 nodes of $[\mu]$ mapped to $i$. Such a tableau is usually represented by drawing $[\la]$ with a box for each node $\fkn$, filled with the integer $T(\fkn)$. $T$ is *row-standard* if the entries in this diagram are weakly increasing along the rows, and is *semistandard* if the entries are weakly increasing along the rows and strictly increasing down the columns.

We write ${\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$ for the set of row-standard $\mu$-tableaux of type $\la$, and ${\calt_{\hspace{-2pt}0}}(\mu,\la)$ for the set of semistandard $\mu$-tableaux of type $\la$. For each $T\in{\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$, James defines a homomorphism $\Theta_T:M^\mu\to M^\la$ (over any field), whose precise definition we do not need here. The restriction of $\Theta_T$ to $S^\mu$ is denoted ${\hat\Theta_{T}}$. Now we have the following.

[ ]{}\[semi\] The set $$\lset{{\hat\Theta_{T}}}{T\in{\calt_{\hspace{-2pt}0}}(\mu,\la)}$$ is linearly independent, and spans ${\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,M^\la)$ if $\mu$ is $2$-regular.

The Kernel Intersection Theorem
-------------------------------

Now return to the assumption that $\la$ is a partition. As a consequence of Theorem \[semi\], in order to compute ${\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^\la)$ when $\mu$ is $2$-regular, we just need to find all linear combinations $\theta$ of the homomorphisms ${\hat\Theta_{T}}$ for $T\in{\calt_{\hspace{-2pt}0}}(\mu,\la)$ for which the image of $\theta$ lies in $S^\la$. Even when $\mu$ is not $2$-regular, homomorphisms from $S^\mu$ to $S^\la$ can very often be expressed in this way.

In order to determine whether the image of such a homomorphism $\theta$ lies in $S^\la$, we use another theorem of James which provides an alternative definition of $S^\la$. For any pair $(d,t)$ with $d\gs 1$ and $1\ls t\ls \la_{d+1}$, there is a homomorphism ${\psi_{d,t}}:M^\la\to M^\nu$, where $\nu$ is a composition depending on $\la,d,t$. Again, we refer the reader to [@j2 §17] for the definition;