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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?

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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 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 and the subsequent papers by Rota, Désarménien, Kung, de Concini, Procesi and others.

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You can construct "canonical" basis for the decomposition of $V^{\otimes n}$ under the symmetric group (and simultaneously under $GL(V)$) through "Young symmetrizers";

see for starters..

I'm sure the map $S^\lambda(M)$ can also be constructed "canonically". You might want to look at this software by Brian Wybourne for details on the calculations...

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Thanks. Could you say how Young symmetrizers give a basis? As far as I understand them, they project $V^{\otimes n}$ to the Schur functor, but they certainly don't project basis vectors to linearly independent elements. Is there some dual way of looking at them that fixes this problem? – Dmitry Vaintrob Feb 4 '13 at 23:40
The Young symmetrizers are just a starting point. They are used to construct special elements in the group algebra of $S_n$. There are still several steps after that to get a basis for the invariant suspaces of $V^{\otimes n}$. The calculations are not for the faint of heart...a lot of manipulations with Young tableaux. I'll try to dig up more reference later. – Y Macdisi Feb 5 '13 at 3:26

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