Background
Seminormal Basis of Specht modules of $\mathfrak{S}_n$
Let $\lambda$ be a partition of $n$. A $\lambda$-tableau is a bijection $\mathfrak{t}:\lambda \to {1,2...,n}$. We say a tableau, $\mathfrak{t}$, s standard if the entries are increasing along the rows and columns. We let $\mathcal{T}_{\lambda}$ denote the set of standard $\lambda$-tableaux.
Let $1\leq i < j\leq n$, we define the axial distance, $a(i,j)$, as follows: if $i$ occurs in row $i_0$ and column $i_1$ and $j$ occurs in row $j_0$ and column $j_1$, then $a(i,j)=(i_0-i_1) -(j_0-j_1)$.
If $\mathfrak{t}$ is a $\lambda$-tableau and $w \in \Sigma_n$ let $w\mathfrak{t}$ be the tableau obtained from $\mathfrak{t}$ by replacing each entry in $\mathfrak{t}$ by its image under $w$. If $\mathfrak{t}$ is a standard $\lambda$-tableau, we set $\mathfrak{t}_{i \leftrightarrow i+1}$ equal to $w \mathfrak{t}$ if this is still a standard $\lambda$-tableau, and 0 otherwise.
For a given partition $\lambda$ of $n$, the
Specht module ${\mathbf{S}(\lambda)}$
has a basis given by the set of standard $\lambda$-tableaux.
With respect to this
basis the generators act as follows
\begin{align*}
{\rho_{\lambda}}(s_{i,i+1})\mathfrak{t} = \frac{1}{a(i,i+1)} \mathfrak{t} + \left(1 + \frac{1}{a(i,i+1)}\right) \mathfrak{t}_{i \leftrightarrow i+1}
\end{align*}
This basis is very compatible with induction and restriction rules (see \emph{Seminormal representations of Weyl groups and Iwahori-Hecke algebras}, Arun Ram).
The Littlewood--Richardson rule
The LR rule describes the coefficients in the restriction $$\mathbf{S}(\nu)\downarrow_{\mathfrak{S}_{r_1}\times \mathfrak{S}_{r_2}} \cong \oplus c^{\nu}_{\lambda,\mu} \mathbf{S}(\lambda) \boxtimes \mathbf{S}(\mu)$$
There are many formulations of this rule. For example, the Jeu de Taquin version maps standard skew-tableaux of shape $\nu/\lambda$ to those of shape $\mu$. The LR coef, $c^{\nu}_{\lambda, \mu}$ is the cardinality of the fiber $f^{-1}(\mathfrak{t})$ for any $\mu$-tableau $\mathfrak{t}$.
So we have a map from $\nu$-tableaux to $\lambda \times \mu$-tableaux. The fibers give the LR coefficients. However, this map is not a homomorphism of Specht modules.
Question:
Is there a reference for an explicit construction of such a homomorphism? I.e. a formulation of the LR rule which is compatible with the seminormal bases of Specht modules.
