10.3 Ordering of Partitions

Definition

Let $\lambda =({\lambda }_1,\cdots ,{\lambda }_{\ell })$ and $\mu =({\mu }_1,\cdots ,{\mu }_k)$ be $\lambda ⊢n$ and $\mu ⊢n$. Then $\lambda <\mu$ in lexicographic order if for some $i$, ${\lambda }_j={\mu }_j$ for all $j<i$ and ${\lambda }_i<{\mu }_i$.

Definition

For any $\lambda ⊢n$ and $\mu ⊢n$, if ${\lambda }_1+\cdots +{\lambda }_i⩽{\mu }_1+\cdots +{\mu }_i$ for any $i⩾1$, then we say $\mu$ dominates $\lambda$, written as $\mu ⊵\lambda$.

Example. Hasse diagram for dominance order for $6$, see here.

dominance lemma

Let $t$ and $s$ be tableaux of shape $\lambda$ and $\mu$ respectively. If for each index $i$, the elements of row $i$ of $s$ are in different columns in $t$, then $\lambda ⊵\mu$.

\begin{proof} We will arrange the tableau $t$ to new form $t^{\prime }$ by permuting entries within each column. Take elements of the first row of $s$ all in different columns of $t$ and move them to the first row in $t$. Continue to do it for the second row for each $i$. The number of elements in the first $i$ rows of $s$ is ${\mu }_1+\cdots +{\mu }_i$ and all elements of first $i$ rows of $\mu$ appears in first $i$ rows of $t^{\prime }$. Thus for any $i⩾1$, ${\lambda }_1+\cdots +{\lambda }_i⩾{\mu }_1+\cdots +{\mu }_i$ and so $\lambda ⊵\mu$. As an example, see here. \end{proof}