# The Unapologetic Mathematician

## Intertwinors from Semistandard Tableaux Span, part 3

Now we are ready to finish our proof that the intertwinors $\bar{\theta}_T:S^\lambda\to M^\mu$ coming from semistandard generalized tableaux $T$ span the space of all intertwinors between these modules.

As usual, pick any intertwinor $\theta:S^\lambda\to M^\mu$ and write $\displaystyle\theta(e_t)=\sum\limits_Tc_TT$

Now define the set $L_\theta$ to consist of those semistandard generalized tableaux $S$ so that $[S]\trianglelefteq[T]$ for some $T$ appearing in this sum with a nonzero coefficient. This is called the “lower order ideal” generated by the $T$ in the sum. We will prove our assertion by induction on the size of this order ideal.

If $L_\theta$ is empty, then $\theta$ must be the zero map. Indeed, our lemmas showed that if $\theta$ is not the zero map, then at least one semistandard $T$ shows up in the above sum, and this $T$ would itself belong to $L_\theta$. And of course the zero map is contained in any span.

Now, if $L_\theta$ is not empty, then there is at least some semistandard $T$ with $c_T\neq0$ in the sum. Our lemmas even show that we can pick one so that $[T]$ is maximal among all the tableaux in the sum. Let’s do that and define a new intertwinor: $\displaystyle \theta' = \theta - c_T\bar{\theta}_T$

I say that $L_{\theta'}$ is $L_\theta$ with $T$ removed.

Every $S$ appearing in $\bar{\theta}_T(e_t)$ has $[S]\trianglelefteq[T]$, since if $T$ is semistandard then $[T]$ is the largest column equivalence class in $\theta_T(\{t\})$. Thus $L_{\theta'}$ must be a subset of $L_\theta$ since we can’t be introducing any new nonzero coefficients.

Our lemmas show that if $[S]=[T]$, then $c_S$ must appear with the same coefficient in both $\theta(e_t)$ and $c_T\bar{\theta}_T(e_t)$. That is, they must be cancelled off by the subtraction. Since $T$ is maximal there’s nothing above it that might keep it inside the ideal, and so $T\notin L_{\theta'}$.

So by induction we conclude that $\theta'$ is contained within the span of the $\bar{\theta}_T$ generated by semistandard tableaux, and thus $\theta$ must be as well.

February 14, 2011