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 - Posted by John Armstrong | Algebra, Representation Theory, Representations of Symmetric Groups

5 Comments »

[…] , and appears in a lower row than in the Young tabloid , then dominates . That is, swapping two entries of so as to move the lower number to a higher row moves the […]

Pingback by The Dominance Lemma for Tabloids « The Unapologetic Mathematician | January 11, 2011 | Reply
[…] of Standard Tableaux Standard tableaux have a certain maximality property with respect to the dominance order on tabloids. Specifically, if is standard and appears as a summand in the polytabloid , then […]

Pingback by The Maximality of Standard Tableaux « The Unapologetic Mathematician | January 13, 2011 | Reply
[…] now we can define the dominance order on column tabloids just like the dominance order for row tabloids. Of course, in doing so we have to alter our definition of the dominance order on […]

Pingback by The Column Dominance Order « The Unapologetic Mathematician | January 20, 2011 | Reply
[…] have predicted this: we’re going to have orders on generalized tabloids analogous to the dominance and column dominance orders for tabloids without repetitions. Each tabloid (or column tabloid) […]

Pingback by Dominance for Generalized Tabloids « The Unapologetic Mathematician | February 9, 2011 | Reply
The definition of the dominant order on tabloids is too complicate.
I wonder if using any one of its linear order extension is completely enough for later discussion.

Comment by Chih-wen Weng | November 29, 2016 | Reply

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