TITLE: Littlewood Richardson rule and seminormal basis of Specht modules
QUESTION [8 upvotes]: 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, $\mathfrak{t}$, is standard if the entries are increasing along the rows and columns.   We let $\mathcal{T}_{\lambda}$ denote the set of standard $\lambda$-tableaux.
Given some tableau $\mathfrak{t}$ and two integers $1\leq i < j\leq n$, 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 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.

REPLY [5 votes]: These references solve the analogous problem for the general linear groups:

MR2166314 (2006h:20062)  Howe, Roger E. ;  Tan, Eng-Chye ;  Willenbring, Jeb F.
A basis for the $\mathrm{GL}_n$ tensor product algebra,
Adv. Math.  196  (2005),  no. 2, 531–564, doi:10.1016/j.aim.2004.09.007, arXiv:math/0407468

MR2888167  Howe, Roger ;  Lee, Soo Teck .
Why should the Littlewood-Richardson rule be true?
Bull. Amer. Math. Soc. (N.S.)  49  (2012),  no. 2, 187–236, doi:10.1090/S0273-0979-2011-01358-1

MR0955587 (89j:20046)  Tokuyama, Takeshi .
Determinantal method and the Littlewood-Richardson rule,
J. Algebra  117  (1988),  no. 1, 1–18, https://doi.org/10.1016/0021-8693(88)90237-2, (pdf)


with some work, using Schur-Weyl duality, you should be able to solve your problem.