Take the tour ×  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?  share|improve this question add comment  2 Answers
up vote 4 down vote accepted  I think this exact question
answered in Theorem 1 of §8.1 of William Fulton's "Young
Tableaux". Lazy as I am, I am unsure whether I have ever read
the proof, but the answer is the following (unless