My post refers to Jerzy Weyman's "Cohomology of vector bundles and syzygies" pag. 37.
Let $R$ be a ring and $E$ a free $R$-module of rank $n$. Let ${e_1, \cdots, e_n}$ a basis of $E$. Let us consider a partition $\lambda=(\lambda_1, \cdots, \lambda_s)$ of $n$ and the Schur functor
$$L_{\lambda}(E):= (\Lambda^{\lambda_1}E \otimes \cdots \otimes \Lambda^{\lambda_s}E ) /R(\lambda,E) ,$$ where $R(\lambda,E)$ is the sum of the following submodules: $$ \Lambda^{\lambda_1}E \otimes \cdots \otimes \Lambda^{\lambda_{a-1}}E \otimes R_{a,a+1}(E) \otimes \Lambda^{\lambda_{a+2}}(E) \otimes \cdots \otimes \Lambda^{\lambda_s}E ,$$ over all $1 \le a \le s-1$. And $R_{a,a+1}(E)$ are the images of the maps $$\theta(\lambda, E, a,u,v): \Lambda^{u}E \otimes \Lambda^{\lambda_a+\lambda_{a+1}-u-v}E \otimes \Lambda^vE \stackrel{Id \otimes \Delta \otimes Id}{\longrightarrow} \Lambda^{u}E \otimes \Lambda^{\lambda_a -u}E \otimes \Lambda^{\lambda_{a+1} - v}E \otimes \Lambda^vE \stackrel{m_{12} \otimes m_{34}}{\longrightarrow} \Lambda^{\lambda_a}E \otimes \Lambda^{\lambda_{a+1}}E. $$ (with $u+v < \lambda_{a+1}$). Here $\Delta : \wedge^m E \to \wedge^p E \otimes \wedge^{m-p} E$ denotes the comultiplication of the Hopf algebra $\wedge E$ restricted to $\wedge^m E$ and with its image projected upon $\wedge^p E \otimes \wedge^{m-p} E$. (There is a Koszul sign in this comultiplication.)
Now we can define the Schur map: $$ \Phi_{\lambda}: \Lambda^{\lambda_1} E \otimes \cdots \otimes \Lambda^{\lambda_s}E \stackrel{\alpha}{\to} \bigotimes_{(i,j) \subset \lambda} E(i,j) \stackrel{\beta}{\to} S_{\lambda_1'}E \otimes \cdots \otimes S_{\lambda_s'}E ,$$ where: by$\lambda'$ I denote the dual partition (aka conjugate partition) of $\lambda$, $\alpha$ is the tensor product of the diagonal $\Delta: \Lambda^{\lambda_i}E \to E(i,1) \otimes \cdots \otimes E(i,\lambda_j)$ and $\beta$ is the tensor product of the multiplication $m: E(1,i) \otimes E(2,i) \otimes \cdots \otimes E(\lambda_i',i) \to S_{\lambda_i'}E.$ I'd like to understand this maps: for example could you describe $\Phi_\lambda$ if $\lambda=(3,2,1)$? How can I describe in terms of tableaux the Schur map? What is the meaning of the notation $\otimes_{(i,j) \subset \lambda}E(i,j)$?