Suppose $V$ is a finite-dimensional vector space (over $\mathbb{C}$) and $\lambda$ is a partition of $n$ (not necessarily the dimension). Let $S^\lambda(V)=(V^{\otimes n})\_\lambda$ be the $\lambda$'th Schur functor applied to $V$ (a.k.a. the $\lambda$-component of the $S\_n$ representation $V^{\otimes n}$). Pick a basis for $V$. Is there a good basis for $S^\lambda(V)$? Even better, is there a good subset of the canonical basis of $V^{\otimes n}$ which projects to a basis of $S^\lambda(V)$? (This is the case for our favorite Schur functors coming from the trivial and alternating reps). 


Further, is there a good combinatorial description for the functoriality property, i.e. $S^\lambda(M)$ where $M:V\to W$ is a matrix?



A basis for Schur functors
I think this exact question answered in Theorem 1 of §8.1 of [William Fulton's "Young Tableaux"](http://books.google.com/books/about/Young_Tableaux.html?id=cYA_RpBLJUkC). Lazy as I am, I am unsure whether I have ever read the proof, but the answer is the following (unless I got Fulton's notations wrong): Whenever $p$ is a filling of the Young diagram of $\lambda$ **with vectors in $V$**, we can define an element $x\_p$ of $S^{\lambda}\left(V\right)$ by taking, for each $i=1,2,...,\lambda\_1$, the wedge product $w\_i$ of the elements of the $i$-th column of $p$ (from top to bottom), and then taking the tensor product $w\_1\otimes w\_2\otimes ...\otimes w\_{\lambda\_1}$ of these $w\_i$, and projecting this tensor product onto $S^{\lambda}\left(V\right)$. Now, letting $e\_1,e\_2,...,e\_n$ be a basis of $V$, we can construct, for every Young tableau $T$ of shape $\lambda$ over the alphabet $\left\lbrace 1,2,...,n\right\rbrace $, an element $e\_T$ of $S^{\lambda}\left(V\right)$ by $e\_T = x\_{p\_T}$, where $p\_T$ is the filling of the shape $\lambda$ in which every cell filled with a letter $j$ in $T$ is filled with the basis vector $e\_j$. Then, Fulton's Theorem 1 claims that the $e\_T$ with $T$ ranging over all **semistandard** tableaux over the alphabet $\left\lbrace 1,2,...,n\right\rbrace $ form a vector space basis of $S^{\lambda}\left(V\right)$.


Nota bene: This is only canonical with respect to a **totally ordered** basis.


Related results exist for so-called "bitableaux" (and probably contain the above result for tableaux?); see [Doubilet-Rota-Stein Foundations IX](http://www.math.ucdavis.edu/~deloera/MISC/BIBLIOTECA/trunk/Rota2.pdf) and the subsequent papers by Rota, Désarménien, Kung, de Concini, Procesi and others.


