ increased by one and part $r$ decreased by one. Our assumption that $k$ is in a lower row than $l$ in $\{t\}$ is that $q>r$. Therefore, since the lower row in $\lambda^i$ is less than in $\mu^i$, we find that $\lambda^i\triangleleft\mu^i$. And we conclude that $\{t\}\trianglelefteq(k\,l)\{t\}$, as asserted.

January 11, 2011