# The Unapologetic Mathematician

## Intertwinors from Generalized Tableaux

Given any generalized Young tableau $T$ with shape $\lambda$ and content $\mu$, we can construct an intertwinor $\theta_T:M^\lambda\to M^\mu$. Actually, we’ll actually go from $M^\lambda$ to $\mathbb{C}[T_{\lambda\mu}$, but since we’ve seen that this is isomorphic to $M^\mu$, it’s good enough. Anyway, first, we have to define the row-equivalence class $\{T\}$ and column-equivalence class $[T]$. These are the same as for regular tableaux.

So, let $t$ be our reference tableau and let $\{t\}$ be the associated tabloid. We define

$\displaystyle\theta_T\left(\{t\}\right)=\sum\limits_{S\in\{T\}}S$

Continuing our example, with

$\displaystyle T=\begin{array}{ccc}2&1&1\\3&2&\end{array}$

we we define

\displaystyle\begin{aligned}\theta_T\left(\begin{array}{ccc}\cline{1-3}1&2&3\\\cline{1-3}4&5&\\\cline{1-2}\end{array}\right)&=\begin{array}{ccc}2&1&1\\3&2&\end{array}+\begin{array}{ccc}1&2&1\\3&2&\end{array}+\begin{array}{ccc}1&1&2\\3&2&\end{array}\\&+\begin{array}{ccc}2&1&1\\2&3&\end{array}+\begin{array}{ccc}1&2&1\\2&3&\end{array}+\begin{array}{ccc}1&1&2\\2&3&\end{array}\end{aligned}

Now, we extend in the only way possible. The module $M^\lambda$ is cyclic, meaning that it can be generated by a single element and the action of $\mathbb{C}[S_n]$. In fact, any single tabloid will do as a generator, and in particular $\{t\}$ generates $M^\lambda$.

So, any other module element in $M^\lambda$ is of the form $\pi\{t\}$ for some $\pi\in\mathbb{C}[S_n]$. And so if $\theta_T$ is to be an intertwinor we must define

$\displaystyle\theta_T\left(\pi\{t\}\right)=\pi\theta_T(\{t\})=\sum\limits_{S\in\{T\}}\pi S$

Remember here that $\pi$ acts on generalized tableaux by shuffling the entries by place, not by value. Thus in our example we find

\displaystyle\begin{aligned}\theta_T\left(\begin{array}{ccc}\cline{1-3}2&4&3\\\cline{1-3}1&5&\\\cline{1-2}\end{array}\right)&=\theta_T\left((1\,2\,4)\begin{array}{ccc}\cline{1-3}1&2&3\\\cline{1-3}4&5&\\\cline{1-2}\end{array}\right)\\&=(1\,2\,4)\theta_T\left(\begin{array}{ccc}\cline{1-3}1&2&3\\\cline{1-3}4&5&\\\cline{1-2}\end{array}\right)\\&=(1\,2\,4)\begin{array}{ccc}2&1&1\\3&2&\end{array}+(1\,2\,4)\begin{array}{ccc}1&2&1\\3&2&\end{array}+(1\,2\,4)\begin{array}{ccc}1&1&2\\3&2&\end{array}\\&+(1\,2\,4)\begin{array}{ccc}2&1&1\\2&3&\end{array}+(1\,2\,4)\begin{array}{ccc}1&2&1\\2&3&\end{array}+(1\,2\,4)\begin{array}{ccc}1&1&2\\2&3&\end{array}\\&=\begin{array}{ccc}3&2&1\\1&2&\end{array}+\begin{array}{ccc}3&1&1\\2&2&\end{array}+\begin{array}{ccc}3&1&2\\1&2&\end{array}\\&+\begin{array}{ccc}2&2&1\\1&3&\end{array}+\begin{array}{ccc}2&1&1\\2&3&\end{array}+\begin{array}{ccc}2&1&2\\1&3&\end{array}\end{aligned}

Now it shouldn’t be a surprise that since so much of our construction to this point has depended on an aribtrary choice of a reference tableau $t$, the linear combination of generalized tableaux on the right doesn’t quite seem like it comes from the tabloid on the left. But this is okay. Just relax and go with it.

February 5, 2011