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

5 Comments »

Hope you don’t mind me posting this Q here John. But I was reading a piece from Keith Devlin on MAA on how we learn math: http://www.maa.org/devlin/devlin_03_06.html. His basic conclusion is that when learning mathematics, understanding of the concepts can come only AFTER mastery of the procedural elements. In other words, much of the mathematics will be carried out with little (if any) understanding of the concepts and initially it will really only be a symbolic manipulation game. Once the rules are fully internalized, only some time after can any understanding eventually arise. As a mathematician, I’d love to hear your take on this. Do you think it’s possible to understand (even if vaguely) concepts first; or must procedural skill always come before understanding?

Comment by Jubayer K | February 5, 2011 | Reply
I think there’s a place for both.

Comment by John Armstrong | February 5, 2011 | Reply
[…] Generalized Tableaux We want to take our intertwinors and restrict them to the Specht modules. If the generalized tableau has shape and content , we […]

Pingback by Semistandard Generalized Tableaux « The Unapologetic Mathematician | February 8, 2011 | Reply
[…] Let’s start with the semistandard generalized tableaux and use them to construct intertwinors . I say that this collection is linearly […]

Pingback by Independence of Intertwinors from Semistandard Tableaux « The Unapologetic Mathematician | February 9, 2011 | Reply
[…] that we’ve shown the intertwinors that come from semistandard tableaux are independent, we want to show that they span the space . […]

Pingback by Intertwinors from Semistandard Tableaux Span, part 1 « The Unapologetic Mathematician | February 11, 2011 | Reply

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