# The Unapologetic Mathematician

## The Dominance Order on Tabloids

Sorry, this should have gone up last Friday.

If $\{t\}$ is a Young tabloid with shape $\lambda\vdash n$, we can define tabloids $\{t^i\}$ for each $i$ from $1$ to $n$ by letting $\{t^i\}$ be formed by the entries in $\{t\}$ less than or equal to $i$. We define $\lambda^i$ to be the shape of $\{t^i\}$ as a composition. For example, if we have

$\displaystyle\{t\}=\begin{array}{cc}\cline{1-2}2&3\\\cline{1-2}1&4\\\cline{1-2}\end{array}$

then we define

\displaystyle\begin{aligned}\{t^1\}&=\begin{array}{c}\cline{1-1}\\\cline{1-1}1\\\cline{1-1}\end{array}\\\lambda^1&=(0,1)\\\{t^2\}&=\begin{array}{c}\cline{1-1}2\\\cline{1-1}1\\\cline{1-1}\end{array}\\\lambda^2&=(1,1)\\\{t^3\}&=\begin{array}{cc}\cline{1-2}2&3\\\cline{1-2}1&\\\cline{1-1}\end{array}\\\lambda^3&=(2,1)\\\{t^4\}&=\begin{array}{cc}\cline{1-2}2&3\\\cline{1-2}1&4\\\cline{1-2}\end{array}\\\lambda^4&=(2,2)\end{aligned}

Along the way we see why we might want to consider a composition like $(0,1)$ with a zero part.

Anyway, now we define a dominance order on tabloids. If $\{s\}$ and $\{t\}$ are two tabloids with composition sequences $\lambda^i$ and $\mu^i$, respectively, then we say $\{s\}$ “dominates” $\{t\}$ — and we write $\{s\}\triangleright\{t\}$ — if $\lambda^i$ dominates $\mu^i$ for all $i$.

As a (big!) example, we can write down the dominance order on all tabloids of shape $(2,2)$:

$\displaystyle\begin{array}{ccccc}&&\begin{array}{cc}\cline{1-2}1&2\\\cline{1-2}3&4\\\cline{1-2}\end{array}&&\\&&\uparrow&&\\&&\begin{array}{cc}\cline{1-2}1&3\\\cline{1-2}2&4\\\cline{1-2}\end{array}&&\\&\nearrow&&\nwarrow&\\\begin{array}{cc}\cline{1-2}2&3\\\cline{1-2}1&4\\\cline{1-2}\end{array}&&&&\begin{array}{cc}\cline{1-2}1&4\\\cline{1-2}2&3\\\cline{1-2}\end{array}\\&\nwarrow&&\nearrow&\\&&\begin{array}{cc}\cline{1-2}2&4\\\cline{1-2}1&3\\\cline{1-2}\end{array}&&\\&&\uparrow&&\\&&\begin{array}{cc}\cline{1-2}3&4\\\cline{1-2}1&2\\\cline{1-2}\end{array}&&\end{array}$

It’s an exercise to verify that these are indeed all the tabloids with this shape. For each arrow, we can verify the dominance. As an example, let’s show that

$\displaystyle\begin{array}{cc}\cline{1-2}1&2\\\cline{1-2}3&4\\\cline{1-2}\end{array}\trianglerighteq\begin{array}{cc}\cline{1-2}2&3\\\cline{1-2}1&4\\\cline{1-2}\end{array}$

First, let’s write down their composition sequences:

$\displaystyle\begin{array}{c|cc}i&\lambda^i&\mu^i\\\cline{1-3}1&(1)&(0,1)\\2&(2)&(1,1)\\3&(2,1)&(2,1)\\4&(2,2)&(2,2)\end{array}$

Now it should be easy to see on each row that $\lambda^i\trianglerighteq\mu^i$. As another example, let’s try to compare $\begin{array}{cc}\cline{1-2}2&3\\\cline{1-2}1&4\\\cline{1-2}\end{array}$ and $\begin{array}{cc}\cline{1-2}1&4\\\cline{1-2}2&3\\\cline{1-2}\end{array}$. Again, we write down their composition sequences:

$\displaystyle\begin{array}{c|cc}i&\lambda^i&\mu^i\\\cline{1-3}1&(0,1)&(1)\\2&(1,1)&(1,1)\\3&(2,1)&(1,2)\\4&(2,2)&(2,2)\end{array}$

We see that $\lambda^1\not\trianglerighteq\mu^1$, but $\mu^3\not\trianglerighteq\lambda^3$. Thus neither tabloid dominates the other. The other examples to verify this diagram are all similarly straightforward.

January 10, 2011